1

Python code for Artificial

Intelligence: Foundations of
Computational Agents

David L. Poole and Alan K. Mackworth

Version 0.7.5 of June 19, 2018.

http://aipython.org http://artint.info
©David L Poole and Alan K Mackworth 2017.
All code is licensed under a Creative Commons Attribution-NonCommercial-
ShareAlike 4.0 International License. See: http://creativecommons.org/licenses/
by-nc-sa/4.0/deed.en US
This document and all the code can be downloaded from
http://artint.info/AIPython/ or from http://aipython.org
The authors and publisher of this book have used their best efforts in prepar-
ing this book. These efforts include the development, research and testing of
the theories and programs to determine their effectiveness. The authors and
publisher make no warranty of any kind, expressed or implied, with regard to
these programs or the documentation contained in this book. The author and
publisher shall not be liable in any event for incidental or consequential dam-
ages in connection with, or arising out of, the furnishing, performance, or use
of these programs.

http://aipython.org Version 0.7.5 June 19, 2018

Contents

Contents 3

1 Python for Artificial Intelligence 7

1.1 Why Python? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Getting Python . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Running Python . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Pitfalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Features of Python . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Useful Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.8 Testing Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Agents and Control 19

2.1 Representing Agents and Environments . . . . . . . . . . . . . 19
2.2 Paper buying agent and environment . . . . . . . . . . . . . . 20
2.3 Hierarchical Controller . . . . . . . . . . . . . . . . . . . . . . . 23

3 Searching for Solutions 31

3.1 Representing Search Problems . . . . . . . . . . . . . . . . . . 31
3.2 Generic Searcher and Variants . . . . . . . . . . . . . . . . . . . 39
3.3 Branch-and-bound Search . . . . . . . . . . . . . . . . . . . . . 44

4 Reasoning with Constraints 49

4.1 Constraint Satisfaction Problems . . . . . . . . . . . . . . . . . 49
4.2 Solving a CSP using Search . . . . . . . . . . . . . . . . . . . . 56
4.3 Consistency Algorithms . . . . . . . . . . . . . . . . . . . . . . 58

3
4 Contents

4.4 Solving CSPs using Stochastic Local Search . . . . . . . . . . . 63

5 Propositions and Inference 73

5.1 Representing Knowledge Bases . . . . . . . . . . . . . . . . . . 73
5.2 Bottom-up Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Top-down Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Assumables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6 Planning with Certainty 81

6.1 Representing Actions and Planning Problems . . . . . . . . . . 81
6.2 Forward Planning . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Regression Planning . . . . . . . . . . . . . . . . . . . . . . . . 89
6.4 Planning as a CSP . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5 Partial-Order Planning . . . . . . . . . . . . . . . . . . . . . . . 95

7 Supervised Machine Learning 103

7.1 Representations of Data and Predictions . . . . . . . . . . . . . 103
7.2 Learning With No Input Features . . . . . . . . . . . . . . . . . 113
7.3 Decision Tree Learning . . . . . . . . . . . . . . . . . . . . . . . 116
7.4 Cross Validation and Parameter Tuning . . . . . . . . . . . . . 120
7.5 Linear Regression and Classification . . . . . . . . . . . . . . . 122
7.6 Deep Neural Network Learning . . . . . . . . . . . . . . . . . 128
7.7 Boosting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

8 Reasoning Under Uncertainty 137

8.1 Representing Probabilistic Models . . . . . . . . . . . . . . . . 137
8.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.3 Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.4 Variable Elimination . . . . . . . . . . . . . . . . . . . . . . . . 145
8.5 Stochastic Simulation . . . . . . . . . . . . . . . . . . . . . . . . 147
8.6 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . 155
8.7 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . . 157
8.8 Dynamic Belief Networks . . . . . . . . . . . . . . . . . . . . . 163

9 Planning with Uncertainty 167

9.1 Decision Networks . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.2 Markov Decision Processes . . . . . . . . . . . . . . . . . . . . 172
9.3 Value Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

10 Learning with Uncertainty 177

10.1 K-means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.2 EM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

11 Multiagent Systems 185

11.1 Minimax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

http://aipython.org Version 0.7.5 June 19, 2018

Contents 5

12 Reinforcement Learning 191

12.1 Representing Agents and Environments . . . . . . . . . . . . . 191
12.2 Q Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.3 Model-based Reinforcement Learner . . . . . . . . . . . . . . . 200
12.4 Reinforcement Learning with Features . . . . . . . . . . . . . . 202
12.5 Learning to coordinate - UNFINISHED!!!! . . . . . . . . . . . . 208

13 Relational Learning 209

13.1 Collaborative Filtering . . . . . . . . . . . . . . . . . . . . . . . 209

Index 217

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 1

Python for Artificial Intelligence

1.1 Why Python?

We use Python because Python programs can be close to pseudo-code. It is
designed for humans to read.
Python is reasonably efficient. Efficiency is usually not a problem for small
examples. If your Python code is not efficient enough, a general procedure
to improve it is to find out what is taking most the time, and implement just
that part more efficiently in some lower-level language. Most of these lower-
level languages interoperate with Python nicely. This will result in much less
programming and more efficient code (because you will have more time to
optimize) than writing everything in a low-level language. You will not have
to do that for the code here if you are using it for course projects.

1.2 Getting Python

You need Python 3 (http://python.org/) and matplotlib (http://matplotlib.
org/) that runs with Python 3. This code is not compatible with Python 2 (e.g.,
with Python 2.7).
Download and istall the latest Python 3 release from http://python.org/.
This should also install pip3. You can install matplotlib using

pip3 install matplotlib

in a terminal shell (not in Python). That should “just work”. If not, try using
pip instead of pip3.
The command python or python3 should then start the interactive python
shell. You can quit Python with a control-D or with quit().

7
8 1. Python for Artificial Intelligence

To upgrade matplotlib to the latest version (which you should do if you

install a new version of Python) do:
pip3 install --upgrade matplotlib
We recommend using the enhanced interactive python ipython (http://
ipython.org/). To install ipython after you have installed python do:
pip3 install ipython

1.3 Running Python

We assume that everything is done with an interactive Python shell. You can
either do this with an IDE, such as IDLE that comes with standard Python
distributions, or just running ipython3 (or perhaps just ipython) from a shell.
Here we describe the most simple version that uses no IDE. If you down-
load the zip file, and cd to the “aipython” folder where the .py files are, you
should be able to do the following, with user input following : . The first
ipython3 command is in the operating system shell (note that the -i is impor-
tant to enter interactive mode):
$ ipython3 -i searchGeneric.py
Python 3.6.5 (v3.6.5:f59c0932b4, Mar 28 2018, 05:52:31)
Type 'copyright', 'credits' or 'license' for more information
IPython 6.2.1 -- An enhanced Interactive Python. Type '?' for help.
Testing problem 1:
7 paths have been expanded and 4 paths remain in the frontier
Path found: a --> b --> c --> d --> g
Passed unit test

In [1]: searcher2 = AStarSearcher(searchProblem.acyclic_delivery_problem) #A*

In [2]: searcher2.search() # find first path

16 paths have been expanded and 5 paths remain in the frontier
Out[2]: o103 --> o109 --> o119 --> o123 --> r123

In [3]: searcher2.search() # find next path

21 paths have been expanded and 6 paths remain in the frontier
Out[3]: o103 --> b3 --> b4 --> o109 --> o119 --> o123 --> r123

In [4]: searcher2.search() # find next path

28 paths have been expanded and 5 paths remain in the frontier
Out[4]: o103 --> b3 --> b1 --> b2 --> b4 --> o109 --> o119 --> o123 --> r123

In [5]: searcher2.search() # find next path

No (more) solutions. Total of 33 paths expanded.

http://aipython.org Version 0.7.5 June 19, 2018

1.4. Pitfalls 9

In [6]:
You can then interact at the last prompt.
There are many textbooks for Python. The best source of information about
python is https://www.python.org/. We will be using Python 3; please down-
load the latest release. The documentation is at https://docs.python.org/3/.
The rest of this chapter is about what is special about the code for AI tools.
We will only use the Standard Python Library and matplotlib. All of the exer-
cises can be done (and should be done) without using other libraries; the aim
is for you to spend your time thinking about how to solve the problem rather
than searching for pre-existing solutions.

1.4 Pitfalls
It is important to know when side effects occur. Often AI programs consider
what would happen or what may have happened. In many such cases, we
don’t want side effects. When an agent acts in the world, side effects are ap-
propriate.
In Python, you need to be careful to understand side effects. For example,
the inexpensive function to add an element to a list, namely append, changes
the list. In a functional language like Lisp, adding a new element to a list,
without changing the original list, is a cheap operation. For example if x is a
list containing n elements, adding an extra element to the list in Python (using
append) is fast, but it has the side effect of changing the list x. To construct a
new list that contains the elements of x plus a new element, without changing
the value of x, entails copying the list, or using a different representation for
lists. In the searching code, we will use a different representation for lists for
this reason.

1.5 Features of Python

1.5.1 Lists, Tuples, Sets, Dictionaries and Comprehensions
We make extensive uses of lists, tuples, sets and dictionaries (dicts). See
https://docs.python.org/3/library/stdtypes.html
One of the nice features of Python is the use of list comprehensions (and
also tuple, set and dictionary comprehensions).

(fe for e in iter if cond)

enumerates the values fe for each e in iter for which cond is true. The “if cond”
part is optional, but the “for” and “in” are not optional. Here e has to be a
variable, iter is an iterator, which can generate a stream of data, such as a list,
a set, a range object (to enumerate integers between ranges) or a file. cond

http://aipython.org Version 0.7.5 June 19, 2018

10 1. Python for Artificial Intelligence

is an expression that evaluates to either True or False for each e, and fe is an

expression that will be evaluated for each value of e for which cond returns
True.
The result can go in a list or used in another iteration, or can be called
directly using next. The procedure next takes an iterator returns the next el-
ement (advancing the iterator) and raises a StopIteration exception if there is
no next element. The following shows a simple example, where user input is
prepended with >>>
>>> [e*e for e in range(20) if e%2==0]
[0, 4, 16, 36, 64, 100, 144, 196, 256, 324]
>>> a = (e*e for e in range(20) if e%2==0)
>>> next(a)
0
>>> next(a)
4
>>> next(a)
16
>>> list(a)
[36, 64, 100, 144, 196, 256, 324]
>>> next(a)
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
StopIteration
Notice how list(a) continued on the enumeration, and got to the end of it.
Comprehensions can also be used for dictionaries. The following code cre-
ates an index for list a:
>>> a = ["a","f","bar","b","a","aaaaa"]
>>> ind = {a[i]:i for i in range(len(a))}
>>> ind
{'a': 4, 'f': 1, 'bar': 2, 'b': 3, 'aaaaa': 5}
>>> ind['b']
3
which means that 'b' is the 3rd element of the list.
The assignment of ind could have also be written as:
>>> ind = {val:i for (i,val) in enumerate(a)}
where enumerate returns an iterator of (index, value) pairs.

1.5.2 Functions as first-class objects

Python can create lists and other data structures that contain functions. There
is an issue that tricks many newcomers to Python. For a local variable in a
function, the function uses the last value of the variable when the function is

http://aipython.org Version 0.7.5 June 19, 2018

 1.5. Features of Python 11

called, not the value of the variable when the function was defined (this is called
“late binding”). This means if you want to use the value a variable has when
the function is created, you need to save the current value of that variable.
Whereas Python uses “late binding” by default, the alternative that newcomers
often expect is “early binding”, where a function uses the value a variable had
when the function was defined, can be easily implemented.
Consider the following programs designed to create a list of 5 functions,
where the ith function in the list is meant to add i to its argument:1
pythonDemo.py — Some tricky examples
11 fun_list1 = []
12 for i in range(5):
13 def fun1(e):
14 return e+i
15 fun_list1.append(fun1)
16
17 fun_list2 = []
18 for i in range(5):
19 def fun2(e,iv=i):
20 return e+iv
21 fun_list2.append(fun2)
22
23 fun_list3 = [lambda e: e+i for i in range(5)]
24
25 fun_list4 = [lambda e,iv=i: e+iv for i in range(5)]
26
27 i=56
Try to predict, and then test to see the output, of the output of the following
calls, remembering that the function uses the latest value of any variable that
is not bound in the function call:
pythonDemo.py — (continued)

29 # in Shell do
30 ## ipython -i pythonDemo.py
31 # Try these (copy text after the comment symbol and paste in the Python prompt):
32 # print([f(10) for f in fun_list1])
33 # print([f(10) for f in fun_list2])
34 # print([f(10) for f in fun_list3])
35 # print([f(10) for f in fun_list4])
In the first for-loop, the function fun uses i, whose value is the last value it was
assigned. In the second loop, the function fun2 uses iv. There is a separate iv
variable for each function, and its value is the value of i when the function was
defined. Thus fun1 uses late binding, and fun2 uses early binding. fun list3
and fun list4 are equivalent to the first two (except fun list4 uses a different i
variable).
1 Numbered lines are Python code available in the code-directory, aipython. The name of

the file is given in the gray text above the listing. The numbers correspond to the line numbers
in that file.

http://aipython.org Version 0.7.5 June 19, 2018

 12 1. Python for Artificial Intelligence

One of the advantages of using the embedded definitions (as in fun1 and
fun2 above) over the lambda is that is it possible to add a __doc__ string, which
is the standard for documenting functions in Python, to the embedded defini-
tions.

1.5.3 Generators and Coroutines

Python has generators which can be used for a form of coroutines.
The yield command returns a value that is obtained with next. It is typically
used to enumerate the values for a for loop or in generators.
A version of the built-in range, with 2 or 3 arguments (and positive steps)
can be implemented as:

pythonDemo.py — (continued)

37 def myrange(start, stop, step=1):

38 """enumerates the values from start in steps of size step that are
39 less than stop.
40 """
41 assert step>0, "only positive steps implemented in myrange"
42 i = start
43 while i<stop:
44 yield i
45 i += step
46
47 print("myrange(2,30,3):",list(myrange(2,30,3)))

Note that the built-in range is unconventional in how it handles a single ar-
gument, as the single argument acts as the second argument of the function.
Note also that the built-in range also allows for indexing (e.g., range(2, 30, 3)[2]
returns 8), which the above implementation does not. However myrange also
works for floats, which the built-in range does not.
Exercise 1.1 Implement a version of myrange that acts like the built-in version
when there is a single argument. (Hint: make the second argument have a default
value that can be recognized in the function.)

Yield can be used to generate the same sequence of values as in the example
of Section 1.5.1:

pythonDemo.py — (continued)

49 def ga(n):
50 """generates square of even nonnegative integers less than n"""
51 for e in range(n):
52 if e%2==0:
53 yield e*e
54 a = ga(20)

The sequence of next(a), and list(a) gives exactly the same results as the com-
prehension in Section 1.5.1.

http://aipython.org Version 0.7.5 June 19, 2018

 1.6. Useful Libraries 13

It is straightforward to write a version of the built-in enumerate. Let’s call it

myenumerate:
pythonDemo.py — (continued)

56 def myenumerate(enum):
57 for i in range(len(enum)):
58 yield i,enum[i]

Exercise 1.2 Write a version of enumerate where the only iteration is “for val in
enum”. Hint: keep track of the index.

1.6 Useful Libraries

1.6.1 Timing Code
In order to compare algorithms, we often want to compute how long a program
takes; this is called the runtime of the program. The most straightforward way
to compute runtime is to use time.perf counter(), as in:
import time
start_time = time.perf_counter()
compute_for_a_while()
end_time = time.perf_counter()
print("Time:", end_time - start_time, "seconds")
If this time is very small (say less than 0.2 second), it is probably very inac-
curate, and it may be better to run your code many times to get a more accu-
rate count. For this you can use timeit (https://docs.python.org/3/library/
timeit.html). To use timeit to time the call to foo.bar(aaa) use:
import timeit
time = timeit.timeit("foo.bar(aaa)",
setup="from __main__ import foo,aaa", number=100)
The setup is needed so that Python can find the meaning of the names in the
string that is called. This returns the number of seconds to execute foo.bar(aaa)
100 times. The variable number should be set so that the runtime is at least 0.2
seconds.
You should not trust a single measurement as that can be confounded by
interference from other processes. timeit.repeat can be used for running timit
a few (say 3) times. Usually the minimum time is the one to report, but you
should be explicit and explain what you are reporting.

1.6.2 Plotting: Matplotlib

The standard plotting for Python is matplotlib (http://matplotlib.org/). We
will use the most basic plotting using the pyplot interface.
Here is a simple example that uses everything we will use.

http://aipython.org Version 0.7.5 June 19, 2018

 14 1. Python for Artificial Intelligence

pythonDemo.py — (continued)

60 import matplotlib.pyplot as plt

61
62 def myplot(min,max,step,fun1,fun2):
63 plt.ion() # make it interactive
64 plt.xlabel("The x axis")
65 plt.ylabel("The y axis")
66 plt.xscale('linear') # Makes a 'log' or 'linear' scale
67 xvalues = range(min,max,step)
68 plt.plot(xvalues,[fun1(x) for x in xvalues],
69 label="The first fun")
70 plt.plot(xvalues,[fun2(x) for x in xvalues], linestyle='--',color='k',
71 label=fun2.__doc__) # use the doc string of the function
72 plt.legend(loc="upper right") # display the legend
73
74 def slin(x):
75 """y=2x+7"""
76 return 2*x+7
77 def sqfun(x):
78 """y=(x-40)ˆ2/10-20"""
79 return (x-40)**2/10-20
80
81 # Try the following:
82 # from pythonDemo import myplot, slin, sqfun
83 # import matplotlib.pyplot as plt
84 # myplot(0,100,1,slin,sqfun)
85 # plt.legend(loc="best")
86 # import math
87 # plt.plot([41+40*math.cos(th/10) for th in range(50)],
88 # [100+100*math.sin(th/10) for th in range(50)])
89 # plt.text(40,100,"ellipse?")
90 # plt.xscale('log')

At the end of the code are some commented-out commands you should try in
interactive mode. Cut from the file and paste into Python (and remember to
remove the comments symbol and leading space).

1.7 Utilities
1.7.1 Display
In this distribution, to keep things simple and to only use standard Python, we
use a text-oriented tracing of the code. A graphical depiction of the code could
override the definition of display (but we leave it as a project).
The method self .display is used to trace the program. Any call

self .display(level, to print . . . )

http://aipython.org Version 0.7.5 June 19, 2018

 1.7. Utilities 15

where the level is less than or equal to the value for max display level will be
printed. The to print . . . can be anything that is accepted by the built-in print
(including any keyword arguments).
The definition of display is:

utilities.py — AIFCA utilities

11 class Displayable(object):
12 max_display_level = 1 # can be overridden in subclasses
13
14 def display(self,level,*args,**nargs):
15 """print the arguments if level is less than or equal to the
16 current max_display_level.
17 level is an integer.
18 the other arguments are whatever arguments print can take.
19 """
20 if level <= self.max_display_level:
21 print(*args, **nargs) ##if error you are using Python2 not Python3

Note that args gets a tuple of the positional arguments, and nargs gets a dictio-
nary of the keyword arguments). This will not work in Python 2, and will give
an error.
Any class that wants to use display can be made a subclass of Displayable.
To change the maximum display level to say 3, for a class do:

Classname.max display level = 3

which will make calls to display in that class print when the value of level is less
than-or-equal to 3. The default display level is 1. It can also be changed for
individual objects (the object value overrides the class value).
The value of max display level by convention is:

0 display nothing

1 display solutions (nothing that happens repeatedly)

2 also display the values as they change (little detail through a loop)

3 also display more details

4 and above even more detail

In order to implement more sophisticated visualizations of the algorithm,

we add a visualize “decorator” to the methods to be visualized. The following
code ignores the decorator:

utilities.py — (continued)

23 def visualize(func):
24 return func

http://aipython.org Version 0.7.5 June 19, 2018

 16 1. Python for Artificial Intelligence

1.7.2 Argmax
Python has a built-in max function that takes a generator (or a list or set) and re-
turns the maximum value. The argmax method returns the index of an element
that has the maximum value. If there are multiple elements with the maximum
value, one if the indexes to that value is returned at random. This assumes a
generator of (element, value) pairs, as for example is generated by the built-in
enumerate.
utilities.py — (continued)

25 import random
26
27 def argmax(gen):
28 """gen is a generator of (element,value) pairs, where value is a real.
29 argmax returns an element with maximal value.
30 If there are multiple elements with the max value, one is returned at random.
31 """
32 maxv = float('-Infinity') # negative infinity
33 maxvals = [] # list of maximal elements
34 for (e,v) in gen:
35 if v>maxv:
36 maxvals,maxv = [e], v
37 elif v==maxv:
38 maxvals.append(e)
39 return random.choice(maxvals)
40
41 # Try:
42 # argmax(enumerate([1,6,3,77,3,55,23]))

Exercise 1.3 Change argmax to have an optinal argument that specifies whether
you want the “first”, “last” or a “random” index of the maximum value returned.
If you want the first or the last, you don’t need to keep a list of the maximum
elements.

1.7.3 Probability
For many of the simulations, we want to make a variable True with some prob-
ability. flip(p) returns True with probability p, and otherwise returns False.
utilities.py — (continued)

44 def flip(prob):
45 """return true with probability prob"""
46 return random.random() < prob

1.7.4 Dictionary Union

The function dict union(d1, d2) returns the union of dictionaries d1 and d2. If
the values for the keys conflict, the values in d2 are used. This is similar to
dict(d1, ∗ ∗ d2), but that only works when the keys of d2 are strings.

http://aipython.org Version 0.7.5 June 19, 2018

 1.8. Testing Code 17

utilities.py — (continued)

48 def dict_union(d1,d2):
49 """returns a dictionary that contains the keys of d1 and d2.
50 The value for each key that is in d2 is the value from d2,
51 otherwise it is the value from d1.
52 This does not have side effects.
53 """
54 d = dict(d1) # copy d1
55 d.update(d2)
56 return d

1.8 Testing Code

It is important to test code early and test it often. We include a simple form of
unit tests. The value of the current module is in __name__ and if the module is
run at the top-level, it’s value is "__main__". See https://docs.python.org/3/
library/ main .html.
The following code tests argmax and dict_union, but only when if utilities
is loaded in the top-level. If it is loaded in a module the test code is not run.
In your code you should do more substantial testing than we do here, in
particular testing the boundary cases.
utilities.py — (continued)

58 def test():
59 """Test part of utilities"""
60 assert argmax(enumerate([1,6,55,3,55,23])) in [2,4]
61 assert dict_union({1:4, 2:5, 3:4},{5:7, 2:9}) == {1:4, 2:9, 3:4, 5:7}
62 print("Passed unit test in utilities")
63
64 if __name__ == "__main__":
65 test()

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 2

Agents and Control

This implements the controllers described in Chapter 2.

In this version the higher-levels call the lower-levels. A more sophisti-
cated version may have them run concurrently (either as coroutines or in paral-
lel). The higher-levels calling the lower-level works in simulated environments
when there is a single agent, and where the lower-level are written to make sure
they return (and don’t go on forever), and the higher level doesn’t take too long
(as the lower-levels will wait until called again).

2.1 Representing Agents and Environments

An agent observes the world, and carries out actions in the environment, it also
has an internal state that it updates. The environment takes in actions of the
agents, updates it internal state and returns the percepts.
In this implementation, the state of the agent and the state of the environ-
ment are represented using standard Python variables, which are updated as
the state changes. The percepts and the actions are represented as variable-
value dictionaries.
An agent implements the go(n) method, where n is an integer. This means
that the agent should run for n time steps.
In the following code raise NotImplementedError() is a way to specify
an abstract method that needs to be overidden in any implemented agent or
environment.
agents.py — Agent and Controllers
11 import random
12
13 class Agent(object):
14 def __init__(self,env):

19
 20 2. Agents and Control

15 """set up the agent"""

16 self.env=env
17
18 def go(self,n):
19 """acts for n time steps"""
20 raise NotImplementedError("go") # abstract method
The environment implements a do(action) method where action is a variable-
value dictionary. This returns a percept, which is also a variable-value dictio-
nary. The use of dictionaries allows for structured actions and percepts.
Note that Environment is a subclass of Displayable so that it can use the
display method described in Section 1.7.1.
agents.py — (continued)

22 from utilities import Displayable

23 class Environment(Displayable):
24 def initial_percepts(self):
25 """returns the initial percepts for the agent"""
26 raise NotImplementedError("initial_percepts") # abstract method
27
28 def do(self,action):
29 """does the action in the environment
30 returns the next percept """
31 raise NotImplementedError("do") # abstract method

2.2 Paper buying agent and environment

To run the demo, in folder ”aipython”, load ”agents.py”, using e.g.,
ipython -i agents.py, and copy and paste the commented-out
commands at the bottom of that file. This requires Python 3 with
matplotlib.

This is an implementation of the paper buying example.

2.2.1 The Environment

The environment state is given in terms of the time and the amount of paper in
stock. It also remembers the in-stock history and the price history. The percepts
are the price and the amount of paper in stock. The action of the agent is the
number to buy.
Here we assume that the prices are obtained from the prices list plus a ran-
dom integer in range [0, max price addon) plus a linear ”inflation”. The agent
cannot access the price model; it just observes the prices and the amount in
stock.
agents.py — (continued)

33 class TP_env(Environment):

http://aipython.org Version 0.7.5 June 19, 2018

 2.2. Paper buying agent and environment 21

34 prices = [234, 234, 234, 234, 255, 255, 275, 275, 211, 211, 211,
35 234, 234, 234, 234, 199, 199, 275, 275, 234, 234, 234, 234, 255,
36 255, 260, 260, 265, 265, 265, 265, 270, 270, 255, 255, 260, 260,
37 265, 265, 150, 150, 265, 265, 270, 270, 255, 255, 260, 260, 265,
38 265, 265, 265, 270, 270, 211, 211, 255, 255, 260, 260, 265, 265,
39 260, 265, 270, 270, 205, 255, 255, 260, 260, 265, 265, 265, 265,
40 270, 270]
41 max_price_addon = 20 # maximum of random value added to get price
42
43 def __init__(self):
44 """paper buying agent"""
45 self.time=0
46 self.stock=20
47 self.stock_history = [] # memory of the stock history
48 self.price_history = [] # memory of the price history
49
50 def initial_percepts(self):
51 """return initial percepts"""
52 self.stock_history.append(self.stock)
53 price = self.prices[0]+random.randrange(self.max_price_addon)
54 self.price_history.append(price)
55 return {'price': price,
56 'instock': self.stock}
57
58 def do(self, action):
59 """does action (buy) and returns percepts (price and instock)"""
60 used = pick_from_dist({6:0.1, 5:0.1, 4:0.2, 3:0.3, 2:0.2, 1:0.1})
61 bought = action['buy']
62 self.stock = self.stock+bought-used
63 self.stock_history.append(self.stock)
64 self.time += 1
65 price = (self.prices[self.time%len(self.prices)] # repeating pattern
66 +random.randrange(self.max_price_addon) # plus randomness
67 +self.time//2) # plus inflation
68 self.price_history.append(price)
69 return {'price': price,
70 'instock': self.stock}

The pick from dist method takes in a item : probability dictionary, and returns
one of the items in proportion to its probability.

agents.py — (continued)

72 def pick_from_dist(item_prob_dist):
73 """ returns a value from a distribution.
74 item_prob_dist is an item:probability dictionary, where the
75 probabilities sum to 1.
76 returns an item chosen in proportion to its probability
77 """
78 ranreal = random.random()
79 for (it,prob) in item_prob_dist.items():
80 if ranreal < prob:

http://aipython.org Version 0.7.5 June 19, 2018

 22 2. Agents and Control

81 return it
82 else:
83 ranreal -= prob
84 raise RuntimeError(str(item_prob_dist)+" is not a probability distribution")

2.2.2 The Agent

The agent does not have access to the price model but can only observe the
current price and the amount in stock. It has to decide how much to buy.
The belief state of the agent is an estimate of the average price of the paper,
and the total amount of money the agent has spent.
agents.py — (continued)

86 class TP_agent(Agent):
87 def __init__(self, env):
88 self.env = env
89 self.spent = 0
90 percepts = env.initial_percepts()
91 self.ave = self.last_price = percepts['price']
92 self.instock = percepts['instock']
93
94 def go(self, n):
95 """go for n time steps
96 """
97 for i in range(n):
98 if self.last_price < 0.9*self.ave and self.instock < 60:
99 tobuy = 48
100 elif self.instock < 12:
101 tobuy = 12
102 else:
103 tobuy = 0
104 self.spent += tobuy*self.last_price
105 percepts = env.do({'buy': tobuy})
106 self.last_price = percepts['price']
107 self.ave = self.ave+(self.last_price-self.ave)*0.05
108 self.instock = percepts['instock']
Set up an environment and an agent. Uncomment the last lines to run the agent
for 90 steps, and determine the average amount spent.
agents.py — (continued)

110 env = TP_env()

111 ag = TP_agent(env)
112 #ag.go(90)
113 #ag.spent/env.time ## average spent per time period

2.2.3 Plotting
The following plots the price and number in stock history:

http://aipython.org Version 0.7.5 June 19, 2018

 2.3. Hierarchical Controller 23

agents.py — (continued)

115 import matplotlib.pyplot as plt

116
117 class Plot_prices(object):
118 """Set up the plot for history of price and number in stock"""
119 def __init__(self, ag,env):
120 self.ag = ag
121 self.env = env
122 plt.ion()
123 plt.xlabel("Time")
124 plt.ylabel("Number in stock. Price.")
125
126 def plot_run(self):
127 """plot history of price and instock"""
128 num = len(env.stock_history)
129 plt.plot(range(num),env.stock_history,label="In stock")
130 plt.plot(range(num),env.price_history,label="Price")
131 #plt.legend(loc="upper left")
132 plt.draw()
133
134 # pl = Plot_prices(ag,env)
135 # ag.go(90); pl.plot_run()

2.3 Hierarchical Controller

To run the hierarchical controller, in folder ”aipython”, load
”agentTop.py”, using e.g., ipython -i agentTop.py, and copy and
paste the commands near the bottom of that file. This requires Python
3 with matplotlib.

In this implementation, each layer, including the top layer, implements the en-
vironment class, because each layer is seen as an environment from the layer
above.
We arbitrarily divide the environment and the body, so that the environ-
ment just defines the walls, and the body includes everything to do with the
agent. Note that the named locations are part of the (top-level of the) agent,
not part of the environment, although they could have been.

2.3.1 Environment
The environment defines the walls.
agentEnv.py — Agent environment
11 import math
12 from agents import Environment
13
14 class Rob_env(Environment):

http://aipython.org Version 0.7.5 June 19, 2018

 24 2. Agents and Control

15 def __init__(self,walls = {}):

16 """walls is a set of line segments
17 where each line segment is of the form ((x0,y0),(x1,y1))
18 """
19 self.walls = walls

2.3.2 Body
The body defines everything about the agent body.
agentEnv.py — (continued)

21 import math
22 from agents import Environment
23 import matplotlib.pyplot as plt
24 import time
25
26 class Rob_body(Environment):
27 def __init__(self, env, init_pos=(0,0,90)):
28 """ env is the current environment
29 init_pos is a triple of (x-position, y-position, direction)
30 direction is in degrees; 0 is to right, 90 is straight-up, etc
31 """
32 self.env = env
33 self.rob_x, self.rob_y, self.rob_dir = init_pos
34 self.turning_angle = 18 # degrees that a left makes
35 self.whisker_length = 6 # length of the whisker
36 self.whisker_angle = 30 # angle of whisker relative to robot
37 self.crashed = False
38 # The following control how it is plotted
39 self.plotting = True # whether the trace is being plotted
40 self.sleep_time = 0.05 # time between actions (for real-time plotting)
41 # The following are data structures maintained:
42 self.history = [(self.rob_x, self.rob_y)] # history of (x,y) positions
43 self.wall_history = [] # history of hitting the wall
44
45 def percepts(self):
46 return {'rob_x_pos':self.rob_x, 'rob_y_pos':self.rob_y,
47 'rob_dir':self.rob_dir, 'whisker':self.whisker() , 'crashed':self.crashed}
48 initial_percepts = percepts # use percept function for initial percepts too
49
50 def do(self,action):
51 """ action is {'steer':direction}
52 direction is 'left', 'right' or 'straight'
53 """
54 if self.crashed:
55 return self.percepts()
56 direction = action['steer']
57 compass_deriv = {'left':1,'straight':0,'right':-1}[direction]*self.turning_angle
58 self.rob_dir = (self.rob_dir + compass_deriv +360)%360 # make in range [0,360)
59 rob_x_new = self.rob_x + math.cos(self.rob_dir*math.pi/180)

http://aipython.org Version 0.7.5 June 19, 2018

 2.3. Hierarchical Controller 25

60 rob_y_new = self.rob_y + math.sin(self.rob_dir*math.pi/180)

61 path = ((self.rob_x,self.rob_y),(rob_x_new,rob_y_new))
62 if any(line_segments_intersect(path,wall) for wall in self.env.walls):
63 self.crashed = True
64 if self.plotting:
65 plt.plot([self.rob_x],[self.rob_y],"r*",markersize=20.0)
66 plt.draw()
67 self.rob_x, self.rob_y = rob_x_new, rob_y_new
68 self.history.append((self.rob_x, self.rob_y))
69 if self.plotting and not self.crashed:
70 plt.plot([self.rob_x],[self.rob_y],"go")
71 plt.draw()
72 plt.pause(self.sleep_time)
73 return self.percepts()

This detects if the whisker and the wall intersect. It’s value is returned as a
percept.

agentEnv.py — (continued)

75 def whisker(self):
76 """returns true whenever the whisker sensor intersects with a wall
77 """
78 whisk_ang_world = (self.rob_dir-self.whisker_angle)*math.pi/180
79 # angle in radians in world coordinates
80 wx = self.rob_x + self.whisker_length * math.cos(whisk_ang_world)
81 wy = self.rob_y + self.whisker_length * math.sin(whisk_ang_world)
82 whisker_line = ((self.rob_x,self.rob_y),(wx,wy))
83 hit = any(line_segments_intersect(whisker_line,wall)
84 for wall in self.env.walls)
85 if hit:
86 self.wall_history.append((self.rob_x, self.rob_y))
87 if self.plotting:
88 plt.plot([self.rob_x],[self.rob_y],"ro")
89 plt.draw()
90 return hit
91
92 def line_segments_intersect(linea,lineb):
93 """returns true if the line segments, linea and lineb intersect.
94 A line segment is represented as a pair of points.
95 A point is represented as a (x,y) pair.
96 """
97 ((x0a,y0a),(x1a,y1a)) = linea
98 ((x0b,y0b),(x1b,y1b)) = lineb
99 da, db = x1a-x0a, x1b-x0b
100 ea, eb = y1a-y0a, y1b-y0b
101 denom = db*ea-eb*da
102 if denom==0: # line segments are parallel
103 return False
104 cb = (da*(y0b-y0a)-ea*(x0b-x0a))/denom # position along line b
105 if cb<0 or cb>1:
106 return False

http://aipython.org Version 0.7.5 June 19, 2018

 26 2. Agents and Control

107 ca = (db*(y0b-y0a)-eb*(x0b-x0a))/denom # position along line a

108 return 0<=ca<=1
109
110 # Test cases:
111 # assert line_segments_intersect(((0,0),(1,1)),((1,0),(0,1)))
112 # assert not line_segments_intersect(((0,0),(1,1)),((1,0),(0.6,0.4)))
113 # assert line_segments_intersect(((0,0),(1,1)),((1,0),(0.4,0.6)))

2.3.3 Middle Layer

The middle layer acts like both a controller (for the environment layer) and an
environment for the upper layer. It has to tell the environment how to steer.
Thus it calls env.do(·). It also is told the position to go to and the timeout. Thus
it also has to implement do(·).

agentMiddle.py — Middle Layer

11 from agents import Environment
12 import math
13
14 class Rob_middle_layer(Environment):
15 def __init__(self,env):
16 self.env=env
17 self.percepts = env.initial_percepts()
18 self.straight_angle = 11 # angle that is close enough to straight ahead
19 self.close_threshold = 2 # distance that is close enough to arrived
20 self.close_threshold_squared = self.close_threshold**2 # just compute it once
21
22 def initial_percepts(self):
23 return {}
24
25 def do(self, action):
26 """action is {'go_to':target_pos,'timeout':timeout}
27 target_pos is (x,y) pair
28 timeout is the number of steps to try
29 returns {'arrived':True} when arrived is true
30 or {'arrived':False} if it reached the timeout
31 """
32 if 'timeout' in action:
33 remaining = action['timeout']
34 else:
35 remaining = -1 # will never reach 0
36 target_pos = action['go_to']
37 arrived = self.close_enough(target_pos)
38 while not arrived and remaining != 0:
39 self.percepts = self.env.do({"steer":self.steer(target_pos)})
40 remaining -= 1
41 arrived = self.close_enough(target_pos)
42 return {'arrived':arrived}

http://aipython.org Version 0.7.5 June 19, 2018

 2.3. Hierarchical Controller 27

This determines how to steer depending on whether the goal is to the right or
the left of where the robot is facing.
agentMiddle.py — (continued)

44 def steer(self,target_pos):
45 if self.percepts['whisker']:
46 self.display(3,'whisker on', self.percepts)
47 return "left"
48 else:
49 gx,gy = target_pos
50 rx,ry = self.percepts['rob_x_pos'],self.percepts['rob_y_pos']
51 goal_dir = math.acos((gx-rx)/math.sqrt((gx-rx)*(gx-rx)
52 +(gy-ry)*(gy-ry)))*180/math.pi
53 if ry>gy:
54 goal_dir = -goal_dir
55 goal_from_rob = (goal_dir - self.percepts['rob_dir']+540)%360-180
56 assert -180 < goal_from_rob <= 180
57 if goal_from_rob > self.straight_angle:
58 return "left"
59 elif goal_from_rob < -self.straight_angle:
60 return "right"
61 else:
62 return "straight"
63
64 def close_enough(self,target_pos):
65 gx,gy = target_pos
66 rx,ry = self.percepts['rob_x_pos'],self.percepts['rob_y_pos']
67 return (gx-rx)**2 + (gy-ry)**2 <= self.close_threshold_squared

2.3.4 Top Layer

The top layer treats the middle layer as its environment. Note that the top layer
is an environment for us to tell it what to visit.
agentTop.py — Top Layer
11 from agentMiddle import Rob_middle_layer
12 from agents import Environment
13
14 class Rob_top_layer(Environment):
15 def __init__(self, middle, timeout=200, locations = {'mail':(-5,10),
16 'o103':(50,10), 'o109':(100,10),'storage':(101,51)} ):
17 """middle is the middle layer
18 timeout is the number of steps the middle layer goes before giving up
19 locations is a loc:pos dictionary
20 where loc is a named location, and pos is an (x,y) position.
21 """
22 self.middle = middle
23 self.timeout = timeout # number of steps before the middle layer should give up
24 self.locations = locations
25

http://aipython.org Version 0.7.5 June 19, 2018

 28 2. Agents and Control

26 def do(self,plan):
27 """carry out actions.
28 actions is of the form {'visit':list_of_locations}
29 It visits the locations in turn.
30 """
31 to_do = plan['visit']
32 for loc in to_do:
33 position = self.locations[loc]
34 arrived = self.middle.do({'go_to':position, 'timeout':self.timeout})
35 self.display(1,"Arrived at",loc,arrived)

2.3.5 Plotting
The following is used to plot the locations, the walls and (eventually) the move-
ment of the robot. It can either plot the movement if the robot as it is go-
ing (with the default env.plotting = True), or not plot it as it is going (setting
env.plotting = False; in this case the trace can be plotted using pl.plot run()).
agentTop.py — (continued)

37 import matplotlib.pyplot as plt

38
39 class Plot_env(object):
40 def __init__(self, body,top):
41 """sets up the plot
42 """
43 self.body = body
44 plt.ion()
45 plt.clf()
46 plt.axes().set_aspect('equal')
47 for wall in body.env.walls:
48 ((x0,y0),(x1,y1)) = wall
49 plt.plot([x0,x1],[y0,y1],"-k",linewidth=3)
50 for loc in top.locations:
51 (x,y) = top.locations[loc]
52 plt.plot([x],[y],"k<")
53 plt.text(x+1.0,y+0.5,loc) # print the label above and to the right
54 plt.plot([body.rob_x],[body.rob_y],"go")
55 plt.draw()
56
57 def plot_run(self):
58 """plots the history after the agent has finished.
59 This is typically only used if body.plotting==False
60 """
61 xs,ys = zip(*self.body.history)
62 plt.plot(xs,ys,"go")
63 wxs,wys = zip(*self.body.wall_history)
64 plt.plot(wxs,wys,"ro")
65 #plt.draw()
The following code plots the agent as it acts in the world:

http://aipython.org Version 0.7.5 June 19, 2018

 2.3. Hierarchical Controller 29

agentTop.py — (continued)

67 from agentEnv import Rob_body, Rob_env

68
69 env = Rob_env({((20,0),(30,20)), ((70,-5),(70,25))})
70 body = Rob_body(env)
71 middle = Rob_middle_layer(body)
72 top = Rob_top_layer(middle)
73
74 # try:
75 # pl=Plot_env(body,top)
76 # top.do({'visit':['o109','storage','o109','o103']})
77 # You can directly control the middle layer:
78 # middle.do({'go_to':(30,-10), 'timeout':200})
79 # Can you make it crash?

Exercise 2.1 The following code implements a robot trap. Write a controller that
can escape the “trap” and get to the goal. See textbook for hints.
agentTop.py — (continued)

81 # Robot Trap for which the current controller cannot escape:

82 trap_env = Rob_env({((10,-21),(10,0)), ((10,10),(10,31)), ((30,-10),(30,0)),
83 ((30,10),(30,20)), ((50,-21),(50,31)), ((10,-21),(50,-21)),
84 ((10,0),(30,0)), ((10,10),(30,10)), ((10,31),(50,31))})
85 trap_body = Rob_body(trap_env,init_pos=(-1,0,90))
86 trap_middle = Rob_middle_layer(trap_body)
87 trap_top = Rob_top_layer(trap_middle,locations={'goal':(71,0)})
88
89 # Robot trap exercise:
90 # pl=Plot_env(trap_body,trap_top)
91 # trap_top.do({'visit':['goal']})

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 3

Searching for Solutions

3.1 Representing Search Problems

A search problem consists of:

• a start node

• a neighbors function that given a node, returns an enumeration of the

arcs from the node

• a specification of a goal in terms of a Boolean function that takes a node

and returns true if the node is a goal

• a (optional) heuristic function that, given a node, returns a non-negative

real number. The heuristic function defaults to zero.

As far as the searcher is concerned a node can be anything. If multiple-path

pruning is used, a node must be hashable. In the simple examples, it is a string,
but in more complicated examples (in later chapters) it can be a tuple, a frozen
set, or a Python object.
In the following code raise NotImplementedError() is a way to specify that
this is an abstract method that needs to be overridden to define an actual search
problem.
searchProblem.py — representations of search problems
11 class Search_problem(object):
12 """A search problem consists of:
13 * a start node
14 * a neighbors function that gives the neighbors of a node
15 * a specification of a goal
16 * a (optional) heuristic function.

31
 32 3. Searching for Solutions

17 The methods must be overridden to define a search problem."""

18
19 def start_node(self):
20 """returns start node"""
21 raise NotImplementedError("start_node") # abstract method
22
23 def is_goal(self,node):
24 """is True if node is a goal"""
25 raise NotImplementedError("is_goal") # abstract method
26
27 def neighbors(self,node):
28 """returns a list of the arcs for the neighbors of node"""
29 raise NotImplementedError("neighbors") # abstract method
30
31 def heuristic(self,n):
32 """Gives the heuristic value of node n.
33 Returns 0 if not overridden."""
34 return 0
The neighbors is a list of arcs. A (directed) arc consists of a from node node
and a to node node. The arc is the pair hfrom node, to nodei, but can also contain
a non-negative cost (which defaults to 1) and can be labeled with an action.
searchProblem.py — (continued)

36 class Arc(object):
37 """An arc has a from_node and a to_node node and a (non-negative) cost"""
38 def __init__(self, from_node, to_node, cost=1, action=None):
39 assert cost >= 0, ("Cost cannot be negative for"+
40 str(from_node)+"->"+str(to_node)+", cost: "+str(cost))
41 self.from_node = from_node
42 self.to_node = to_node
43 self.action = action
44 self.cost=cost
45
46 def __repr__(self):
47 """string representation of an arc"""
48 if self.action:
49 return str(self.from_node)+" --"+str(self.action)+"--> "+str(self.to_node)
50 else:
51 return str(self.from_node)+" --> "+str(self.to_node)

3.1.1 Explicit Representation of Search Graph

The first representation of a search problem is from an explicit graph (as op-
posed to one that is generated as needed).
An explicit graph consists of

• a list or set of nodes

• a list or set of arcs

http://aipython.org Version 0.7.5 June 19, 2018

 3.1. Representing Search Problems 33

• a start node

• a list or set of goal nodes

• (optionally) a dictionary that maps a node to a heuristic value for that

node

To define a search problem, we need to define the start node, the goal predicate,
the neighbors function and the heuristic function.
searchProblem.py — (continued)

53 class Search_problem_from_explicit_graph(Search_problem):
54 """A search problem consists of:
55 * a list or set of nodes
56 * a list or set of arcs
57 * a start node
58 * a list or set of goal nodes
59 * a dictionary that maps each node into its heuristic value.
60 """
61
62 def __init__(self, nodes, arcs, start=None, goals=set(), hmap={}):
63 self.neighs = {}
64 self.nodes = nodes
65 for node in nodes:
66 self.neighs[node]=[]
67 self.arcs = arcs
68 for arc in arcs:
69 self.neighs[arc.from_node].append(arc)
70 self.start = start
71 self.goals = goals
72 self.hmap = hmap
73
74 def start_node(self):
75 """returns start node"""
76 return self.start
77
78 def is_goal(self,node):
79 """is True if node is a goal"""
80 return node in self.goals
81
82 def neighbors(self,node):
83 """returns the neighbors of node"""
84 return self.neighs[node]
85
86 def heuristic(self,node):
87 """Gives the heuristic value of node n.
88 Returns 0 if not overridden in the hmap."""
89 if node in self.hmap:
90 return self.hmap[node]
91 else:
92 return 0

http://aipython.org Version 0.7.5 June 19, 2018

 34 3. Searching for Solutions

93
94 def __repr__(self):
95 """returns a string representation of the search problem"""
96 res=""
97 for arc in self.arcs:
98 res += str(arc)+". "
99 return res
The following is used for the depth-first search implementation below.
searchProblem.py — (continued)

101 def neighbor_nodes(self,node):

102 """returns an iterator over the neighbors of node"""
103 return (path.to_node for path in self.neighs[node])

3.1.2 Paths
A searcher will return a path from the start node to a goal node. A Python list
is not a suitable representation for a path, as many search algorithms consider
multiple paths at once, and these paths should share initial parts of the path.
If we wanted to do this with Python lists, we would need to keep copying the
list, which can be expensive if the list is long. An alternative representation is
used here in terms of a recursive data structure that can share subparts.
A path is either:

• a node (representing a path of length 0) or

• a path, initial and an arc, where the from node of the arc is the node at the
end of initial.

These cases are distinguished in the following code by having arc = None if the
path has length 0, in which case initial is the node of the path.
searchProblem.py — (continued)

105 class Path(object):

106 """A path is either a node or a path followed by an arc"""
107
108 def __init__(self,initial,arc=None):
109 """initial is either a node (in which case arc is None) or
110 a path (in which case arc is an object of type Arc)"""
111 self.initial = initial
112 self.arc=arc
113 if arc is None:
114 self.cost=0
115 else:
116 self.cost = initial.cost+arc.cost
117
118 def end(self):
119 """returns the node at the end of the path"""
120 if self.arc is None:

http://aipython.org Version 0.7.5 June 19, 2018

 3.1. Representing Search Problems 35

121 return self.initial

122 else:
123 return self.arc.to_node
124
125 def nodes(self):
126 """enumerates the nodes for the path.
127 This starts at the end and enumerates nodes in the path backwards."""
128 current = self
129 while current.arc is not None:
130 yield current.arc.to_node
131 current = current.initial
132 yield current.initial
133
134 def initial_nodes(self):
135 """enumerates the nodes for the path before the end node.
136 This starts at the end and enumerates nodes in the path backwards."""
137 if self.arc is not None:
138 for nd in self.initial.nodes(): yield nd # could be "yield from"
139
140 def __repr__(self):
141 """returns a string representation of a path"""
142 if self.arc is None:
143 return str(self.initial)
144 elif self.arc.action:
145 return (str(self.initial)+"\n --"+str(self.arc.action)
146 +"--> "+str(self.arc.to_node))
147 else:
148 return str(self.initial)+" --> "+str(self.arc.to_node)

3.1.3 Example Search Problems

The first search problem is one with 5 nodes where the least-cost path is one
with many arcs. See Figure 3.1. Note that this example is used for the unit tests,
so the test (in searchGeneric) will need to be changed if this is changed.
searchProblem.py — (continued)

150 problem1 = Search_problem_from_explicit_graph(

151 {'a','b','c','d','g'},
152 [Arc('a','b',1), Arc('a','c',3), Arc('b','d',3), Arc('b','c',1),
153 Arc('c','d',1), Arc('c','g',3), Arc('d','g',1)],
154 start = 'a',
155 goals = {'g'})
The second search problem is one with 8 nodes where many paths do not lead
to the goal. See Figure 3.2.
searchProblem.py — (continued)

157 problem2 = Search_problem_from_explicit_graph(

158 {'a','b','c','d','e','g','h','j'},
159 [Arc('a','b',1), Arc('b','c',3), Arc('b','d',1), Arc('d','e',3),

http://aipython.org Version 0.7.5 June 19, 2018

 36 3. Searching for Solutions

a 3
1
b
3
1 c 1
3 d
1
g

Figure 3.1: problem1

a
3
1
h
b 1 1 j
d 1
3 g
3

c
g

Figure 3.2: problem2

160 Arc('d','g',1), Arc('a','h',3), Arc('h','j',1)],

161 start = 'a',
162 goals = {'g'})
The third search problem is a disconnected graph (contains no arcs), where the
start node is a goal node. This is a boundary case to make sure that weird cases
work.
searchProblem.py — (continued)

164 problem3 = Search_problem_from_explicit_graph(

165 {'a','b','c','d','e','g','h','j'},
166 [],
167 start = 'g',
168 goals = {'k','g'})
The acyclic delivery problem is the delivery problem described in Example
3.4 and shown in Figure 3.2 of the textbook.
searchProblem.py — (continued)

http://aipython.org Version 0.7.5 June 19, 2018

 3.1. Representing Search Problems 37

170 acyclic_delivery_problem = Search_problem_from_explicit_graph(

171 {'mail','ts','o103','o109','o111','b1','b2','b3','b4','c1','c2','c3',
172 'o125','o123','o119','r123','storage'},
173 [Arc('ts','mail',6),
174 Arc('o103','ts',8),
175 Arc('o103','b3',4),
176 Arc('o103','o109',12),
177 Arc('o109','o119',16),
178 Arc('o109','o111',4),
179 Arc('b1','c2',3),
180 Arc('b1','b2',6),
181 Arc('b2','b4',3),
182 Arc('b3','b1',4),
183 Arc('b3','b4',7),
184 Arc('b4','o109',7),
185 Arc('c1','c3',8),
186 Arc('c2','c3',6),
187 Arc('c2','c1',4),
188 Arc('o123','o125',4),
189 Arc('o123','r123',4),
190 Arc('o119','o123',9),
191 Arc('o119','storage',7)],
192 start = 'o103',
193 goals = {'r123'},
194 hmap = {
195 'mail' : 26,
196 'ts' : 23,
197 'o103' : 21,
198 'o109' : 24,
199 'o111' : 27,
200 'o119' : 11,
201 'o123' : 4,
202 'o125' : 6,
203 'r123' : 0,
204 'b1' : 13,
205 'b2' : 15,
206 'b3' : 17,
207 'b4' : 18,
208 'c1' : 6,
209 'c2' : 10,
210 'c3' : 12,
211 'storage' : 12
212 }
213 )
The cyclic delivery problem is the delivery problem described in Example
3.8 and shown in Figure 3.6 of the textbook. This is the same as acyclic delivery problem,
but almost every arc also has its inverse.
searchProblem.py — (continued)

215 cyclic_delivery_problem = Search_problem_from_explicit_graph(

http://aipython.org Version 0.7.5 June 19, 2018

 38 3. Searching for Solutions

216 {'mail','ts','o103','o109','o111','b1','b2','b3','b4','c1','c2','c3',
217 'o125','o123','o119','r123','storage'},
218 [ Arc('ts','mail',6), Arc('mail','ts',6),
219 Arc('o103','ts',8), Arc('ts','o103',8),
220 Arc('o103','b3',4),
221 Arc('o103','o109',12), Arc('o109','o103',12),
222 Arc('o109','o119',16), Arc('o119','o109',16),
223 Arc('o109','o111',4), Arc('o111','o109',4),
224 Arc('b1','c2',3),
225 Arc('b1','b2',6), Arc('b2','b1',6),
226 Arc('b2','b4',3), Arc('b4','b2',3),
227 Arc('b3','b1',4), Arc('b1','b3',4),
228 Arc('b3','b4',7), Arc('b4','b3',7),
229 Arc('b4','o109',7),
230 Arc('c1','c3',8), Arc('c3','c1',8),
231 Arc('c2','c3',6), Arc('c3','c2',6),
232 Arc('c2','c1',4), Arc('c1','c2',4),
233 Arc('o123','o125',4), Arc('o125','o123',4),
234 Arc('o123','r123',4), Arc('r123','o123',4),
235 Arc('o119','o123',9), Arc('o123','o119',9),
236 Arc('o119','storage',7), Arc('storage','o119',7)],
237 start = 'o103',
238 goals = {'r123'},
239 hmap = {
240 'mail' : 26,
241 'ts' : 23,
242 'o103' : 21,
243 'o109' : 24,
244 'o111' : 27,
245 'o119' : 11,
246 'o123' : 4,
247 'o125' : 6,
248 'r123' : 0,
249 'b1' : 13,
250 'b2' : 15,
251 'b3' : 17,
252 'b4' : 18,
253 'c1' : 6,
254 'c2' : 10,
255 'c3' : 12,
256 'storage' : 12
257 }
258 )

http://aipython.org Version 0.7.5 June 19, 2018

 3.2. Generic Searcher and Variants 39

3.2 Generic Searcher and Variants

To run the search demos, in folder “aipython”, load
“searchGeneric.py” , using e.g., ipython -i searchGeneric.py,
and copy and paste the example queries at the bottom of that file. This
requires Python 3.

3.2.1 Searcher
A Searcher for a problem can be asked repeatedly for the next path. To solve a
problem, we can construct a Searcher object for the problem and then repeatedly
ask for the next path using search. If there are no more paths, None is returned.
searchGeneric.py — Generic Searcher, including depth-first and A*
11 from utilities import Displayable, visualize
12
13 class Searcher(Displayable):
14 """returns a searcher for a problem.
15 Paths can be found by repeatedly calling search().
16 This does depth-first search unless overridden
17 """
18 def __init__(self, problem):
19 """creates a searcher from a problem
20 """
21 self.problem = problem
22 self.initialize_frontier()
23 self.num_expanded = 0
24 self.add_to_frontier(Path(problem.start_node()))
25 super().__init__()
26
27 def initialize_frontier(self):
28 self.frontier = []
29
30 def empty_frontier(self):
31 return self.frontier == []
32
33 def add_to_frontier(self,path):
34 self.frontier.append(path)
35
36 @visualize
37 def search(self):
38 """returns (next) path from the problem's start node
39 to a goal node.
40 Returns None if no path exists.
41 """
42 while not self.empty_frontier():
43 path = self.frontier.pop()
44 self.display(2, "Expanding:",path,"(cost:",path.cost,")")
45 self.num_expanded += 1

http://aipython.org Version 0.7.5 June 19, 2018

 40 3. Searching for Solutions

46 if self.problem.is_goal(path.end()): # solution found

47 self.display(1, self.num_expanded, "paths have been expanded and",
48 len(self.frontier), "paths remain in the frontier")
49 self.solution = path # store the solution found
50 return path
51 else:
52 neighs = self.problem.neighbors(path.end())
53 self.display(3,"Neighbors are", neighs)
54 for arc in reversed(neighs):
55 self.add_to_frontier(Path(path,arc))
56 self.display(3,"Frontier:",self.frontier)
57 self.display(1,"No (more) solutions. Total of",
58 self.num_expanded,"paths expanded.")
Note that this reverses the neigbours so that it implements depth-first search
in an intutive manner (expanding the first neighbor first). This might not be
required for other methods.
Exercise 3.1 When it returns a path, the algorithm can be used to find another
path by calling search() again. However, it does not find other paths that go
through one goal node to another. Explain why, and change the code so that it
can find such paths when search() is called again.

3.2.2 Frontier as a Priority Queue

In many of the search algorithms, such as A∗ and other best-first searchers, the
frontier is implemented as a priority queue. Here we use the Python’s built-in
priority queue implementations, heapq.
Following the lead of the Python documentation, http://docs.python.org/
3.3/library/heapq.html, a frontier is a list of triples. The first element of each
triple is the value to be minimized. The second element is a unique index which
specifies the order when the first elements are the same, and the third element
is the path that is on the queue. The use of the unique index ensures that the
priority queue implementation does not compare paths; whether one path is
less than another is not defined. It also lets us control what sort of search (e.g.,
depth-first or breadth-first) occurs when the value to be minimized does not
give a unique next path.
The variable frontier index is the total number of elements of the frontier
that have been created. As well as being used as a unique index, it is useful for
statistics, particularly in conjunction with the current size of the frontier.
searchGeneric.py — (continued)

60 import heapq # part of the Python standard library

61 from searchProblem import Path
62
63 class FrontierPQ(object):
64 """A frontier consists of a priority queue (heap), frontierpq, of
65 (value, index, path) triples, where
66 * value is the value we want to minimize (e.g., path cost + h).

http://aipython.org Version 0.7.5 June 19, 2018

 3.2. Generic Searcher and Variants 41

67 * index is a unique index for each element

68 * path is the path on the queue
69 Note that the priority queue always returns the smallest element.
70 """
71
72 def __init__(self):
73 """constructs the frontier, initially an empty priority queue
74 """
75 self.frontier_index = 0 # the number of items ever added to the frontier
76 self.frontierpq = [] # the frontier priority queue
77
78 def empty(self):
79 """is True if the priority queue is empty"""
80 return self.frontierpq == []
81
82 def add(self, path, value):
83 """add a path to the priority queue
84 value is the value to be minimized"""
85 self.frontier_index += 1 # get a new unique index
86 heapq.heappush(self.frontierpq,(value, -self.frontier_index, path))
87
88 def pop(self):
89 """returns and removes the path of the frontier with minimum value.
90 """
91 (_,_,path) = heapq.heappop(self.frontierpq)
92 return path

The following methods are used for finding and printing information about
the frontier.

searchGeneric.py — (continued)

94 def count(self,val):
95 """returns the number of elements of the frontier with value=val"""
96 return sum(1 for e in self.frontierpq if e[0]==val)
97
98 def __repr__(self):
99 """string representation of the frontier"""
100 return str([(n,c,str(p)) for (n,c,p) in self.frontierpq])
101
102 def __len__(self):
103 """length of the frontier"""
104 return len(self.frontierpq)
105
106 def __iter__(self):
107 """iterate through the paths in the frontier"""
108 for (_,_,path) in self.frontierpq:
109 yield path

http://aipython.org Version 0.7.5 June 19, 2018

 42 3. Searching for Solutions

3.2.3 A∗ Search
For an A∗ Search the frontier is implemented using the FrontierPQ class.
searchGeneric.py — (continued)

111 class AStarSearcher(Searcher):

112 """returns a searcher for a problem.
113 Paths can be found by repeatedly calling search().
114 """
115
116 def __init__(self, problem):
117 super().__init__(problem)
118
119 def initialize_frontier(self):
120 self.frontier = FrontierPQ()
121
122 def empty_frontier(self):
123 return self.frontier.empty()
124
125 def add_to_frontier(self,path):
126 """add path to the frontier with the appropriate cost"""
127 value = path.cost+self.problem.heuristic(path.end())
128 self.frontier.add(path, value)
Testing:
searchGeneric.py — (continued)

130 import searchProblem as searchProblem

131
132 def test(SearchClass):
133 print("Testing problem 1:")
134 schr1 = SearchClass(searchProblem.problem1)
135 path1 = schr1.search()
136 print("Path found:",path1)
137 assert list(path1.nodes()) == ['g','d','c','b','a'], "Shortest path not found in problem1"
138 print("Passed unit test")
139
140 if __name__ == "__main__":
141 #test(Searcher)
142 test(AStarSearcher)
143
144 # example queries:
145 # searcher1 = Searcher(searchProblem.acyclic_delivery_problem) # DFS
146 # searcher1.search() # find first path
147 # searcher1.search() # find next path
148 # searcher2 = AStarSearcher(searchProblem.acyclic_delivery_problem) # A*
149 # searcher2.search() # find first path
150 # searcher2.search() # find next path
151 # searcher3 = Searcher(searchProblem.cyclic_delivery_problem) # DFS
152 # searcher3.search() # find first path with DFS. What do you expect to happen?
153 # searcher4 = AStarSearcher(searchProblem.cyclic_delivery_problem) # A*
154 # searcher4.search() # find first path

http://aipython.org Version 0.7.5 June 19, 2018

 3.2. Generic Searcher and Variants 43

Exercise 3.2 Change the code so that it implements (i) best-first search and (ii)
lowest-cost-first search. For each of these methods compare it to A∗ in terms of the
number of paths expanded, and the path found.
Exercise 3.3 In the add method in FrontierPQ what does the ”-” in front of frontier index
do? When there are multiple paths with the same f -value, which search method
does this act like? What happens if the ”-” is removed? When there are multiple
paths with the same value, which search method does this act like? Does it work
better with or without the ”-”? What evidence did you base your conclusion on?
Exercise 3.4 The searcher acts like a Python iterator, in that it returns one value
(here a path) and then returns other values (paths) on demand, but does not imple-
ment the iterator interface. Change the code so it implements the iterator interface.
What does this enable us to do?

3.2.4 Multiple Path Pruning

To run the multiple-path pruning demo, in folder “aipython”, load
“searchMPP.py” , using e.g., ipython -i searchMPP.py, and copy and
paste the example queries at the bottom of that file.

The following implements A∗ with multiple-path pruning. It overrides search()

in Searcher.
searchMPP.py — Searcher with multiple-path pruning
11 from searchGeneric import AStarSearcher, visualize
12 from searchProblem import Path
13
14 class SearcherMPP(AStarSearcher):
15 """returns a searcher for a problem.
16 Paths can be found by repeatedly calling search().
17 """
18 def __init__(self, problem):
19 super().__init__(problem)
20 self.explored = set()
21
22 @visualize
23 def search(self):
24 """returns next path from an element of problem's start nodes
25 to a goal node.
26 Returns None if no path exists.
27 """
28 while not self.empty_frontier():
29 path = self.frontier.pop()
30 if path.end() not in self.explored:
31 self.display(2, "Expanding:",path,"(cost:",path.cost,")")
32 self.explored.add(path.end())
33 self.num_expanded += 1
34 if self.problem.is_goal(path.end()):
35 self.display(1, self.num_expanded, "paths have been expanded and",

http://aipython.org Version 0.7.5 June 19, 2018

 44 3. Searching for Solutions

36 len(self.frontier), "paths remain in the frontier")

37 self.solution = path # store the solution found
38 return path
39 else:
40 neighs = self.problem.neighbors(path.end())
41 self.display(3,"Neighbors are", neighs)
42 for arc in neighs:
43 self.add_to_frontier(Path(path,arc))
44 self.display(3,"Frontier:",self.frontier)
45 self.display(1,"No (more) solutions. Total of",
46 self.num_expanded,"paths expanded.")
47
48 from searchGeneric import test
49 if __name__ == "__main__":
50 test(SearcherMPP)
51
52 import aispace.searchProblem as searchProblem
53 # searcherMPPcdp = SearcherMPP(searchProblem.cyclic_delivery_problem)
54 # print(searcherMPPcdp.search()) # find first path

Exercise 3.5 Implement a searcher that implements cycle pruning instead of

multiple-path pruning. You need to decide whether to check for cycles when paths
are added to the frontier or when they are removed. (Hint: either method can be
implemented by only changing one or two lines in SearcherMPP.) Compare no
pruning, multiple path pruning and cycle pruning for the cyclic delivery problem.
Which works better in terms of number of paths expanded, computational time or
space?

3.3 Branch-and-bound Search

To run the demo, in folder “aipython”, load
“searchBranchAndBound.py”, and copy and paste the example queries
at the bottom of that file.

Depth-first search methods do not need an a priority queue, but can use
a list as a stack. In this implementation of branch-and-bound search, we call
search to find an optimal solution with cost less than bound. This uses depth-
first search to find a path to a goal that extends path with cost less than the
bound. Once a path to a goal has been found, that path is remembered as the
best path, the bound is reduced, and the search continues.
searchBranchAndBound.py — Branch and Bound Search
11 from searchProblem import Path
12 from searchGeneric import Searcher
13 from utilities import Displayable, visualize
14
15 class DF_branch_and_bound(Searcher):
16 """returns a branch and bound searcher for a problem.

http://aipython.org Version 0.7.5 June 19, 2018

 3.3. Branch-and-bound Search 45

17 An optimal path with cost less than bound can be found by calling search()
18 """
19 def __init__(self, problem, bound=float("inf")):
20 """creates a searcher than can be used with search() to find an optimal path.
21 bound gives the initial bound. By default this is infinite - meaning there
22 is no initial pruning due to depth bound
23 """
24 super().__init__(problem)
25 self.best_path = None
26 self.bound = bound
27
28 @visualize
29 def search(self):
30 """returns an optimal solution to a problem with cost less than bound.
31 returns None if there is no solution with cost less than bound."""
32 self.frontier = [Path(self.problem.start_node())]
33 self.num_expanded = 0
34 while self.frontier:
35 path = self.frontier.pop()
36 if path.cost+self.problem.heuristic(path.end()) < self.bound:
37 self.display(3,"Expanding:",path,"cost:",path.cost)
38 self.num_expanded += 1
39 if self.problem.is_goal(path.end()):
40 self.best_path = path
41 self.bound = path.cost
42 self.display(2,"New best path:",path," cost:",path.cost)
43 else:
44 neighs = self.problem.neighbors(path.end())
45 self.display(3,"Neighbors are", neighs)
46 for arc in reversed(list(neighs)):
47 self.add_to_frontier(Path(path, arc))
48 self.display(1,"Number of paths expanded:",self.num_expanded)
49 self.solution = self.best_path
50 return self.best_path
Note that this code used reversed in order to expand the neighbors of a node
in the left-to-right order one might expect. It does this because pop() removes
the rightmost element of the list. Note that reversed only works on lists and
tuples, but the neighbours can be generated.
Here is a unit test and some queries:
searchBranchAndBound.py — (continued)

52 from searchGeneric import test

53 if __name__ == "__main__":
54 test(DF_branch_and_bound)
55
56 # Example queries:
57 import aispace.searchProblem as searchProblem
58 # searcherb1 = DF_branch_and_bound(searchProblem.acyclic_delivery_problem)
59 # print(searcherb1.search()) # find optimal path
60 # searcherb2 = DF_branch_and_bound(searchProblem.cyclic_delivery_problem, bound=100)

http://aipython.org Version 0.7.5 June 19, 2018

 46 3. Searching for Solutions

61 # print(searcherb2.search()) # find optimal path

Exercise 3.6 Implement a branch-and-bound search uses recursion. Hint: you

don’t need an explicit frontier, but can do a recursive call for the children.
Exercise 3.7 After the branch-and-bound search found a solution, Sam ran search
again, and noticed a different count. Sam hypothesized that this count was related
to the number of nodes that an A∗ search would use (either expand or be added to
the frontier). Or maybe, Sam thought, the count for a number of nodes when the
bound is slightly above the optimal path case is related to how A∗ would work.
Is there relationship between these counts? Are there different things that it could
count so they are related? Try to find the most specific statement that is true, and
explain why it is true.
To test the hypothesis, Sam wrote the following code, but isn’t sure it is helpful:

searchTest.py — code that may be useful to compare A* and branch-and-bound

11 from searchGeneric import Searcher, AStarSearcher
12 from searchBranchAndBound import DF_branch_and_bound
13 from searchMPP import SearcherMPP
14
15 DF_branch_and_bound.max_display_level = 1
16 Searcher.max_display_level = 1
17
18 def run(problem,name):
19 print("\n\n*******",name)
20
21 print("\nA*:")
22 asearcher = AStarSearcher(problem)
23 print("Path found:",asearcher.search()," cost=",asearcher.solution.cost)
24 print("there are",asearcher.frontier.count(asearcher.solution.cost),
25 "elements remaining on the queue with f-value=",asearcher.solution.cost)
26
27 print("\nA* with MPP:"),
28 msearcher = SearcherMPP(problem)
29 print("Path found:",msearcher.search()," cost=",msearcher.solution.cost)
30 print("there are",msearcher.frontier.count(msearcher.solution.cost),
31 "elements remaining on the queue with f-value=",msearcher.solution.cost)
32
33 bound = asearcher.solution.cost+0.01
34 print("\nBranch and bound (with too-good initial bound of", bound,")")
35 tbb = DF_branch_and_bound(problem,bound) # cheating!!!!
36 print("Path found:",tbb.search()," cost=",tbb.solution.cost)
37 print("Rerunning B&B")
38 print("Path found:",tbb.search())
39
40 bbound = asearcher.solution.cost*2+10
41 print("\nBranch and bound (with not-very-good initial bound of", bbound, ")")
42 tbb2 = DF_branch_and_bound(problem,bbound) # cheating!!!!
43 print("Path found:",tbb2.search()," cost=",tbb2.solution.cost)
44 print("Rerunning B&B")

http://aipython.org Version 0.7.5 June 19, 2018

 3.3. Branch-and-bound Search 47

45 print("Path found:",tbb2.search())
46
47 print("\nDepth-first search: (Use ˆC if it goes on forever)")
48 tsearcher = Searcher(problem)
49 print("Path found:",tsearcher.search()," cost=",tsearcher.solution.cost)
50
51
52 import aispace.searchProblem as searchProblem
53 from searchTest import run
54 if __name__ == "__main__":
55 run(searchProblem.problem1,"Problem 1")
56 # run(searchProblem.acyclic_delivery_problem,"Acyclic Delivery")
57 # run(searchProblem.cyclic_delivery_problem,"Cyclic Delivery")
58 # also test some graphs with cycles, and some with multiple least-cost paths

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 4

Reasoning with Constraints

4.1 Constraint Satisfaction Problems

4.1.1 Constraints
A variable is a string or any value that is printable and can be the key of a
Python dictionary.
A constraint consists of a tuple (or list) of variables and a condition.

• The tuple (or list) of variables is called the scope.

• The condition is a Boolean function that takes the same number of ar-
guments as there are variables in the scope. The condition must have a
__name__ property that gives a printable name of the function; built-in
functions and functions that are defined using def have such a property;
for other functions you may need to define this property.

cspProblem.py — Representations of a Constraint Satisfaction Problem

11 from utilities import Displayable, dict_union
12
13 class Constraint(object):
14 """A Constraint consists of
15 * scope: a tuple of variables
16 * condition: a function that can applied to a tuple of values
17 for the variables
18 """
19 def __init__(self, scope, condition):
20 self.scope = scope
21 self.condition = condition
22
23 def __repr__(self):

49
 50 4. Reasoning with Constraints

24 return self.condition.__name__ + str(self.scope)

An assignment is a variable:value dictionary.
If con is a constraint, con.holds(assignment) returns True or False depending
on whether the condition is true or false for that assignment. The assignment
assignment must assigns a value to every variable in the scope of the constraint
con (and could also assign values other variables); con.holds gives an error if
not all variables in the scope of con are assigned in the assignment. It ignores
variables in assignment that are not in the scope of the constraint.
In Python, the ∗ notation is used for unpacking a tuple. For example,
F(∗(1, 2, 3)) is the same as F(1, 2, 3). So if t has value (1, 2, 3), then F(∗t) is
the same as F(1, 2, 3).
cspProblem.py — (continued)

26 def holds(self,assignment):
27 """returns the value of Constraint con evaluated in assignment.
28
29 precondition: all variables are assigned in assignment
30 """
31 return self.condition(*tuple(assignment[v] for v in self.scope))

4.1.2 CSPs
A constraint satisfaction problem (CSP) requires:

• domains: a dictionary that maps variables to the set of possible values.

Thus domains[var] is the domain of variable var.

• constaraints: a set or list of constraints.

Other properties are inferred from these:

• variables is the set of variables. The variables can be enumerated by using

“for var in domains” because iterating over a dictionary gives the keys,
which in this case are the variables.

• var to const is a mapping from variables to set of constraints, such that

var to const[var] is the set of constraints with var in the scope.

cspProblem.py — (continued)

33 class CSP(Displayable):
34 """A CSP consists of
35 * domains, a dictionary that maps each variable to its domain
36 * constraints, a list of constraints
37 * variables, a set of variables
38 * var_to_const, a variable to set of constraints dictionary
39 """
40 def __init__(self,domains,constraints):

http://aipython.org Version 0.7.5 June 19, 2018

 4.1. Constraint Satisfaction Problems 51

41 """domains is a variable:domain dictionary

42 constraints is a list of constriants
43 """
44 self.variables = set(domains)
45 self.domains = domains
46 self.constraints = constraints
47 self.var_to_const = {var:set() for var in self.variables}
48 for con in constraints:
49 for var in con.scope:
50 self.var_to_const[var].add(con)
51
52 def __str__(self):
53 """string representation of CSP"""
54 return str(self.domains)
55
56 def __repr__(self):
57 """more detailed string representation of CSP"""
58 return "CSP("+str(self.domains)+", "+str([str(c) for c in self.constraints])+")"

csp.consistent(assignment) returns true if the assignment is consistent with each

of the constraints in csp (i.e., all of the constraints that can be evaluated evaluate
to true). Note that this is a local consistency with each constraint; it does not
imply the CSP is consistent or has a solution.
cspProblem.py — (continued)

60 def consistent(self,assignment):
61 """assignment is a variable:value dictionary
62 returns True if all of the constraints that can be evaluated
63 evaluate to True given assignment.
64 """
65 return all(con.holds(assignment)
66 for con in self.constraints
67 if all(v in assignment for v in con.scope))

4.1.3 Examples
In the following code ne , when given a number, returns a function that is true
when its argument is not that number. For example, if f = ne (3), then f (2)
is True and f (3) is False. That is, ne (x)(y) is true when x 6= y. Allowing
a function of multiple arguments to use its arguments one at a time is called
currying, after the logician Haskell Curry. The use of a condition in constraints
requires that the function with a single argument has a name.
cspExamples.py — Example CSPs
11 from cspProblem import CSP, Constraint
12 from operator import lt,ne,eq,gt
13
14 def ne_(val):
15 """not equal value"""

http://aipython.org Version 0.7.5 June 19, 2018

 52 4. Reasoning with Constraints

16 # nev = lambda x: x != val # alternative definition

17 # nev = partial(neq,val) # another alternative definition
18 def nev(x):
19 return val != x
20 nev.__name__ = str(val)+"!=" # name of the function
21 return nev
Similarly is (x)(y) is true when x = y.
cspExamples.py — (continued)

23 def is_(val):
24 """is a value"""
25 # isv = lambda x: x == val # alternative definition
26 # isv = partial(eq,val) # another alternative definition
27 def isv(x):
28 return val == x
29 isv.__name__ = str(val)+"=="
30 return isv
The CSP, csp0 has variables X, Y and Z, each with domain {1, 2, 3}. The con-
straints are X < Y and Y < Z.
cspExamples.py — (continued)

32 csp0 = CSP({'X':{1,2,3},'Y':{1,2,3}, 'Z':{1,2,3}},

33 [ Constraint(('X','Y'),lt),
34 Constraint(('Y','Z'),lt)])
The CSP, csp1 has variables A, B and C, each with domain {1, 2, 3, 4}. The con-
straints are A < B, B 6= 2 and B < C. This is slightly more interesting than csp0
as it has more solutions. This example is used in the unit tests, and so if it is
changed, the unit tests need to be changed.
cspExamples.py — (continued)

36 C0 = Constraint(('A','B'),lt)
37 C1 = Constraint(('B',),ne_(2))
38 C2 = Constraint(('B','C'),lt)
39 csp1 = CSP({'A':{1,2,3,4},'B':{1,2,3,4}, 'C':{1,2,3,4}},
40 [C0, C1, C2])
The next CSP, csp2 is Example 4.9 of the textbook; the domain consistent
network is shown in Figure 4.1.
cspExamples.py — (continued)

42 csp2 = CSP({'A':{1,2,3,4},'B':{1,2,3,4}, 'C':{1,2,3,4},

43 'D':{1,2,3,4}, 'E':{1,2,3,4}},
44 [ Constraint(('B',),ne_(3)),
45 Constraint(('C',),ne_(2)),
46 Constraint(('A','B'),ne),
47 Constraint(('B','C'),ne),
48 Constraint(('C','D'),lt),
49 Constraint(('A','D'),eq),
50 Constraint(('A','E'),gt),

http://aipython.org Version 0.7.5 June 19, 2018

 4.1. Constraint Satisfaction Problems 53

A≠B
A B
{1,2,3,4} {1,2,4}

A=D B≠C
B≠D

D C E<B
E<A {1,2,3,4} {1,3,4}
C<D

E<D E<C

E
{1,2,3,4}

Figure 4.1: Domain-consistent constraint network (csp2).

51 Constraint(('B','E'),gt),
52 Constraint(('C','E'),gt),
53 Constraint(('D','E'),gt),
54 Constraint(('B','D'),ne)])

The following example is another scheduling problem (but with multiple an-
swers). This is the same a scheduling 2 in the original AIspace.org consistency
app.

cspExamples.py — (continued)

56 csp3 = CSP({'A':{1,2,3,4},'B':{1,2,3,4}, 'C':{1,2,3,4},

57 'D':{1,2,3,4}, 'E':{1,2,3,4}},
58 [Constraint(('A','B'), ne),
59 Constraint(('A','D'), lt),
60 Constraint(('A','E'), lambda a,e: (a-e)%2 == 1), # A-E is odd
61 Constraint(('B','E'), lt),
62 Constraint(('D','C'), lt),
63 Constraint(('C','E'), ne),
64 Constraint(('D','E'), ne)])

The following example is another abstract scheduling problem. What are

the solutions?

cspExamples.py — (continued)

http://aipython.org Version 0.7.5 June 19, 2018

 54 4. Reasoning with Constraints

1 2

Words:
3
ant, big, bus, car, has,
book, buys, hold, lane,
year, ginger, search,
symbol, syntax.
4

Figure 4.2: A crossword puzzle to be solved

66 def adjacent(x,y):
67 """True when x and y are adjacent numbers"""
68 return abs(x-y) == 1
69
70 csp4 = CSP({'A':{1,2,3,4,5},'B':{1,2,3,4,5}, 'C':{1,2,3,4,5},
71 'D':{1,2,3,4,5}, 'E':{1,2,3,4,5}},
72 [Constraint(('A','B'),adjacent),
73 Constraint(('B','C'),adjacent),
74 Constraint(('C','D'),adjacent),
75 Constraint(('D','E'),adjacent),
76 Constraint(('A','C'),ne),
77 Constraint(('B','D'),ne),
78 Constraint(('C','E'),ne)])
The following examples represent the crossword shown in Figure 4.2.
cspExamples.py — (continued)

80 def meet_at(p1,p2):
81 """returns a function that is true when the words meet at the postions p1, p2
82 """
83 def meets(w1,w2):
84 return w1[p1] == w2[p2]
85 meets.__name__ = "meet_at("+str(p1)+','+str(p2)+')'
86 return meets
87
88 crossword1 = CSP({'one_across':{'ant', 'big', 'bus', 'car', 'has'},
89 'one_down':{'book', 'buys', 'hold', 'lane', 'year'},
90 'two_down':{'ginger', 'search', 'symbol', 'syntax'},
91 'three_across':{'book', 'buys', 'hold', 'land', 'year'},
92 'four_across':{'ant', 'big', 'bus', 'car', 'has'}},
93 [Constraint(('one_across','one_down'),meet_at(0,0)),
94 Constraint(('one_across','two_down'),meet_at(2,0)),
95 Constraint(('three_across','two_down'),meet_at(2,2)),

http://aipython.org Version 0.7.5 June 19, 2018

 4.1. Constraint Satisfaction Problems 55

96 Constraint(('three_across','one_down'),meet_at(0,2)),
97 Constraint(('four_across','two_down'),meet_at(0,4))])

In an alternative representation of a crossword (the “dual” representation),

the variables represent letters, and the constraints are that adjacent sequences
of letters form words.
cspExamples.py — (continued)

99 words = {'ant', 'big', 'bus', 'car', 'has','book', 'buys', 'hold',

100 'lane', 'year', 'ginger', 'search', 'symbol', 'syntax'}
101
102 def is_word(*letters, words=words):
103 """is true if the letters concatenated form a word in words"""
104 return "".join(letters) in words
105
106 letters = ["a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l",
107 "m", "n", "o", "p", "q", "r", "s", "t", "u", "v", "w", "x", "y",
108 "z"]
109
110 crossword1d = CSP({'p00':letters, 'p10':letters, 'p20':letters, # first row
111 'p01':letters, 'p21':letters, # second row
112 'p02':letters, 'p12':letters, 'p22':letters, 'p32':letters, # third row
113 'p03':letters, 'p23':letters, #fourth row
114 'p24':letters, 'p34':letters, 'p44':letters, # fifth row
115 'p25':letters # sixth row
116 },
117 [Constraint(('p00', 'p10', 'p20'), is_word), #1-across
118 Constraint(('p00', 'p01', 'p02', 'p03'), is_word), # 1-down
119 Constraint(('p02', 'p12', 'p22', 'p32'), is_word), # 3-across
120 Constraint(('p20', 'p21', 'p22', 'p23', 'p24', 'p25'), is_word), # 2-down
121 Constraint(('p24', 'p34', 'p44'), is_word) # 4-across
122 ])

Unit tests
The following defines a unit test for solvers on example csp1.
cspExamples.py — (continued)

124 def test(CSP_solver, csp=csp1,

125 solutions=[{'A': 1, 'B': 3, 'C': 4}, {'A': 2, 'B': 3, 'C': 4}]):
126 """CSP_solver is a solver that finds a solution to a CSP.
127 CSP_solver takes a csp and returns a solution.
128 csp has to be a CSP, where solutions is the list of all solutions.
129 This tests whether the solution returned by CSP_solver is a solution.
130 """
131 print("Testing csp with",CSP_solver.__doc__)
132 sol0 = CSP_solver(csp)
133 print("Solution found:",sol0)
134 assert sol0 in solutions, "Solution not found for "+str(csp)
135 print("Passed unit test")

http://aipython.org Version 0.7.5 June 19, 2018

 56 4. Reasoning with Constraints

Exercise 4.1 Modify test so that instead of taking in a list of solutions, it checks
whether the returned solution actually is a solution.
Exercise 4.2 Propose a test that is appropriate for CSPs with no solutions. As-
sume that the test designer knows there are no solutions. Consider what a CSP
solver should return if there are no solutions to the CSP.

4.2 Solving a CSP using Search

To run the demo, in folder ”aipython”, load ”cspSearch.py”, and copy
and paste the example queries at the bottom of that file.

The first solver searches through the space of partial assignments. This
takes in a CSP problem and an optional variable ordering, which is a list of the
variables in the CSP. It then constructs a search space that can be solved using
the search methods of the previous chapter. In this search space:

• A node is a variable : value dictionary.

• An arc corresponds to an assignment of a value to the next variable. This

assumes a static ordering; the next variable chosen to split does not de-
pend on the context. If no variable ordering is given, this makes no at-
tempt to choose a good ordering.

cspSearch.py — Representations of a Search Problem from a CSP.

11 from cspProblem import CSP, Constraint
12 from searchProblem import Arc, Search_problem
13 from utilities import dict_union
14
15 class Search_from_CSP(Search_problem):
16 """A search problem directly from the CSP.
17
18 A node is a variable:value dictionary"""
19 def __init__(self, csp, variable_order=None):
20 self.csp=csp
21 if variable_order:
22 assert set(variable_order) == set(csp.variables)
23 assert len(variable_order) == len(csp.variables)
24 self.variables = variable_order
25 else:
26 self.variables = list(csp.variables)
27
28 def is_goal(self, node):
29 return len(node)==len(self.csp.variables)
30
31 def start_node(self):
32 return {}

http://aipython.org Version 0.7.5 June 19, 2018

 4.2. Solving a CSP using Search 57

The neighbors(node) method uses the fact that the length of the node, which
is the number of variables already assigned, is the index of the next variable to
split on.

cspSearch.py — (continued)

34 def neighbors(self, node):

35 """returns a list of the neighboring nodes of node.
36 """
37 var = self.variables[len(node)] # the next variable
38 res = []
39 for val in self.csp.domains[var]:
40 new_env = dict_union(node,{var:val}) #dictionary union
41 if self.csp.consistent(new_env):
42 res.append(Arc(node,new_env))
43 return res

cspSearch.py — (continued)

45 from cspExamples import csp1,csp2,test, crossword1, crossword1d

46 from searchGeneric import Searcher
47
48 def dfs_solver(csp):
49 """depth-first search solver"""
50 path = Searcher(Search_from_CSP(csp)).search()
51 if path is not None:
52 return path.end()
53 else:
54 return None
55
56 if __name__ == "__main__":
57 test(dfs_solver)
58
59 ## Test Solving CSPs with Search:
60 searcher1 = Searcher(Search_from_CSP(csp1))
61 #print(searcher1.search()) # get next solution
62 searcher2 = Searcher(Search_from_CSP(csp2))
63 #print(searcher2.search()) # get next solution
64 searcher3 = Searcher(Search_from_CSP(crossword1))
65 #print(searcher3.search()) # get next solution
66 searcher4 = Searcher(Search_from_CSP(crossword1d))
67 #print(searcher4.search()) # get next solution (warning: slow)

Exercise 4.3 What would happen if we constructed the new assignment by as-
signing node[var] = val (with side effects) instead of using dictionary union? Give
an example of where this could give a wrong answer. How could the algorithm be
changed to work with side effects? (Hint: think about what information needs to
be in a node).
Exercise 4.4 Change neighbors so that it returns an iterator of values rather than
a list. (Hint: use yield.)

http://aipython.org Version 0.7.5 June 19, 2018

 58 4. Reasoning with Constraints

4.3 Consistency Algorithms

To run the demo, in folder ”aipython”, load ”cspConsistency.py”, and
copy and paste the commented-out example queries at the bottom of
that file.

A Con solver is used to simplify a CSP using arc consistency.

cspConsistency.py — Arc Consistency and Domain splitting for solving a CSP
11 from utilities import Displayable
12
13 class Con_solver(Displayable):
14 def __init__(self, csp, **kwargs):
15 """a CSP solver that uses arc consistency
16 * csp is the CSP to be solved
17 * kwargs is the keyword arguments for Displayable superclass
18 """
19 super().__init__(**kwargs) # Or Displayable.__init__(self,**kwargs)
20 self.csp = csp

The following implementation of arc consistency maintains the set to do of

(variable, constraint) pairs that are to be checked. It takes in a domain dic-
tionary and returns a new domain dictionary. It needs to be careful to avoid
side effects (by copying the domains dictionary and the to do set).
cspConsistency.py — (continued)

22 def make_arc_consistent(self, orig_domains=None, to_do=None):

23 """Makes this CSP arc-consistent using generalized arc consistency
24 orig_domains is the original domains
25 to_do is a set of (variable,constraint) pairs
26 returns the reduced domains (an arc-consistent variable:domain dictionary)
27 """
28 if orig_domains is None:
29 orig_domains = self.csp.domains
30 if to_do is None:
31 to_do = {(var, const) for const in self.csp.constraints
32 for var in const.scope}
33 else:
34 to_do = to_do.copy() # use a copy of to_do
35 domains = orig_domains.copy()
36 self.display(2,"Performing AC with domains", domains)
37 while to_do:
38 var, const = self.select_arc(to_do)
39 self.display(3, "Processing arc (", var, ",", const, ")")
40 other_vars = [ov for ov in const.scope if ov != var]
41 if len(other_vars)==0:
42 new_domain = {val for val in domains[var]
43 if const.holds({var:val})}
44 elif len(other_vars)==1:
45 other = other_vars[0]

http://aipython.org Version 0.7.5 June 19, 2018

 4.3. Consistency Algorithms 59

46 new_domain = {val for val in domains[var]

47 if any(const.holds({var: val,other:other_val})
48 for other_val in domains[other])}
49 else: # general case
50 new_domain = {val for val in domains[var]
51 if self.any_holds(domains, const, {var: val}, other_vars)}
52 if new_domain != domains[var]:
53 self.display(4, "Arc: (", var, ",", const, ") is inconsistent")
54 self.display(3, "Domain pruned", "dom(", var, ") =", new_domain,
55 " due to ", const)
56 domains[var] = new_domain
57 add_to_do = self.new_to_do(var, const) - to_do
58 to_do |= add_to_do # set union
59 self.display(3, " adding", add_to_do if add_to_do else "nothing", "to to_do.")
60 self.display(4, "Arc: (", var, ",", const, ") now consistent")
61 self.display(2, "AC done. Reduced domains", domains)
62 return domains
63
64 def new_to_do(self, var, const):
65 """returns new elements to be added to to_do after assigning
66 variable var in constraint const.
67 """
68 return {(nvar, nconst) for nconst in self.csp.var_to_const[var]
69 if nconst != const
70 for nvar in nconst.scope
71 if nvar != var}
The following selects an arc. Any element of to do can be selected. The selected
element needs to be removed from to do. The default implementation just se-
lects which ever element pop method for sets returns. A user interface could
allow the user to select an arc. Alternatively a more sophisticated selection
could be employed (or just a stack or a queue).
cspConsistency.py — (continued)

73 def select_arc(self, to_do):

74 """Selects the arc to be taken from to_do .
75 * to_do is a set of arcs, where an arc is a (variable,constraint) pair
76 the element selected must be removed from to_do.
77 """
78 return to_do.pop()
The function any holds is useful to go beyond unary and binary constraints. It
allows us to use constraints involving an arbitrary number of variables. (Note
that it also works for unary and binary constraints; the cases where len(other vars)
is 0 or 1 are not actually required, but are there for efficiency and because they
are easier to understand.) any holds is a recursive function that tries to finds an
assignment of values to the other variables (other vars) that satisfies constraint
const given the assignment in env. The integer variable ind specifies which in-
dex to other vars needs to be checked next. As soon as one assignment returns
True, the algorithm returns True. Note that it has side effects with respect to

http://aipython.org Version 0.7.5 June 19, 2018

 60 4. Reasoning with Constraints

env; it changes the values of the variables in other vars. It should only be called
when the side effects have no ill effects.

cspConsistency.py — (continued)

80 def any_holds(self, domains, const, env, other_vars, ind=0):

81 """returns True if Constraint const holds for an assignment
82 that extends env with the variables in other_vars[ind:]
83 env is a dictionary
84 Warning: this has side effects and changes the elements of env
85 """
86 if ind == len(other_vars):
87 return const.holds(env)
88 else:
89 var = other_vars[ind]
90 for val in domains[var]:
91 # env = dict_union(env,{var:val}) # no side effects!
92 env[var] = val
93 if self.any_holds(domains, const, env, other_vars, ind + 1):
94 return True
95 return False

4.3.1 Direct Implementation of Domain Splitting

The following is a direct implementation of domain splitting with arc consis-
tency that uses recursion. It finds one solution if one exists or returns False if
there are no solutions.

cspConsistency.py — (continued)

97 def solve_one(self, domains=None, to_do=None):

98 """return a solution to the current CSP or False if there are no solutions
99 to_do is the list of arcs to check
100 """
101 if domains is None:
102 domains = self.csp.domains
103 new_domains = self.make_arc_consistent(domains, to_do)
104 if any(len(new_domains[var]) == 0 for var in domains):
105 return False
106 elif all(len(new_domains[var]) == 1 for var in domains):
107 self.display(2, "solution:", {var: select(
108 new_domains[var]) for var in new_domains})
109 return {var: select(new_domains[var]) for var in domains}
110 else:
111 var = self.select_var(x for x in self.csp.variables if len(new_domains[x]) > 1)
112 if var:
113 dom1, dom2 = partition_domain(new_domains[var])
114 self.display(3, "...splitting", var, "into", dom1, "and", dom2)
115 new_doms1 = copy_with_assign(new_domains, var, dom1)
116 new_doms2 = copy_with_assign(new_domains, var, dom2)
117 to_do = self.new_to_do(var, None)

http://aipython.org Version 0.7.5 June 19, 2018

 4.3. Consistency Algorithms 61

118 self.display(3, " adding", to_do if to_do else "nothing", "to to_do.")
119 return self.solve_one(new_doms1, to_do) or self.solve_one(new_doms2, to_do)
120
121 def select_var(self, iter_vars):
122 """return the next variable to split"""
123 return select(iter_vars)
124
125 def partition_domain(dom):
126 """partitions domain dom into two.
127 """
128 split = len(dom) // 2
129 dom1 = set(list(dom)[:split])
130 dom2 = dom - dom1
131 return dom1, dom2

The domains are implemented as a dictionary that maps each variables to

its domain. Assigning a value in Python has side effects which we want to
avoid. copy with assign takes a copy of the domains dictionary, perhaps al-
lowing for a new domain for a variable. It creates a copy of the CSP with an
(optional) assignment of a new domain to a variable. Only the domains are
copied.

cspConsistency.py — (continued)

133 def copy_with_assign(domains, var=None, new_domain={True, False}):

134 """create a copy of the domains with an assignment var=new_domain
135 if var==None then it is just a copy.
136 """
137 newdoms = domains.copy()
138 if var is not None:
139 newdoms[var] = new_domain
140 return newdoms

cspConsistency.py — (continued)

142 def select(iterable):

143 """select an element of iterable. Returns None if there is no such element.
144
145 This implementation just picks the first element.
146 For many of the uses, which element is selected does not affect correctness,
147 but may affect efficiency.
148 """
149 for e in iterable:
150 return e # returns first element found

Exercise 4.5 Implement of solve all that is like solve one but returns the set of all
solutions.
Exercise 4.6 Implement solve enum that enumerates the solutions. It should use
Python’s yield (and perhaps yield from).

Unit test:

http://aipython.org Version 0.7.5 June 19, 2018

 62 4. Reasoning with Constraints

cspConsistency.py — (continued)

152 from cspExamples import test

153 def ac_solver(csp):
154 "arc consistency (solve_one)"
155 return Con_solver(csp).solve_one()
156 if __name__ == "__main__":
157 test(ac_solver)

4.3.2 Domain Splitting as an interface to graph searching

An alternative implementation is to implement domain splitting in terms of
the search abstraction of Chapter 3.
A node is domains dictionary.

cspConsistency.py — (continued)

159 from searchProblem import Arc, Search_problem

160
161 class Search_with_AC_from_CSP(Search_problem,Displayable):
162 """A search problem with arc consistency and domain splitting
163
164 A node is a CSP """
165 def __init__(self, csp):
166 self.cons = Con_solver(csp) #copy of the CSP
167 self.domains = self.cons.make_arc_consistent()
168
169 def is_goal(self, node):
170 """node is a goal if all domains have 1 element"""
171 return all(len(node[var])==1 for var in node)
172
173 def start_node(self):
174 return self.domains
175
176 def neighbors(self,node):
177 """returns the neighboring nodes of node.
178 """
179 neighs = []
180 var = select(x for x in node if len(node[x])>1)
181 if var:
182 dom1, dom2 = partition_domain(node[var])
183 self.display(2,"Splitting", var, "into", dom1, "and", dom2)
184 to_do = self.cons.new_to_do(var,None)
185 for dom in [dom1,dom2]:
186 newdoms = copy_with_assign(node,var,dom)
187 cons_doms = self.cons.make_arc_consistent(newdoms,to_do)
188 if all(len(cons_doms[v])>0 for v in cons_doms):
189 # all domains are non-empty
190 neighs.append(Arc(node,cons_doms))
191 else:
192 self.display(2,"...",var,"in",dom,"has no solution")

http://aipython.org Version 0.7.5 June 19, 2018

 4.4. Solving CSPs using Stochastic Local Search 63

193 return neighs

Exercise 4.7 When splitting a domain, this code splits the domain into half, ap-
proximately in half (without any effort to make a sensible choice). Does it work
better to split one element from a domain?
Unit test:
cspConsistency.py — (continued)

195 from cspExamples import test

196 from searchGeneric import Searcher
197
198 def ac_search_solver(csp):
199 """arc consistency (search interface)"""
200 sol = Searcher(Search_with_AC_from_CSP(csp)).search()
201 if sol:
202 return {v:select(d) for (v,d) in sol.end().items()}
203
204 if __name__ == "__main__":
205 test(ac_search_solver)
Testing:
cspConsistency.py — (continued)

207 from cspExamples import csp1, csp2, crossword1, crossword1d

208
209 ## Test Solving CSPs with Arc consistency and domain splitting:
210 #Con_solver.max_display_level = 4 # display details of AC (0 turns off)
211 #Con_solver(csp1).solve_one()
212 #searcher1d = Searcher(Search_with_AC_from_CSP(csp1))
213 #print(searcher1d.search())
214 #Searcher.max_display_level = 2 # display search trace (0 turns off)
215 #searcher2c = Searcher(Search_with_AC_from_CSP(csp2))
216 #print(searcher2c.search())
217 #searcher3c = Searcher(Search_with_AC_from_CSP(crossword1))
218 #print(searcher3c.search())
219 #searcher5c = Searcher(Search_with_AC_from_CSP(crossword1d))
220 #print(searcher5c.search())

4.4 Solving CSPs using Stochastic Local Search

To run the demo, in folder ”aipython”, load ”cspSLS.py”, and copy
and paste the commented-out example queries at the bottom of that
file. This assumes Python 3. Some of the queries require matplotlib.

This implements both the two-stage choice, the any-conflict algorithm and
a random choice of variable (and a probabilistic mix of the three).
Given a CSP, the stochastic local searcher (SLSearcher) creates the data struc-
tures:

http://aipython.org Version 0.7.5 June 19, 2018

 64 4. Reasoning with Constraints

• variables to select is the set of all of the variables with domain-size greater
than one. For a variable not in this set, we cannot pick another value from
that variable.

• var to constraints maps from a variable into the set of constraints it is in-
volved in. Note that the inverse mapping from constraints into variables
is part of the definition of a constraint.

cspSLS.py — Stochastic Local Search for Solving CSPs

11 from cspProblem import CSP, Constraint
12 from searchProblem import Arc, Search_problem
13 from utilities import Displayable
14 import random
15 import heapq
16
17 class SLSearcher(Displayable):
18 """A search problem directly from the CSP..
19
20 A node is a variable:value dictionary"""
21 def __init__(self, csp):
22 self.csp = csp
23 self.variables_to_select = {var for var in self.csp.variables
24 if len(self.csp.domains[var]) > 1}
25 # Create assignment and conflicts set
26 self.current_assignment = None # this will trigger a random restart
27 self.number_of_steps = 1 #number of steps after the initialization
restart creates a new total assignment, and constructs the set of conflicts (the
constraints that are false in this assignment).
cspSLS.py — (continued)

29 def restart(self):
30 """creates a new total assignment and the conflict set
31 """
32 self.current_assignment = {var:random_sample(dom) for
33 (var,dom) in self.csp.domains.items()}
34 self.display(2,"Initial assignment",self.current_assignment)
35 self.conflicts = set()
36 for con in self.csp.constraints:
37 if not con.holds(self.current_assignment):
38 self.conflicts.add(con)
39 self.display(2,"Number of conflicts",len(self.conflicts))
40 self.variable_pq = None
The search method is the top-level searching algorithm. It can either be used
to start the search or to continue searching. If there is no current assignment,
it must create one. Note that, when counting steps, a restart is counted as one
step.
This method selects one of two implementations. The argument pob best
is the probability of selecting a best variable (one involving the most conflicts).

http://aipython.org Version 0.7.5 June 19, 2018

 4.4. Solving CSPs using Stochastic Local Search 65

When the value of prob best is positive, the algorithm needs to maintain a prior-
ity queue of variables and the number of conflicts (using search with var pq). If
the probability of selecting a best variable is zero, it does not need to maintain
this priority queue (as implemented in search with any conflict).
The argument prob anycon is the probability that the any-conflict strategy is
used (which selects a variable at random that is in a conflict), assuming that
it is not picking a best variable. Note that for the probability parameters, any
value less that zero acts like probability zero and any value greater than 1 acts
like probability 1. This means that when prob anycon = 1.0, a best variable is
chosen with probability prob best, otherwise a variable in any conflict is chosen.
A variable is chosen at random with probability 1 − prob anycon − prob best as
long as that is positive.
This returns the number of steps needed to find a solution, or None if no
solution is found. If there is a solution, it is in self .current assignment.
cspSLS.py — (continued)

42 def search(self,max_steps, prob_best=1.0, prob_anycon=1.0):

43 """
44 returns the number of steps or None if these is no solution.
45 If there is a solution, it can be found in self.current_assignment
46
47 max_steps is the maximum number of steps it will try before giving up
48 prob_best is the probability that a best varaible (one in most conflict) is selected
49 prob_anycon is the probability that a variabe in any conflict is selected
50 (otherwise a variable is chosen at random)
51 """
52 if self.current_assignment is None:
53 self.restart()
54 self.number_of_steps += 1
55 if not self.conflicts:
56 return self.number_of_steps
57 if prob_best > 0: # we need to maintain a variable priority queue
58 return self.search_with_var_pq(max_steps, prob_best, prob_anycon)
59 else:
60 return self.search_with_any_conflict(max_steps, prob_anycon)

Exercise 4.8 This does an initial random assignment but does not do any random
restarts. Implement a searcher that takes in the maximum number of walk steps
(corresponding to existing max steps) and the maximum number of restarts, and
returns the total number of steps for the first solution found. (As in search, the
solution found can be extracted from the variable self .current assignment).

4.4.1 Any-conflict
If the probability of picking a best variable is zero, the implementation need to
keeps track of which variables are in conflicts.
cspSLS.py — (continued)

http://aipython.org Version 0.7.5 June 19, 2018

 66 4. Reasoning with Constraints

62 def search_with_any_conflict(self, max_steps, prob_anycon=1.0):

63 """Searches with the any_conflict heuristic.
64 This relies on just maintaining the set of conflicts;
65 it does not maintain a priority queue
66 """
67 self.variable_pq = None # we are not maintaining the priority queue.
68 # This ensures it is regenerated if needed.
69 for i in range(max_steps):
70 self.number_of_steps +=1
71 if random.random() < prob_anycon:
72 con = random_sample(self.conflicts) # pick random conflict
73 var = random_sample(con.scope) # pick variable in conflict
74 else:
75 var = random_sample(self.variables_to_select)
76 if len(self.csp.domains[var]) > 1:
77 val = random_sample(self.csp.domains[var] -
78 {self.current_assignment[var]})
79 self.display(2,self.number_of_steps,": Assigning",var,"=",val)
80 self.current_assignment[var]=val
81 for varcon in self.csp.var_to_const[var]:
82 if varcon.holds(self.current_assignment):
83 if varcon in self.conflicts:
84 self.conflicts.remove(varcon)
85 else:
86 if varcon not in self.conflicts:
87 self.conflicts.add(varcon)
88 self.display(2," Number of conflicts",len(self.conflicts))
89 if not self.conflicts:
90 self.display(1,"Solution found:", self.current_assignment,
91 "in", self.number_of_steps,"steps")
92 return self.number_of_steps
93 self.display(1,"No solution in",self.number_of_steps,"steps",
94 len(self.conflicts),"conflicts remain")
95 return None

Exercise 4.9 This makes no attempt to find the best alternative value for a vari-
able. Modify the code so that after selecting a variable it selects a value the reduces
the number of conflicts by the most. Have a parameter that specifies the probabil-
ity that the best value is chosen.

4.4.2 Two-Stage Choice

This is the top-level searching algorithm that maintains a priority queue of vari-
ables ordered by (the negative of) the number of conflicts, so that the variable
with the most conflicts is selected first. If there is no current priority queue of
variables, one is created.
The main complexity here is to maintain the priority queue. This uses
the dictionary var differential which specifies how much the values of variables

http://aipython.org Version 0.7.5 June 19, 2018

 4.4. Solving CSPs using Stochastic Local Search 67

should change. This is used with the updatable queue (page 68) to find a vari-
able with the most conflicts.
cspSLS.py — (continued)

97 def search_with_var_pq(self,max_steps, prob_best=1.0, prob_anycon=1.0):

98 """search with a priority queue of variables.
99 This is used to select a variable with the most conflicts.
100 """
101 if not self.variable_pq:
102 self.create_pq()
103 pick_best_or_con = prob_best + prob_anycon
104 for i in range(max_steps):
105 self.number_of_steps +=1
106 randnum = random.random()
107 ## Pick a variable
108 if randnum < prob_best: # pick best variable
109 var,oldval = self.variable_pq.top()
110 elif randnum < pick_best_or_con: # pick a variable in a conflict
111 con = random_sample(self.conflicts)
112 var = random_sample(con.scope)
113 else: #pick any variable that can be selected
114 var = random_sample(self.variables_to_select)
115 if len(self.csp.domains[var]) > 1: # var has other values
116 ## Pick a value
117 val = random_sample(self.csp.domains[var] -
118 {self.current_assignment[var]})
119 self.display(2,"Assigning",var,val)
120 ## Update the priority queue
121 var_differential = {}
122 self.current_assignment[var]=val
123 for varcon in self.csp.var_to_const[var]:
124 self.display(3,"Checking",varcon)
125 if varcon.holds(self.current_assignment):
126 if varcon in self.conflicts: #was incons, now consis
127 self.display(3,"Became consistent",varcon)
128 self.conflicts.remove(varcon)
129 for v in varcon.scope: # v is in one fewer conflicts
130 var_differential[v] = var_differential.get(v,0)-1
131 else:
132 if varcon not in self.conflicts: # was consis, not now
133 self.display(3,"Became inconsistent",varcon)
134 self.conflicts.add(varcon)
135 for v in varcon.scope: # v is in one more conflicts
136 var_differential[v] = var_differential.get(v,0)+1
137 self.variable_pq.update_each_priority(var_differential)
138 self.display(2,"Number of conflicts",len(self.conflicts))
139 if not self.conflicts: # no conflicts, so solution found
140 self.display(1,"Solution found:", self.current_assignment,"in",
141 self.number_of_steps,"steps")
142 return self.number_of_steps
143 self.display(1,"No solution in",self.number_of_steps,"steps",

http://aipython.org Version 0.7.5 June 19, 2018

 68 4. Reasoning with Constraints

144 len(self.conflicts),"conflicts remain")

145 return None
create pq creates an updatable priority queue of the variables, ordered by the
number of conflicts they participate in. The priority queue only includes vari-
ables in conflicts and the value of a variable is the negative of the number of
conflicts the variable is in. This ensures that the priority queue, which picks
the minimum value, picks a variable with the most conflicts.
cspSLS.py — (continued)

147 def create_pq(self):

148 """Create the variable to number-of-conflicts priority queue.
149 This is needed to select the variable in the most conflicts.
150
151 The value of a variable in the priority queue is the negative of the
152 number of conflicts the variable appears in.
153 """
154 self.variable_pq = Updatable_priority_queue()
155 var_to_number_conflicts = {}
156 for con in self.conflicts:
157 for var in con.scope:
158 var_to_number_conflicts[var] = var_to_number_conflicts.get(var,0)+1
159 for var,num in var_to_number_conflicts.items():
160 if num>0:
161 self.variable_pq.add(var,-num)

cspSLS.py — (continued)

163 def random_sample(st):

164 """selects a random element from set st"""
165 return random.sample(st,1)[0]

Exercise 4.10 This makes no attempt to find the best alternative value for a vari-
able. Modify the code so that after selecting a variable it selects a value the reduces
the number of conflicts by the most. Have a parameter that specifies the probabil-
ity that the best value is chosen.
Exercise 4.11 These implementations always select a value for the variable se-
lected that is different from its current value (if that is possible). Change the code
so that it does not have this restriction (so it can leave the value the same). Would
you expect this code to be faster? Does it work worse (or better)?

4.4.3 Updatable Priority Queues

An updatable priority queue is a priority queue, where key-value pairs can be
stored, and the pair with the smallest key can be found and removed quickly,
and where the values can be updated. This implementation follows the idea
of http://docs.python.org/3.5/library/heapq.html, where the updated ele-
ments are marked as removed. This means that the priority queue can be used
unmodified. However, this might be expensive if changes are more common

http://aipython.org Version 0.7.5 June 19, 2018

 4.4. Solving CSPs using Stochastic Local Search 69

than popping (as might happen if the probability of choosing the best is close
to zero).
In this implementation, the equal values are sorted randomly. This is achieved
by having the elements of the heap being [val, rand, elt] triples, where the sec-
ond element is a random number. Note that Python requires this to be a list,
not a tuple, as the tuple cannot be modified.

cspSLS.py — (continued)

167 class Updatable_priority_queue(object):

168 """A priority queue where the values can be updated.
169 Elements with the same value are ordered randomly.
170
171 This code is based on the ideas described in
172 http://docs.python.org/3.3/library/heapq.html
173 It could probably be done more efficiently by
174 shuffling the modified element in the heap.
175 """
176 def __init__(self):
177 self.pq = [] # priority queue of [val,rand,elt] triples
178 self.elt_map = {} # map from elt to [val,rand,elt] triple in pq
179 self.REMOVED = "*removed*" # a string that won't be a legal element
180 self.max_size=0
181
182 def add(self,elt,val):
183 """adds elt to the priority queue with priority=val.
184 """
185 assert val <= 0,val
186 assert elt not in self.elt_map, elt
187 new_triple = [val, random.random(),elt]
188 heapq.heappush(self.pq, new_triple)
189 self.elt_map[elt] = new_triple
190
191 def remove(self,elt):
192 """remove the element from the priority queue"""
193 if elt in self.elt_map:
194 self.elt_map[elt][2] = self.REMOVED
195 del self.elt_map[elt]
196
197 def update_each_priority(self,update_dict):
198 """update values in the priority queue by subtracting the values in
199 update_dict from the priority of those elements in priority queue.
200 """
201 for elt,incr in update_dict.items():
202 if incr != 0:
203 newval = self.elt_map.get(elt,[0])[0] - incr
204 assert newval <= 0, str(elt)+":"+str(newval+incr)+"-"+str(incr)
205 self.remove(elt)
206 if newval != 0:
207 self.add(elt,newval)
208

http://aipython.org Version 0.7.5 June 19, 2018

 70 4. Reasoning with Constraints

209 def pop(self):

210 """Removes and returns the (elt,value) pair with minimal value.
211 If the priority queue is empty, IndexError is raised.
212 """
213 self.max_size = max(self.max_size, len(self.pq)) # keep statistics
214 triple = heapq.heappop(self.pq)
215 while triple[2] == self.REMOVED:
216 triple = heapq.heappop(self.pq)
217 del self.elt_map[triple[2]]
218 return triple[2], triple[0] # elt, value
219
220 def top(self):
221 """Returns the (elt,value) pair with minimal value, without removing it.
222 If the priority queue is empty, IndexError is raised.
223 """
224 self.max_size = max(self.max_size, len(self.pq)) # keep statistics
225 triple = self.pq[0]
226 while triple[2] == self.REMOVED:
227 heapq.heappop(self.pq)
228 triple = self.pq[0]
229 return triple[2], triple[0] # elt, value
230
231 def empty(self):
232 """returns True iff the priority queue is empty"""
233 return all(triple[2] == self.REMOVED for triple in self.pq)

4.4.4 Plotting Runtime Distributions

Runtime distribution uses matplotlib to plot runtime distributions. Here the
runtime is a misnomer as we are only plotting the number of steps, not the
time. Computing the runtime is non-trivial as many of the runs have a very
short runtime. To compute the time accurately would require running the same
code, with the same random seed, multiple times to get a good estimate of the
runtime. This is left as an exercise.
cspSLS.py — (continued)

235 import matplotlib.pyplot as plt

236
237 class Runtime_distribution(object):
238 def __init__(self, csp, xscale='log'):
239 """Sets up plotting for csp
240 xscale is either 'linear' or 'log'
241 """
242 self.csp = csp
243 plt.ion()
244 plt.xlabel("Number of Steps")
245 plt.ylabel("Cumulative Number of Runs")
246 plt.xscale(xscale) # Makes a 'log' or 'linear' scale
247

http://aipython.org Version 0.7.5 June 19, 2018

 4.4. Solving CSPs using Stochastic Local Search 71

248 def plot_run(self,num_runs=100,max_steps=1000, prob_best=1.0, prob_anycon=1.0):

249 stats = []
250 SLSearcher.max_display_level, temp_mdl = 0, SLSearcher.max_display_level # no display
251 for i in range(num_runs):
252 searcher = SLSearcher(self.csp)
253 num_steps = searcher.search(max_steps, prob_best, prob_anycon)
254 if num_steps:
255 stats.append(num_steps)
256 stats.sort()
257 if prob_best >= 1.0:
258 label = "P(best)=1.0"
259 else:
260 p_ac = min(prob_anycon, 1-prob_best)
261 label = "P(best)=%.2f, P(ac)=%.2f" % (prob_best, p_ac)
262 plt.plot(stats,range(len(stats)),label=label)
263 plt.legend(loc="upper left")
264 #plt.draw()
265 SLSearcher.max_display_level= temp_mdl #restore display

4.4.5 Testing
cspSLS.py — (continued)

267 from cspExamples import test

268 def sls_solver(csp,prob_best=0.7):
269 """stochastic local searcher (prob_best=0.7)"""
270 se0 = SLSearcher(csp)
271 se0.search(1000,prob_best)
272 return se0.current_assignment
273 def any_conflict_solver(csp):
274 """stochastic local searcher (any-conflict)"""
275 return sls_solver(csp,0)
276
277 if __name__ == "__main__":
278 test(sls_solver)
279 test(any_conflict_solver)
280
281 from cspExamples import csp1, csp2, crossword1
282
283 ## Test Solving CSPs with Search:
284 #se1 = SLSearcher(csp1); print(se1.search(100))
285 #se2 = SLSearcher(csp2); print(se2.search(1000,1.0)) # greedy
286 #se2 = SLSearcher(csp2); print(se2.search(1000,0)) # any_conflict
287 #se2 = SLSearcher(csp2); print(se2.search(1000,0.7)) # 70% greedy; 30% any_conflict
288 #SLSearcher.max_display_level=2 #more detailed display
289 #se3 = SLSearcher(crossword1); print(se3.search(100),0.7)
290 #p = Runtime_distribution(csp2)
291 #p.plot_run(1000,1000,0) # any_conflict
292 #p.plot_run(1000,1000,1.0) # greedy
293 #p.plot_run(1000,1000,0.7) # 70% greedy; 30% any_conflict

http://aipython.org Version 0.7.5 June 19, 2018

72 4. Reasoning with Constraints

Exercise 4.12 Modify this to plot the runtime, instead of the number of steps.
To measure runtime use timeit (https://docs.python.org/3.5/library/timeit.
html). Small runtimes are inaccurate, so timeit can run the same code multi-
ple times. Stochastic local algorithms give different runtimes each time called.
To make the timing meaningful, you need to make sure the random seed is the
same for each repeated call (see random.getstate and random.setstate in https:
//docs.python.org/3.5/library/random.html). Because the runtime for different
seeds can vary a great deal, for each seed, you should start with 1 iteration and
multiplying it by, say 10, until the time is greater than 0.2 seconds. Make sure you
plot the average time for each run. Before you start, try to estimate the total run-
time, so you will be able to tell if there is a problem with the algorithm stopping.

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 5

Propositions and Inference

5.1 Representing Knowledge Bases

A clause consists of a head (an atom) and a body. A body is represented as a
list of atoms. Atoms are represented as strings.
logicProblem.py — Representations Logics
11 class Clause(object):
12 """A definite clause"""
13
14 def __init__(self,head,body=[]):
15 """clause with atom head and lost of atoms body"""
16 self.head=head
17 self.body = body
18
19 def __str__(self):
20 """returns the string representation of a clause.
21 """
22 if self.body:
23 return self.head + " <- " + " & ".join(self.body) + "."
24 else:
25 return self.head + "."

An askable atom can be asked of the user. The user can respond in English or
French or just with a “y”.
logicProblem.py — (continued)

27 class Askable(object):
28 """An askable atom"""
29
30 def __init__(self,atom):
31 """clause with atom head and lost of atoms body"""

73
 74 5. Propositions and Inference

32 self.atom=atom
33
34 def __str__(self):
35 """returns the string representation of a clause."""
36 return "askable " + self.atom + "."
37
38 def yes(ans):
39 """returns true if the answer is yes in some form"""
40 return ans.lower() in ['yes', 'yes.', 'oui', 'oui.', 'y', 'y.'] # bilingual
A knowledge base is a list of clauses and askables. In order to make top-down
inference faster, this creates a dictionary that maps each atoms into the set of
clauses with that atom in the head.
logicProblem.py — (continued)

42 from utilities import Displayable

43
44 class KB(Displayable):
45 """A knowledge base consists of a set of clauses.
46 This also creates a dictionary to give fast access to the clauses with an atom in head.
47 """
48 def __init__(self, statements=[]):
49 self.statements = statements
50 self.clauses = [c for c in statements if isinstance(c, Clause)]
51 self.askables = [c.atom for c in statements if isinstance(c, Askable)]
52 self.atom_to_clauses = {} # dictionary giving clauses with atom as head
53 for c in self.clauses:
54 if c.head in self.atom_to_clauses:
55 self.atom_to_clauses[c.head].add(c)
56 else:
57 self.atom_to_clauses[c.head] = {c}
58
59 def clauses_for_atom(self,a):
60 """returns set of clauses with atom a as the head"""
61 if a in self.atom_to_clauses:
62 return self.atom_to_clauses[a]
63 else:
64 return set()
65
66 def __str__(self):
67 """returns a string representation of this knowledge base.
68 """
69 return '\n'.join([str(c) for c in self.statements])
Here is a trivial example (I think therefore I am) using in the unit tests:
logicProblem.py — (continued)

71 triv_KB = KB([
72 Clause('i_am', ['i_think']),
73 Clause('i_think'),
74 Clause('i_smell', ['i_exist'])
75 ])

http://aipython.org Version 0.7.5 June 19, 2018

 5.2. Bottom-up Proofs 75

Here is a representation of the electrical domain of the textbook:

logicProblem.py — (continued)

77 elect = KB([
78 Clause('light_l1'),
79 Clause('light_l2'),
80 Clause('ok_l1'),
81 Clause('ok_l2'),
82 Clause('ok_cb1'),
83 Clause('ok_cb2'),
84 Clause('live_outside'),
85 Clause('live_l1', ['live_w0']),
86 Clause('live_w0', ['up_s2','live_w1']),
87 Clause('live_w0', ['down_s2','live_w2']),
88 Clause('live_w1', ['up_s1', 'live_w3']),
89 Clause('live_w2', ['down_s1','live_w3' ]),
90 Clause('live_l2', ['live_w4']),
91 Clause('live_w4', ['up_s3','live_w3' ]),
92 Clause('live_p_1', ['live_w3']),
93 Clause('live_w3', ['live_w5', 'ok_cb1']),
94 Clause('live_p_2', ['live_w6']),
95 Clause('live_w6', ['live_w5', 'ok_cb2']),
96 Clause('live_w5', ['live_outside']),
97 Clause('lit_l1', ['light_l1', 'live_l1', 'ok_l1']),
98 Clause('lit_l2', ['light_l2', 'live_l2', 'ok_l2']),
99 Askable('up_s1'),
100 Askable('down_s1'),
101 Askable('up_s2'),
102 Askable('down_s2'),
103 Askable('up_s3'),
104 Askable('down_s2')
105 ])
106
107 # print(kb)

5.2 Bottom-up Proofs

fixed point computes the fixed point of the knowledge base kb.
logicBottomUp.py — Bottom-up Proof Procedure for Definite Clauses
11 from logicProblem import yes
12
13 def fixed_point(kb):
14 """Returns the fixed point of knowledge base kb.
15 """
16 fp = ask_askables(kb)
17 added = True
18 while added:
19 added = False # added is true when an atom was added to fp this iteration

http://aipython.org Version 0.7.5 June 19, 2018

 76 5. Propositions and Inference

20 for c in kb.clauses:
21 if c.head not in fp and all(b in fp for b in c.body):
22 fp.add(c.head)
23 added = True
24 kb.display(2,c.head,"added to fp due to clause",c)
25 return fp
26
27 def ask_askables(kb):
28 return {at for at in kb.askables if yes(input("Is "+at+" true? "))}

Testing:

logicBottomUp.py — (continued)

30 from logicProblem import triv_KB

31 def test():
32 fp = fixed_point(triv_KB)
33 assert fp == {'i_am','i_think'}, "triv_KB gave result "+str(fp)
34 print("Passed unit test")
35 if __name__ == "__main__":
36 test()
37
38 from logicProblem import elect
39 # elect.max_display_level=3 # give detailed trace
40 # fixed_point(elect)

Exercise 5.1 It is not very user-friendly to ask all of the askables up-front. Imple-
ment ask-the-user so that questions are only asked if useful, and are not re-asked.
For example, if there is a clause h ← a ∧ b ∧ c ∧ d ∧ e, where c and e are askable,
c and e only need to be asked if a, b, d are all in fp and they have not been asked
before. Askable e only needs to be asked if the user says “yes” to c. Askable c
doesn’t need to be asked if the user previously replied “no” to e.
This form of ask-the-user can ask a different set of questions than the top-
down interpreter that asks questions when encountered. Give an example where
they ask different questions (neither set of questions asked is a subset of the other).
Exercise 5.2 This algorithm runs in time O(n2 ), where n is the number of clauses,
for a bounded number of elements in the body; each iteration goes through each
of the clauses, and in the worst case, it will do an iteration for each clause. It is
possible to implement this in time O(n) time by creating an index that maps an
atom to the set of clauses with that atom in the body. Implement this. What is its
complexity as a function of n and b, the maximum number of atoms in the body of
a clause?
Exercise 5.3 It is possible to be asymptitocally more efficient (in terms of b) than
the method in the previous question by noticing that each element of the body of
clause only needs to be checked once. For example, the clause a ← b ∧ c ∧ d, needs
only be considered when b is added to fp. Once b is added to fp, if c is already in
pf , we know that a can be added as soon as d is added. Implement this. What is its
complexity as a function of n and b, the maximum number of atoms in the body of
a clause?

http://aipython.org Version 0.7.5 June 19, 2018

 5.3. Top-down Proofs 77

5.3 Top-down Proofs

prove(kb, goal) is used to prove goal from a knowledge base, kb, where a goal is
a list of atoms. It returns True if kb ` goal. The indent is used when tracing the
code (and doesn’t need to have a non-default value).

logicTopDown.py — Top-down Proof Procedure for Definite Clauses

11 from logicProblem import yes
12
13 def prove(kb, ans_body, indent=""):
14 """returns True if kb |- ans_body
15 ans_body is a list of atoms to be proved
16 """
17 kb.display(2,indent,'yes <-',' & '.join(ans_body))
18 if ans_body:
19 selected = ans_body[0] # select first atom from ans_body
20 if selected in kb.askables:
21 return (yes(input("Is "+selected+" true? "))
22 and prove(kb,ans_body[1:],indent+" "))
23 else:
24 return any(prove(kb,cl.body+ans_body[1:],indent+" ")
25 for cl in kb.clauses_for_atom(selected))
26 else:
27 return True # empty body is true

Testing:

logicTopDown.py — (continued)

29 from logicProblem import triv_KB

30 def test():
31 a1 = prove(triv_KB,['i_am'])
32 assert a1, "triv_KB proving i_am gave "+str(a1)
33 a2 = prove(triv_KB,['i_smell'])
34 assert not a2, "triv_KB proving i_smell gave "+str(a2it)
35 print("Passed unit tests")
36 if __name__ == "__main__":
37 test()
38 # try
39 from logicProblem import elect
40 # elect.max_display_level=3 # give detailed trace
41 # prove(elect,['live_w6'])
42 # prove(elect,['lit_l1'])

Exercise 5.4 This code can re-ask a question multiple times. Implement this code
so that it only asks a question once and remembers the answer. Also implement a
function to forget the answers.
Exercise 5.5 What search method is this using? Implement the search interface
so that it can use A∗ or other searching methods. Define an admissible heuristic
that is not always 0.

http://aipython.org Version 0.7.5 June 19, 2018

 78 5. Propositions and Inference

5.4 Assumables
Atom a can be made assumable by including Assumable(a) in the knowledge
base. A knowledge base that can include assumables is declared with KBA.
logicAssumables.py — Definite clauses with assumables
11 from logicProblem import Clause, Askable, KB, yes
12
13 class Assumable(object):
14 """An askable atom"""
15
16 def __init__(self,atom):
17 """clause with atom head and lost of atoms body"""
18 self.atom = atom
19
20 def __str__(self):
21 """returns the string representation of a clause.
22 """
23 return "assumable " + self.atom + "."
24
25 class KBA(KB):
26 """A knowledge base that can include assumables"""
27 def __init__(self,statements):
28 self.assumables = [c.atom for c in statements if isinstance(c, Assumable)]
29 KB.__init__(self,statements)
The top-down Horn clause interpreter, prove all ass returns a list of the sets of
assumables that imply ans body. This list will contain all of the minimal sets
of assumables, but can also find non-minimal sets, and repeated sets, if they
can be generated with separate proofs. The set assumed is the set of assumables
already assumed.
logicAssumables.py — (continued)

31 def prove_all_ass(self, ans_body, assumed=set()):

32 """returns a list of sets of assumables that extends assumed
33 to imply ans_body from self.
34 ans_body is a list of atoms (it is the body of the answer clause).
35 assumed is a set of assumables already assumed
36 """
37 if ans_body:
38 selected = ans_body[0] # select first atom from ans_body
39 if selected in self.askables:
40 if yes(input("Is "+selected+" true? ")):
41 return self.prove_all_ass(ans_body[1:],assumed)
42 else:
43 return [] # no answers
44 elif selected in self.assumables:
45 return self.prove_all_ass(ans_body[1:],assumed|{selected})
46 else:
47 return [ass
48 for cl in self.clauses_for_atom(selected)

http://aipython.org Version 0.7.5 June 19, 2018

 5.4. Assumables 79

49 for ass in self.prove_all_ass(cl.body+ans_body[1:],assumed)

50 ] # union of answers for each clause with head=selected
51 else: # empty body
52 return [assumed] # one answer
53
54 def conflicts(self):
55 """returns a list of minimal conflicts"""
56 return minsets(self.prove_all_ass(['false']))

Given a list of sets, minsets returns a list of the minimal sets in the list. For
example, minsets([{2, 3, 4}, {2, 3}, {6, 2, 3}, {2, 3}, {2, 4, 5}]) returns [{2, 3}, {2, 4, 5}].
logicAssumables.py — (continued)

58 def minsets(ls):
59 """ls is a list of sets
60 returns a list of minimal sets in ls
61 """
62 ans = [] # elements known to be minimal
63 for c in ls:
64 if not any(c1<c for c1 in ls) and not any(c1 <= c for c1 in ans):
65 ans.append(c)
66 return ans
67
68 # minsets([{2, 3, 4}, {2, 3}, {6, 2, 3}, {2, 3}, {2, 4, 5}])

Warning: minsets works for a list of sets or for a set of (frozen) sets, but it does
not work for a generator of sets. For example, try to predict and then test:
minsets(e for e in [{2, 3, 4}, {2, 3}, {6, 2, 3}, {2, 3}, {2, 4, 5}])
The diagnoses can be constructed from the (minimal) conflicts as follows.
This also works if there are non-minimal conflicts, but is not as efficient.
logicAssumables.py — (continued)

69 def diagnoses(cons):
70 """cons is a list of (minimal) conflicts.
71 returns a list of diagnoses."""
72 if cons == []:
73 return [set()]
74 else:
75 return minsets([({e}|d) # | is set union
76 for e in cons[0]
77 for d in diagnoses(cons[1:])])

Test cases:
logicAssumables.py — (continued)

80 electa = KBA([
81 Clause('light_l1'),
82 Clause('light_l2'),
83 Assumable('ok_l1'),
84 Assumable('ok_l2'),

http://aipython.org Version 0.7.5 June 19, 2018

 80 5. Propositions and Inference

85 Assumable('ok_s1'),
86 Assumable('ok_s2'),
87 Assumable('ok_s3'),
88 Assumable('ok_cb1'),
89 Assumable('ok_cb2'),
90 Assumable('live_outside'),
91 Clause('live_l1', ['live_w0']),
92 Clause('live_w0', ['up_s2','ok_s2','live_w1']),
93 Clause('live_w0', ['down_s2','ok_s2','live_w2']),
94 Clause('live_w1', ['up_s1', 'ok_s1', 'live_w3']),
95 Clause('live_w2', ['down_s1', 'ok_s1','live_w3' ]),
96 Clause('live_l2', ['live_w4']),
97 Clause('live_w4', ['up_s3','ok_s3','live_w3' ]),
98 Clause('live_p_1', ['live_w3']),
99 Clause('live_w3', ['live_w5', 'ok_cb1']),
100 Clause('live_p_2', ['live_w6']),
101 Clause('live_w6', ['live_w5', 'ok_cb2']),
102 Clause('live_w5', ['live_outside']),
103 Clause('lit_l1', ['light_l1', 'live_l1', 'ok_l1']),
104 Clause('lit_l2', ['light_l2', 'live_l2', 'ok_l2']),
105 Askable('up_s1'),
106 Askable('down_s1'),
107 Askable('up_s2'),
108 Askable('down_s2'),
109 Askable('up_s3'),
110 Askable('down_s2'),
111 Askable('dark_l1'),
112 Askable('dark_l2'),
113 Clause('false', ['dark_l1', 'lit_l1']),
114 Clause('false', ['dark_l2', 'lit_l2'])
115 ])
116 # electa.prove_all_ass(['false'])
117 # cs=electa.conflicts()
118 # print(cs)
119 # diagnoses(cs) # diagnoses from conflicts

Exercise 5.6 To implement a version of conflicts that never generates non-minimal

conflicts, modify prove all ass to implement iterative deepening on the number of
assumables used in a proof, and prune any set of assumables that is a superset of
a conflict.
Exercise 5.7 Implement explanations(self , body), where body is a list of atoms, that
returns the a list of the minimal explanations of the body. This does not require
modification of prove all ass.
Exercise 5.8 Implement explanations, as in the previous question, so that it never
generates non-minimal explanations. Hint: modify prove all ass to implement iter-
ative deepening on the number of assumptions, generating conflicts and explana-
tions together, and pruning as early as possible.

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 6

Planning with Certainty

6.1 Representing Actions and Planning Prob-

lems
The STRIPS representation of an action consists of:
• preconditions: a dictionary of feature:value pairs that specifies that the
feature must have this value for the action to be possible.

• effects: a dictionary of feature:value pairs that are made true by this action.
In particular, a feature in the dictionary has the corresponding value (and
not its previous value) after the action, and a feature not in the dictionary
keeps its old value.

stripsProblem.py — STRIPS Representations of Actions

11 class Strips(object):
12 def __init__(self, preconditions, effects, cost=1):
13 """
14 defines the STRIPS represtation for an action:
15 * preconditions is feature:value dictionary that must hold
16 for the action to be carried out
17 * effects is a feature:value map that this action makes
18 true. The action changes the value of any feature specified
19 here, and leaves other properties unchanged.
20 * cost is the cost of the action
21 """
22 self.preconditions = preconditions
23 self.effects = effects
24 self.cost = cost
A STRIPS domain consists of:

81
 82 6. Planning with Certainty

• A set of actions.

• A dictionary that maps each feature into a set of possible values for the
feature.

• A dictionary that maps each action into a STRIPS representation of the

action.

stripsProblem.py — (continued)

26 class STRIPS_domain(object):
27 def __init__(self, feats_vals, strips_map):
28 """Problem domain
29 feats_vals is a feature:domain dictionary,
30 mapping each feature to its domain
31 strips_map is an action:strips dictionary,
32 mapping each action to its Strips representation
33 """
34 self.actions = set(strips_map) # set of all actions
35 self.feats_vals = feats_vals
36 self.strips_map = strips_map

6.1.1 Robot Delivery Domain

The following specifies the robot delivery domain of Chapter 8.
stripsProblem.py — (continued)

38 boolean = {True, False}

39 delivery_domain = STRIPS_domain(
40 {'RLoc':{'cs', 'off', 'lab', 'mr'}, 'RHC':boolean, 'SWC':boolean,
41 'MW':boolean, 'RHM':boolean}, #feaures:values dictionary
42 {'mc_cs': Strips({'RLoc':'cs'}, {'RLoc':'off'}),
43 'mc_off': Strips({'RLoc':'off'}, {'RLoc':'lab'}),
44 'mc_lab': Strips({'RLoc':'lab'}, {'RLoc':'mr'}),
45 'mc_mr': Strips({'RLoc':'mr'}, {'RLoc':'cs'}),
46 'mcc_cs': Strips({'RLoc':'cs'}, {'RLoc':'mr'}),
47 'mcc_off': Strips({'RLoc':'off'}, {'RLoc':'cs'}),
48 'mcc_lab': Strips({'RLoc':'lab'}, {'RLoc':'off'}),
49 'mcc_mr': Strips({'RLoc':'mr'}, {'RLoc':'lab'}),
50 'puc': Strips({'RLoc':'cs', 'RHC':False}, {'RHC':True}),
51 'dc': Strips({'RLoc':'off', 'RHC':True}, {'RHC':False, 'SWC':False}),
52 'pum': Strips({'RLoc':'mr','MW':True}, {'RHM':True,'MW':False}),
53 'dm': Strips({'RLoc':'off', 'RHM':True}, {'RHM':False})
54 } )

A planning problem consists of a planning domain, an initial state, and a

goal. The goal does not need to fully specify the final state.
stripsProblem.py — (continued)

56 class Planning_problem(object):

http://aipython.org Version 0.7.5 June 19, 2018

 6.1. Representing Actions and Planning Problems 83

b move(b,c,a) b
a c a c

move(b,c,table)

a c b

Figure 6.1: Blocks world with two actions

57 def __init__(self, prob_domain, initial_state, goal):

58 """
59 a planning problem consists of
60 * a planning domain
61 * the initial state
62 * a goal
63 """
64 self.prob_domain = prob_domain
65 self.initial_state = initial_state
66 self.goal = goal
67
68 problem0 = Planning_problem(delivery_domain,
69 {'RLoc':'lab', 'MW':True, 'SWC':True, 'RHC':False,
70 'RHM':False},
71 {'RLoc':'off'})
72 problem1 = Planning_problem(delivery_domain,
73 {'RLoc':'lab', 'MW':True, 'SWC':True, 'RHC':False,
74 'RHM':False},
75 {'SWC':False})
76 problem2 = Planning_problem(delivery_domain,
77 {'RLoc':'lab', 'MW':True, 'SWC':True, 'RHC':False,
78 'RHM':False},
79 {'SWC':False, 'MW':False, 'RHM':False})

6.1.2 Blocks World

The blocks world consist of blocks and a table. Each block can be on the table
or on another block. A block can only have one other block on top of it. Fig-
ure 6.1 shows 3 states with some of the actions between them. The following

http://aipython.org Version 0.7.5 June 19, 2018

 84 6. Planning with Certainty

represents the blocks world. Note that the actions and the conditions are all
strings.
stripsProblem.py — (continued)

81 ### blocks world

82 def move(x,y,z):
83 """string for the 'move' action"""
84 return 'move_'+x+'_from_'+y+'_to_'+z
85 def on(x,y):
86 """string for the 'on' feature"""
87 return x+'_on_'+y
88 def clear(x):
89 """string for the 'clear' feature"""
90 return 'clear_'+x
91 def create_blocks_world(blocks = ['a','b','c','d']):
92 blocks_and_table = blocks+['table']
93 stmap = {move(x,y,z):Strips({on(x,y):True, clear(x):True, clear(z):True},
94 {on(x,z):True, on(x,y):False, clear(y):True, clear(z):False})
95 for x in blocks
96 for y in blocks_and_table
97 for z in blocks
98 if x!=y and y!=z and z!=x}
99 stmap.update({move(x,y,'table'):Strips({on(x,y):True, clear(x):True},
100 {on(x,'table'):True, on(x,y):False, clear(y):True})
101 for x in blocks
102 for y in blocks
103 if x!=y})
104 feats_vals = {on(x,y):boolean for x in blocks for y in blocks_and_table}
105 feats_vals.update({clear(x):boolean for x in blocks_and_table})
106 return STRIPS_domain(feats_vals, stmap)
This is a classic example, with 3 blocks, and the goal consists of two conditions.
stripsProblem.py — (continued)

108 blocks1dom = create_blocks_world(['a','b','c'])

109 blocks1 = Planning_problem(blocks1dom,
110 {on('a','table'):True,clear('a'):True, clear('b'):True,on('b','c'):True,
111 on('c','table'):True,clear('c'):False}, # initial state
112 {on('a','b'):True, on('c','a'):True}) #goal
This is a problem of inverting a tower of size 4.
stripsProblem.py — (continued)

114 blocks2dom = create_blocks_world(['a','b','c','d'])

115 tower4 = {clear('a'):True, on('a','b'):True, clear('b'):False,
116 on('b','c'):True, clear('c'):False, on('c','d'):True,
117 clear('b'):False, on('d','table'):True}
118 blocks2 = Planning_problem(blocks2dom,
119 tower4, # initial state
120 {on('d','c'):True,on('c','b'):True,on('b','a'):True}) #goal
Moving bottom block to top of a tower of size 4.

http://aipython.org Version 0.7.5 June 19, 2018

 6.2. Forward Planning 85

stripsProblem.py — (continued)

122 blocks3 = Planning_problem(blocks2dom,

123 tower4, # initial state
124 {on('d','a'):True, on('a','b'):True, on('b','c'):True}) #goal

Exercise 6.1 Represent the problem of given a tower of 4 blocks (a on b on c on

d on table), the goal is to have a tower with the previous top block on the bottom
(b on c on d on a). Do not include the table in your goal (the goal does not care
whether a is on the table). [Before you run the program, estimate how many steps
it will take to solve this.] How many steps does an optimal planner take?
Exercise 6.2 The representation of the state does not include negative on facts.
Does it need to? Why or why not? (Note that this may depend on the planner;
write your answer with respect to particular planners.)
Exercise 6.3 It is possible to write the representation of the problem without
using clear, where clear(x) means nothing is on x. Change the definition of the
blocks world so that it does not use clear but uses on being false instead. Does this
work better for any of the planners? (Does this change an answer to the previous
question?)

6.2 Forward Planning

To run the demo, in folder ”aipython”, load
”stripsForwardPlanner.py”, and copy and paste the commented-
out example queries at the bottom of that file.

In a forward planner, a node is a state. A state consists of an assignment,

which is a variable:value dictionary. In order to be able to do multiple-path
pruning, we need to define a hash function, and equality between states.

stripsForwardPlanner.py — Forward Planner with STRIPS actions

11 from searchProblem import Arc, Search_problem
12 from stripsProblem import Strips, STRIPS_domain
13
14 class State(object):
15 def __init__(self,assignment):
16 self.assignment = assignment
17 self.hash_value = None
18 def __hash__(self):
19 if self.hash_value is None:
20 self.hash_value = hash(frozenset(self.assignment.items()))
21 return self.hash_value
22 def __eq__(self,st):
23 return self.assignment == st.assignment
24 def __str__(self):
25 return str(self.assignment)

http://aipython.org Version 0.7.5 June 19, 2018

 86 6. Planning with Certainty

In order to define a search problem (page 31), we need to define the goal
condition, the start nodes, the neighbours, and (optionally) a heuristic function.
Here zero is the default heuristic function.

stripsForwardPlanner.py — (continued)

27 def zero(*args,**nargs):
28 """always returns 0"""
29 return 0
30
31 class Forward_STRIPS(Search_problem):
32 """A search problem from a planning problem where:
33 * a node is a state object.
34 * the dynamics are specified by the STRIPS representation of actions
35 """
36 def __init__(self, planning_problem, heur=zero):
37 """creates a forward seach space from a planning problem.
38 heur(state,goal) is a heuristic function,
39 an underestimate of the cost from state to goal, where
40 both state and goals are feature:value dictionaries.
41 """
42 self.prob_domain = planning_problem.prob_domain
43 self.initial_state = State(planning_problem.initial_state)
44 self.goal = planning_problem.goal
45 self.heur = heur
46
47 def is_goal(self, state):
48 """is True if node is a goal.
49
50 Every goal feature has the same value in the state and the goal."""
51 state_asst = state.assignment
52 return all(prop in state_asst and state_asst[prop]==self.goal[prop]
53 for prop in self.goal)
54
55 def start_node(self):
56 """returns start node"""
57 return self.initial_state
58
59 def neighbors(self,state):
60 """returns neighbors of state in this problem"""
61 cost=1
62 state_asst = state.assignment
63 return [ Arc(state,self.effect(act,state_asst),cost,act)
64 for act in self.prob_domain.actions
65 if self.possible(act,state_asst)]
66
67 def possible(self,act,state_asst):
68 """True if act is possible in state.
69 act is possible if all of its preconditions have the same value in the state"""
70 preconds = self.prob_domain.strips_map[act].preconditions
71 return all(pre in state_asst and state_asst[pre]==preconds[pre]

http://aipython.org Version 0.7.5 June 19, 2018

 6.2. Forward Planning 87

72 for pre in preconds)

73
74 def effect(self,act,state_asst):
75 """returns the state that is the effect of doing act given state_asst"""
76 new_state_asst = self.prob_domain.strips_map[act].effects.copy()
77 for prop in state_asst:
78 if prop not in new_state_asst:
79 new_state_asst[prop]=state_asst[prop]
80 return State(new_state_asst)
81
82 def heuristic(self,state):
83 """in the forward planner a node is a state.
84 the heuristic is an (under)estimate of the cost
85 of going from the state to the top-level goal.
86 """
87 return self.heur(state.assignment, self.goal)

Here are some test cases to try.

stripsForwardPlanner.py — (continued)

89 from searchBranchAndBound import DF_branch_and_bound

90 from searchGeneric import AStarSearcher
91 from searchMPP import SearcherMPP
92 from stripsProblem import problem0, problem1, problem2, blocks1, blocks2, blocks3
93
94 # AStarSearcher(Forward_STRIPS(problem1)).search() #A*
95 # SearcherMPP(Forward_STRIPS(problem1)).search() #A* with MPP
96 # DF_branch_and_bound(Forward_STRIPS(problem1),10).search() #B&B
97 # To find more than one plan:
98 # s1 = SearcherMPP(Forward_STRIPS(problem1)) #A*
99 # s1.search() #find another plan

6.2.1 Defining Heuristics for a Planner

Each planning domain requires its own heuristics. If you change the actions,
you will need to reconsider the heuristic function, as there might then be a
lower-cost path, which might make the heuristic non-admissible.
Here is an example of defining a (not very good) heuristic for the coffee
delivery planning domain.
First we define the distance between two locations, which is used for the
heuristics.

stripsHeuristic.py — Planner with Heursitic Function

11 def dist(loc1, loc2):
12 """returns the distance from location loc1 to loc2
13 """
14 if loc1==loc2:
15 return 0
16 if {loc1,loc2} in [{'cs','lab'},{'mr','off'}]:

http://aipython.org Version 0.7.5 June 19, 2018

 88 6. Planning with Certainty

17 return 2
18 else:
19 return 1
Note that the current state is a complete description; there is a value for
every feature. However the goal need not be complete; it does not need to
define a value for every feature. Before checking the value for a feature in the
goal, a heuristic needs to define whether the feature is defined in the goal.
stripsHeuristic.py — (continued)

21 def h1(state,goal):
22 """ the distance to the goal location, if there is one"""
23 if 'RLoc' in goal:
24 return dist(state['RLoc'], goal['RLoc'])
25 else:
26 return 0
27
28 def h2(state,goal):
29 """ the distance to the coffee shop plus getting coffee and delivering it
30 if the robot needs to get coffee
31 """
32 if ('SWC' in goal and goal['SWC']==False
33 and state['SWC']==True
34 and state['RHC']==False):
35 return dist(state['RLoc'],'cs')+3
36 else:
37 return 0
The maximum of the values of a set of admissible heuristics is also an admis-
sible heuristic. The function maxh takes a number of heuristic functions as ar-
guments, and returns a new heuristic function that takes the maximum of the
values of the heuristics. For example, h1 and h2 are heuristic functions and so
maxh(h1,h2) is also. maxh can take an arbitrary number of arguments.
stripsHeuristic.py — (continued)

39 def maxh(*heuristics):
40 """Returns a new heuristic function that is the maximum of the functions in heuristics.
41 heuristics is the list of arguments which must be heuristic functions.
42 """
43 return lambda state,goal: max(h(state,goal) for h in heuristics)
The following runs the example with and without the heuristic. (Also try using
AStarSearcher instead of SearcherMPP.)
stripsHeuristic.py — (continued)

45 ##### Forward Planner #####

46 from searchGeneric import AStarSearcher
47 from searchMPP import SearcherMPP
48 from stripsForwardPlanner import Forward_STRIPS
49 from stripsProblem import problem0, problem1, problem2
50

http://aipython.org Version 0.7.5 June 19, 2018

 6.3. Regression Planning 89

51 def test_forward_heuristic(thisproblem=problem1):
52 print("\n***** FORWARD NO HEURISTIC")
53 print(SearcherMPP(Forward_STRIPS(thisproblem)).search())
54
55 print("\n***** FORWARD WITH HEURISTIC h1")
56 print(SearcherMPP(Forward_STRIPS(thisproblem,h1)).search())
57
58 print("\n***** FORWARD WITH HEURISTICs h1 and h2")
59 print(SearcherMPP(Forward_STRIPS(thisproblem,maxh(h1,h2))).search())
60
61 if __name__ == "__main__":
62 test_forward_heuristic()

Exercise 6.4 Try the forward planner with a heuristic function of just h1, with
just h2 and with both. Explain how each one prunes or doesn’t prune the search
space.
Exercise 6.5 Create a better heuristic than maxh(h1, h2). Try it for a number of
different problems.
Exercise 6.6 Create an admissible heuristic for the blocks world.

6.3 Regression Planning

To run the demo, in folder ”aipython”, load
”stripsRegressionPlanner.py”, and copy and paste the commented-
out example queries at the bottom of that file.

In a regression planner a node is a subgoal that need to be achieved.

A Subgoal object consists of an assignment, which is variable:value dictionary.
We make it hashable so that multiple path pruning can work. The hash is only
computed when necessary (and only once).

stripsRegressionPlanner.py — Regression Planner with STRIPS actions

11 from searchProblem import Arc, Search_problem
12
13 class Subgoal(object):
14 def __init__(self,assignment):
15 self.assignment = assignment
16 self.hash_value = None
17 def __hash__(self):
18 if self.hash_value is None:
19 self.hash_value = hash(frozenset(self.assignment.items()))
20 return self.hash_value
21 def __eq__(self,st):
22 return self.assignment == st.assignment
23 def __str__(self):
24 return str(self.assignment)

http://aipython.org Version 0.7.5 June 19, 2018

 90 6. Planning with Certainty

A regression search has subgoals as nodes. The initial node is the top-level goal
of the planner. The goal for the search (when the search can stop) is a subgoal
that holds in the initial state.

stripsRegressionPlanner.py — (continued)

26 from stripsForwardPlanner import zero

27
28 class Regression_STRIPS(Search_problem):
29 """A search problem where:
30 * a node is a goal to be achieved, represented by a set of propositions.
31 * the dynamics are specified by the STRIPS representation of actions
32 """
33
34 def __init__(self, planning_problem, heur=zero):
35 """creates a regression seach space from a planning problem.
36 heur(state,goal) is a heuristic function;
37 an underestimate of the cost from state to goal, where
38 both state and goals are feature:value dictionaries
39 """
40 self.prob_domain = planning_problem.prob_domain
41 self.top_goal = Subgoal(planning_problem.goal)
42 self.initial_state = planning_problem.initial_state
43 self.heur = heur
44
45 def is_goal(self, subgoal):
46 """if subgoal is true in the initial state, a path has been found"""
47 goal_asst = subgoal.assignment
48 return all((g in self.initial_state) and (self.initial_state[g]==goal_asst[g])
49 for g in goal_asst)
50
51 def start_node(self):
52 """the start node is the top-level goal"""
53 return self.top_goal
54
55 def neighbors(self,subgoal):
56 """returns a list of the arcs for the neighbors of subgoal in this problem"""
57 cost = 1
58 goal_asst = subgoal.assignment
59 return [ Arc(subgoal,self.weakest_precond(act,goal_asst),cost,act)
60 for act in self.prob_domain.actions
61 if self.possible(act,goal_asst)]
62
63 def possible(self,act,goal_asst):
64 """True if act is possible to achieve goal_asst.
65
66 the action achieves an element of the effects and
67 the action doesn't delete something that needs to be achieved and
68 the precoditions are consistent with other subgoals that need to be achieved
69 """
70 effects = self.prob_domain.strips_map[act].effects

http://aipython.org Version 0.7.5 June 19, 2018

 6.3. Regression Planning 91

71 preconds = self.prob_domain.strips_map[act].preconditions
72 return ( any(goal_asst[prop]==effects[prop]
73 for prop in effects if prop in goal_asst)
74 and all(goal_asst[prop]==effects[prop]
75 for prop in effects if prop in goal_asst)
76 and all(goal_asst[prop]==preconds[prop]
77 for prop in preconds if prop not in effects and prop in goal_asst)
78 )
79
80 def weakest_precond(self,act,goal_asst):
81 """returns the subgoal that must be true so goal_asst holds after act"""
82 new_asst = self.prob_domain.strips_map[act].preconditions.copy()
83 for g in goal_asst:
84 if g not in self.prob_domain.strips_map[act].effects:
85 new_asst[g] = goal_asst[g]
86 return Subgoal(new_asst)
87
88 def heuristic(self,subgoal):
89 """in the regression planner a node is a subgoal.
90 the heuristic is an (under)estimate of the cost of going from the initial state to subgoal.
91 """
92 return self.heur(self.initial_state, subgoal.assignment)

stripsRegressionPlanner.py — (continued)

94 from searchBranchAndBound import DF_branch_and_bound

95 from searchGeneric import AStarSearcher
96 from searchMPP import SearcherMPP
97 from stripsProblem import problem0, problem1, problem2
98
99 # AStarSearcher(Regression_STRIPS(problem1)).search() #A*
100 # SearcherMPP(Regression_STRIPS(problem1)).search() #A* with MPP
101 # DF_branch_and_bound(Regression_STRIPS(problem1),10).search() #B&B

Exercise 6.7 Multiple path pruning could be used to prune more than the current
code. In particular, if the current node contains more conditions than a previously
visited node, it can be pruned. For example, if {a : True, b : False} has been visited,
then any node that is a superset, e.g., {a : True, b : False, d : True}, need not be
expanded. If the simpler subgoal does not lead to a solution, the more complicated
one wont either. Implement this more severe pruning. (Hint: This may require
modifications to the searcher.)
Exercise 6.8 It is possible that, as knowledge of the domain, that some as-
signment of values to variables can never be achieved. For example, the robot
cannot be holding mail when there is mail waiting (assuming it isn’t holding
mail initially). An assignment of values to (some of the) variables is incompat-
ible if no possible (reachable) state can include that assignment. For example,
{0 MW 0 : True,0 RHM0 : True} is an incompatible assignment. This information may
be useful information for a planner; there is no point in trying to achieve these
together. Define a subclass of STRIPS domain that can accept a list of incompatible

http://aipython.org Version 0.7.5 June 19, 2018

 92 6. Planning with Certainty

assignments. Modify the regression planner code to use such a list of incompatible
assignments. Give an example where the search space is smaller.
Exercise 6.9 After completing the previous exercise, design incompatible assign-
ments for the blocks world. (This should result in dramatic search improvements.)

6.3.1 Defining Heuristics for a Regression Planner

The regression planner can use the same heuristic function as the forward plan-
ner. However, just because a heuristic is useful for a forward planner does
not mean it is useful for a regression planner, and vice versa. you should ex-
periment with whether the same heuristic works well for both a a regression
planner and a forward planner.
The following runs the same example as the forward planner with and
without the heuristic defined for the forward planner:
stripsHeuristic.py — (continued)

64 ##### Regression Planner

65 from stripsRegressionPlanner import Regression_STRIPS
66
67 def test_regression_heuristic(thisproblem=problem1):
68 print("\n***** REGRESSION NO HEURISTIC")
69 print(SearcherMPP(Regression_STRIPS(thisproblem)).search())
70
71 print("\n***** REGRESSION WITH HEURISTICs h1 and h2")
72 print(SearcherMPP(Regression_STRIPS(thisproblem,maxh(h1,h2))).search())
73
74 if __name__ == "__main__":
75 test_regression_heuristic()

Exercise 6.10 Try the regression planner with a heuristic function of just h1 and
with just h2 (defined in Section 6.2.1). Explain how each one prunes or doesn’t
prune the search space.
Exercise 6.11 Create a better heuristic than heuristic fun defined in Section 6.2.1.

6.4 Planning as a CSP

To run the demo, in folder ”aipython”, load ”stripsCSPPlanner.py”,
and copy and paste the commented-out example queries at the bottom
of that file. This assumes Python 3.

Here we implement the CSP planner assuming there is a single action at

each step. This creates a CSP that can use any of the CSP algorithms to solve
(e.g., stochastic local search or arc consistency with domain splitting).
This assumes the same action representation as before; we do not consider
factored actions (action features), nor do we implement state constraints.

http://aipython.org Version 0.7.5 June 19, 2018

 6.4. Planning as a CSP 93

stripsCSPPlanner.py — CSP planner where actions are represented using STRIPS

11 from cspProblem import CSP, Constraint
12
13 class CSP_from_STRIPS(CSP):
14 """A CSP where:
15 * a CSP variable is constructed by st(var,stage).
16 * the dynamics are specified by the STRIPS representation of actions
17 """
18
19 def __init__(self, planning_problem, number_stages=2):
20 prob_domain = planning_problem.prob_domain
21 initial_state = planning_problem.initial_state
22 goal = planning_problem.goal
23 self.act_vars = [st('action',stage) for stage in range(number_stages)]
24 domains = {av:prob_domain.actions for av in self.act_vars}
25 domains.update({ st(var,stage):dom
26 for (var,dom) in prob_domain.feats_vals.items()
27 for stage in range(number_stages+1)})
28 # intial state constraints:
29 constraints = [Constraint((st(var,0),), is_(val))
30 for (var,val) in initial_state.items()]
31 # goal constraints on the final state:
32 constraints += [Constraint((st(var,number_stages),),
33 is_(val))
34 for (var,val) in goal.items()]
35 # precondition constraints:
36 constraints += [Constraint((st(var,stage), st('action',stage)),
37 if_(val,act)) # st(var,stage)==val if st('action',stage)=act
38 for act,strps in prob_domain.strips_map.items()
39 for var,val in strps.preconditions.items()
40 for stage in range(number_stages)]
41 # effect constraints:
42 constraints += [Constraint((st(var,stage+1), st('action',stage)),
43 if_(val,act)) # st(var,stage+1)==val if st('action',stage)==act
44 for act,strps in prob_domain.strips_map.items()
45 for var,val in strps.effects.items()
46 for stage in range(number_stages)]
47 # frame constraints:
48 constraints += [Constraint((st(var,stage), st('action',stage), st(var,stage+1)),
49 eq_if_not_in_({act for act in prob_domain.actions
50 if var in prob_domain.strips_map[act].effects}))
51 for var in prob_domain.feats_vals
52 for stage in range(number_stages) ]
53 CSP.__init__(self, domains, constraints)
54
55 def extract_plan(self,soln):
56 return [soln[a] for a in self.act_vars]
57
58 def st(var,stage):
59 """returns a string for the var-stage pair that can be used as a variable"""

http://aipython.org Version 0.7.5 June 19, 2018

 94 6. Planning with Certainty

60 return str(var)+"_"+str(stage)
The following methods return methods which can be applied to the particular
environment.
For example, is (3) returns a function that when applied to 3, returns True
and when aplied to any other value returns False. So is (3)(3) trurns True and
is (3)(7) returns False.
Note that the underscore (’ ’) is part of the name; here we use it as the
convention that it is a function that returns a function. This uses two different
styles to define is and if ; returning a function defined by lambda is equivaent
to returning the embedded function, except that the embedded function has a
name. The embedded function can also be given a docstring.
stripsCSPPlanner.py — (continued)

62 def is_(val):
63 """returns a function that is true when it is it applied to val.
64 """
65 return lambda x: x == val
66
67 def if_(v1,v2):
68 """if the second argument is v2, the first argument must be v1"""
69 #return lambda x1,x2: x1==v1 if x2==v2 else True
70 def if_fun(x1,x2):
71 return x1==v1 if x2==v2 else True
72 if_fun.__doc__ = "if x2 is "+str(v2)+" then x1 is "+str(v1)
73 return if_fun
74
75 def eq_if_not_in_(actset):
76 """first and third arguments are equal if action is not in actset"""
77 return lambda x1, a, x2: x1==x2 if a not in actset else True
Putting it together, this returns a list of actions that solves the problem prob
for a given horizon. If you want to do more than just return the list of actions,
you might want to get it to return the solution. Or even enumerate the solutions
(by using Search with AC from CSP).
stripsCSPPlanner.py — (continued)

79 def con_plan(prob,horizon):
80 """finds a plan for problem prob given horizon.
81 """
82 csp = CSP_from_STRIPS(prob, horizon)
83 sol = Con_solver(csp).solve_one()
84 return csp.extract_plan(sol) if sol else sol
The following are some example queries.
stripsCSPPlanner.py — (continued)

86 from searchGeneric import Searcher

87 from stripsProblem import delivery_domain
88 from cspConsistency import Search_with_AC_from_CSP, Con_solver
89 from stripsProblem import Planning_problem, problem0, problem1, problem2

http://aipython.org Version 0.7.5 June 19, 2018

 6.5. Partial-Order Planning 95

90
91 # Problem 0
92 # con_plan(problem0,1) # should it succeed?
93 # con_plan(problem0,2) # should it succeed?
94 # con_plan(problem0,3) # should it succeed?
95 # To use search to enumerate solutions
96 #searcher0a = Searcher(Search_with_AC_from_CSP(CSP_from_STRIPS(problem0, 1)))
97 #print(searcher0a.search())
98
99 ## Problem 1
100 # con_plan(problem1,5) # should it succeed?
101 # con_plan(problem1,4) # should it succeed?
102 ## To use search to enumerate solutions:
103 #searcher15a = Searcher(Search_with_AC_from_CSP(CSP_from_STRIPS(problem1, 5)))
104 #print(searcher15a.search())
105
106 ## Problem 2
107 #con_plan(problem2, 6) # should fail??
108 #con_plan(problem2, 7) # should succeed???
109
110 ## Example 6.13
111 problem3 = Planning_problem(delivery_domain,
112 {'SWC':True, 'RHC':False}, {'SWC':False})
113 #con_plan(problem3,2) # Horizon of 2
114 #con_plan(problem3,3) # Horizon of 3
115
116 problem4 = Planning_problem(delivery_domain,{'SWC':True},
117 {'SWC':False, 'MW':False, 'RHM':False})
118
119 # For the stochastic local search:
120 #from cspSLS import SLSearcher, Runtime_distribution
121 # cspplanning15 = CSP_from_STRIPS(problem1, 5) # should succeed
122 #se0 = SLSearcher(cspplanning15); print(se0.search(100000,0.5))
123 #p = Runtime_distribution(cspplanning15)
124 #p.plot_run(1000,1000,0.7) # warning will take a few minutes

6.5 Partial-Order Planning

To run the demo, in folder ”aipython”, load ”stripsPOP.py”, and copy
and paste the commented-out example queries at the bottom of that
file.

A partial order planner maintains a partial order of action instances. An

action instance consists of a name and an index. We need action instances
because the same action could be carried out at different times.

stripsPOP.py — Partial-order Planner using STRIPS representation

11 from searchProblem import Arc, Search_problem

http://aipython.org Version 0.7.5 June 19, 2018

 96 6. Planning with Certainty

12 import random
13
14 class Action_instance(object):
15 next_index = 0
16 def __init__(self,action,index=None):
17 if index is None:
18 index = Action_instance.next_index
19 Action_instance.next_index += 1
20 self.action = action
21 self.index = index
22
23 def __str__(self):
24 return str(self.action)+"#"+str(self.index)
25
26 __repr__ = __str__ # __repr__ function is the same as the __str__ function
A node (as in the abstraction of search space) in a partial-order planner
consists of:

• actions: a set of action instances.

• constraints: a set of (a1 , a2 ) pairs, where a1 and a2 are action instances,

which represents that a1 must come before a2 in the partial order. There
are a number of ways that this could be represented. Here we represent
the set of pairs that are in transitive closure of the before relation. This lets
us quickly determine whether some before relation is consistent with the
current constraints.

• agenda: a list of (s, a) pairs, where s is a (var, val) pair and a is an action
instance. This means that variable var must have value val before a can
occur.

• causal links: a set of (a0, g, a1) triples, where a1 and a2 are action instances
and g is a (var, val) pair. This holds when action a0 makes g true for action
a1 .

stripsPOP.py — (continued)

28 class POP_node(object):
29 """a (partial) partial-order plan. This is a node in the search space."""
30 def __init__(self, actions, constraints, agenda, causal_links):
31 """
32 * actions is a set of action instances
33 * constraints a set of (a0,a1) pairs, representing a0<a1,
34 closed under transitivity
35 * agenda list of (subgoal,action) pairs to be achieved, where
36 subgoal is a (variable,value) pair
37 * causal_links is a set of (a0,g,a1) triples,
38 where ai are action instances, and g is a (variable,value) pair
39 """
40 self.actions = actions # a set of action instances

http://aipython.org Version 0.7.5 June 19, 2018

 6.5. Partial-Order Planning 97

41 self.constraints = constraints # a set of (a0,a1) pairs

42 self.agenda = agenda # list of (subgoal,action) pairs to be achieved
43 self.causal_links = causal_links # set of (a0,g,a1) triples
44
45 def __str__(self):
46 return ("actions: "+str({str(a) for a in self.actions})+
47 "\nconstraints: "+
48 str({(str(a1),str(a2)) for (a1,a2) in self.constraints})+
49 "\nagenda: "+
50 str([(str(s),str(a)) for (s,a) in self.agenda])+
51 "\ncausal_links:"+
52 str({(str(a0),str(g),str(a2)) for (a0,g,a2) in self.causal_links})
)
extract plan constructs a total order of action instances that is consistent with
the partial order.
stripsPOP.py — (continued)

54 def extract_plan(self):
55 """returns a total ordering of the action instances consistent
56 with the constraints.
57 raises IndexError if there is no choice.
58 """
59 sorted_acts = []
60 other_acts = set(self.actions)
61 while other_acts:
62 a = random.choice([a for a in other_acts if
63 all(((a1,a) not in self.constraints) for a1 in other_acts)])
64 sorted_acts.append(a)
65 other_acts.remove(a)
66 return sorted_acts
POP search from STRIPS is an instance of a search problem. As such, we
need to define the start nodes, the goal, and the neighbors of a node.
stripsPOP.py — (continued)

68 from utilities import Displayable

69
70 class POP_search_from_STRIPS(Search_problem, Displayable):
71 def __init__(self,planning_problem):
72 Search_problem.__init__(self)
73 self.planning_problem = planning_problem
74 self.start = Action_instance("start")
75 self.finish = Action_instance("finish")
76
77 def is_goal(self, node):
78 return node.agenda == []
79
80 def start_node(self):
81 constraints = {(self.start, self.finish)}
82 agenda = [(g, self.finish) for g in self.planning_problem.goal.items()]
83 return POP_node([self.start,self.finish], constraints, agenda, [] )

http://aipython.org Version 0.7.5 June 19, 2018

 98 6. Planning with Certainty

The neighbors method is a coroutine that enumerates the neighbors of a

given node.
stripsPOP.py — (continued)

85 def neighbors(self, node):

86 """enumerates the neighbors of node"""
87 self.display(3,"finding neighbors of\n",node)
88 if node.agenda:
89 subgoal,act1 = node.agenda[0]
90 self.display(2,"selecting",subgoal,"for",act1)
91 new_agenda = node.agenda[1:]
92 for act0 in node.actions:
93 if (self.achieves(act0, subgoal) and
94 self.possible((act0,act1),node.constraints)):
95 self.display(2," reusing",act0)
96 consts1 = self.add_constraint((act0,act1),node.constraints)
97 new_clink = (act0,subgoal,act1)
98 new_cls = node.causal_links + [new_clink]
99 for consts2 in self.protect_cl_for_actions(node.actions,consts1,new_clink):
100 yield Arc(node,
101 POP_node(node.actions,consts2,new_agenda,new_cls),
102 cost=0)
103 for a0 in self.planning_problem.prob_domain.strips_map: #a0 is an action
104 if self.achieves(a0, subgoal):
105 #a0 acheieves subgoal
106 new_a = Action_instance(a0)
107 self.display(2," using new action",new_a)
108 new_actions = node.actions + [new_a]
109 consts1 = self.add_constraint((self.start,new_a),node.constraints)
110 consts2 = self.add_constraint((new_a,act1),consts1)
111 preconds = self.planning_problem.prob_domain.strips_map[a0].preconditions
112 new_agenda = new_agenda + [(pre,new_a) for pre in preconds.items()]
113 new_clink = (new_a,subgoal,act1)
114 new_cls = node.causal_links + [new_clink]
115 for consts3 in self.protect_all_cls(node.causal_links,new_a,consts2):
116 for consts4 in self.protect_cl_for_actions(node.actions,consts3,new_clink):
117 yield Arc(node,
118 POP_node(new_actions,consts4,new_agenda,new_cls),
119 cost=1)
Given a casual link (a0, subgoal, a1), the following method protects the causal
link from each action in actions. Whenever an action deletes subgoal, the action
needs to be before a0 or after a1. This method enumerates all constraints that
result from protecting the causal link from all actions.
stripsPOP.py — (continued)

121 def protect_cl_for_actions(self, actions, constrs, clink):

122 """yields constriants that extend constrs and
123 protect causal link (a0, subgoal, a1)
124 for each action in actions
125 """

http://aipython.org Version 0.7.5 June 19, 2018

 6.5. Partial-Order Planning 99

126 if actions:
127 a = actions[0]
128 rem_actions = actions[1:]
129 a0, subgoal, a1 = clink
130 if a != a0 and a != a1 and self.deletes(a,subgoal):
131 if self.possible((a,a0),constrs):
132 new_const = self.add_constraint((a,a0),constrs)
133 for e in self.protect_cl_for_actions(rem_actions,new_const,clink): yield e
# could be "yield from"
134 if self.possible((a1,a),constrs):
135 new_const = self.add_constraint((a1,a),constrs)
136 for e in self.protect_cl_for_actions(rem_actions,new_const,clink): yield e
137 else:
138 for e in self.protect_cl_for_actions(rem_actions,constrs,clink): yield e
139 else:
140 yield constrs

Given an action act, the following method protects all the causal links in
clinks from act. Whenever act deletes subgoal from some causal link (a0, subgoal, a1),
the action act needs to be before a0 or after a1. This method enumerates all con-
straints that result from protecting the causal links from act.
stripsPOP.py — (continued)

142 def protect_all_cls(self, clinks, act, constrs):

143 """yields constraints that protect all causal links from act"""
144 if clinks:
145 (a0,cond,a1) = clinks[0] # select a causal link
146 rem_clinks = clinks[1:] # remaining causal links
147 if act != a0 and act != a1 and self.deletes(act,cond):
148 if self.possible((act,a0),constrs):
149 new_const = self.add_constraint((act,a0),constrs)
150 for e in self.protect_all_cls(rem_clinks,act,new_const): yield e
151 if self.possible((a1,act),constrs):
152 new_const = self.add_constraint((a1,act),constrs)
153 for e in self.protect_all_cls(rem_clinks,act,new_const): yield e
154 else:
155 for e in self.protect_all_cls(rem_clinks,act,constrs): yield e
156 else:
157 yield constrs

The following methods check whether an action (or action instance) achives
or deletes some subgoal.
stripsPOP.py — (continued)

159 def achieves(self,action,subgoal):

160 var,val = subgoal
161 return var in self.effects(action) and self.effects(action)[var] == val
162
163 def deletes(self,action,subgoal):
164 var,val = subgoal
165 return var in self.effects(action) and self.effects(action)[var] != val

http://aipython.org Version 0.7.5 June 19, 2018

 100 6. Planning with Certainty

166
167 def effects(self,action):
168 """returns the variable:value dictionary of the effects of action.
169 works for both actions and action instances"""
170 if isinstance(action, Action_instance):
171 action = action.action
172 if action == "start":
173 return self.planning_problem.initial_state
174 elif action == "finish":
175 return {}
176 else:
177 return self.planning_problem.prob_domain.strips_map[action].effects

The constriants are represented as a set of pairs closed under transitivity.

Thus if (a, b) and (b, c) are the list, then (a, c) must also be in the list. This means
that adding a new constraint means adding the implied pairs, but querying
whether some order is consistent is quick.

stripsPOP.py — (continued)

179 def add_constraint(self, pair, const):

180 if pair in const:
181 return const
182 todo = [pair]
183 newconst = const.copy()
184 while todo:
185 x0,x1 = todo.pop()
186 newconst.add((x0,x1))
187 for x,y in newconst:
188 if x==x1 and (x0,y) not in newconst:
189 todo.append((x0,y))
190 if y==x0 and (x,x1) not in newconst:
191 todo.append((x,x1))
192 return newconst
193
194 def possible(self,pair,constraint):
195 (x,y) = pair
196 return (y,x) not in constraint

Some code for testing:

stripsPOP.py — (continued)

198 from searchBranchAndBound import DF_branch_and_bound

199 from searchGeneric import AStarSearcher
200 from searchMPP import SearcherMPP
201 from stripsProblem import problem0, problem1, problem2
202
203 rplanning0 = POP_search_from_STRIPS(problem0)
204 rplanning1 = POP_search_from_STRIPS(problem1)
205 rplanning2 = POP_search_from_STRIPS(problem2)
206 searcher0 = DF_branch_and_bound(rplanning0,5)
207 searcher0a = AStarSearcher(rplanning0)

http://aipython.org Version 0.7.5 June 19, 2018

 6.5. Partial-Order Planning 101

208 searcher1 = DF_branch_and_bound(rplanning1,10)

209 searcher1a = AStarSearcher(rplanning1)
210 searcher2 = DF_branch_and_bound(rplanning2,10)
211 searcher2a = AStarSearcher(rplanning2)
212 # Try one of the following searchers
213 # a = searcher0.search()
214 # a = searcher0a.search()
215 # a.end().extract_plan() # print a plan found
216 # a.end().constraints # print the constraints
217 # AStarSearcher.max_display_level = 0 # less detailed display
218 # DF_branch_and_bound.max_display_level = 0 # less detailed display
219 # a = searcher1.search()
220 # a = searcher1a.search()
221 # a = searcher2.search()
222 # a = searcher2a.search()

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 7

Supervised Machine Learning

A good source of datasets is the UCI machine Learning Repository [?]; the
SPECT and car datasets are from this repository.

7.1 Representations of Data and Predictions

A data set is an enumeration of examples. Each example is a list (or tuple)
of feature values. The feature values can be numbers or strings. A feature is
a function from the examples into the range of the feature. We assume each
feature has a variable frange that gives the range of the feature. A Boolean
feature is a function from the examples into {False, True}. So f (e) is either True
or False, where f is a feature and e is an example. The __doc__ variable contains
the docstring, a string description of the function.
learnProblem.py — A Learning Problem
11 import math, random
12 import csv
13 from utilities import Displayable
14
15 boolean = [False, True]
When creating a data set, we partition the data into a training set (train) and
a test set (test). The target feature is the feature that we are making a prediction
of.
learnProblem.py — (continued)

17 class Data_set(Displayable):
18 """ A data set consists of a list of training data and a list of test data.
19 """
20 seed = None #123456 # make it None for a different test set each time
21

103
 104 7. Supervised Machine Learning

22 def __init__(self, train, test=None, prob_test=0.30, target_index=0, header=None):

23 """A dataset for learning.
24 train is a list of tuples representing the training examples
25 test is the list of tuples representing the test examples
26 if test is None, a test set is created by selecting each
27 example with probability prob_test
28 target_index is the index of the target. If negative, it counts from right.
29 If target_index is larger than the number of properties,
30 there is no target (for unsupervised learning)
31 header is a list of names for the features
32 """
33 if test is None:
34 train,test = partition_data(train, prob_test, seed=self.seed)
35 self.train = train
36 self.test = test
37 self.display(1,"Tuples read. \nTraining set", len(train),
38 "examples. Number of columns:",{len(e) for e in train},
39 "\nTest set", len(test),
40 "examples. Number of columns:",{len(e) for e in test}
41 )
42 self.prob_test = prob_test
43 self.num_properties = len(self.train[0])
44 if target_index < 0: #allows for -1, -2, etc.
45 target_index = self.num_properties + target_index
46 self.target_index = target_index
47 self.header = header
48 self.create_features()
49 self.display(1,"There are",len(self.input_features),"input features")
Initially we assume that all of the properties can be mapped directly into fea-
tures. If all values are 0 or 1 they can be used as Boolean features. This will be
overridden to allow for more general features.
learnProblem.py — (continued)

51 def create_features(self):
52 """create the input features and target feature.
53 This assumes that the features all have domain {0,1}.
54 This should be overridden if the features have a different domain.
55 """
56 self.input_features = []
57 for i in range(self.num_properties):
58 def feat(e,index=i):
59 return e[index]
60 if self.header:
61 feat.__doc__ = self.header[i]
62 else:
63 feat.__doc__ = "e["+str(i)+"]"
64 feat.frange = [0,1]
65 if i == self.target_index:
66 self.target = feat
67 else:

http://aipython.org Version 0.7.5 June 19, 2018

 7.1. Representations of Data and Predictions 105

68 self.input_features.append(feat)

7.1.1 Evaluating Predictions

A predictor is a function that takes an example and makes a prediction on the
value of the target feature. A predictor can be judged according to a number
of evaluation criteria. The function evaluate dataset returns the average error
for each example, where the error for each example depends on the evaluation
criteria. Here we consider three evaluation criteria, the sum-of-squares, the
sum of absolute errors and the logloss (the negative log-likelihood, which is
the number of bits to describe the data using a code based on the prediction
treated as a probability).
learnProblem.py — (continued)

70 evaluation_criteria = ["sum-of-squares","sum_absolute","logloss"]
71
72 def evaluate_dataset(self, data, predictor, evaluation_criterion):
73 """Evaluates predictor on data according to the evaluation_criterion.
74 predictor is a function that takes an example and returns a
75 prediction for the target feature.
76 evaluation_criterion is one of the evaluation_criteria.
77 """
78 assert evaluation_criterion in self.evaluation_criteria,"given: "+str(evaluation_criterion)
79 if data:
80 try:
81 error = sum(error_example(predictor(example), self.target(example),
82 evaluation_criterion)
83 for example in data)/len(data)
84 except ValueError:
85 return float("inf") # infinity
86 return error
error example is used to evaluate a single example, based on the predicted value,
the actual value and the evaluation criterion. Note that for logloss, the actual
value must be 0 or 1.
learnProblem.py — (continued)

88 def error_example(predicted, actual, evaluation_criterion):

89 """returns the error of the for the predicted value given the actual value
90 according to evaluation_criterion.
91 Throws ValueError if the error is infinite (log(0))
92 """
93 if evaluation_criterion=="sum-of-squares":
94 return (predicted-actual)**2
95 elif evaluation_criterion=="sum_absolute":
96 return abs(predicted-actual)
97 elif evaluation_criterion=="logloss":
98 assert actual in [0,1], "actual="+str(actual)
99 if actual==0:

http://aipython.org Version 0.7.5 June 19, 2018

 106 7. Supervised Machine Learning

100 return -math.log2(1-predicted)

101 else:
102 return -math.log2(predicted)
103 elif evaluation_criterion=="characteristic_ss":
104 return sum((1-predicted[i])**2 if actual==i else predicted[i]**2
105 for i in range(len(predicted)))
106 else:
107 raise RuntimeError("Not evaluation criteria: "+str(evaluation_criterion))

7.1.2 Creating Test and Training Sets

The following method partitions the data into a training set and a test set. Note
that this does not guarantee that the test set will contain exactly a proportion
of the data equal to prob test.
[An alternative is to use random.sample() which can guarantee that the test
set will contain exactly a particular proportion of the data. However this would
require knowing how many elements are in the data set, which we may not
know, as data may just be a generator of the data (e.g., when reading the data
from a file).]
learnProblem.py — (continued)

109 def partition_data(data, prob_test=0.30, seed=None):

110 """partitions the data into a training set and a test set, where
111 prob_test is the probability of each example being in the test set.
112 """
113 train = []
114 test = []
115 if seed: # given seed makes the partition consistent from run-to-run
116 random.seed(seed)
117 for example in data:
118 if random.random() < prob_test:
119 test.append(example)
120 else:
121 train.append(example)
122 return train, test

7.1.3 Importing Data From File

A data set is typically loaded from a file. The default here is that it loaded
from a CSV (comma separated values) file, although the default separator can
be changed. This assumes that all lines that contain the separator are valid
data (so we only include those data items that contain more than one element).
This allows for blank lines and comment lines that do not contain the separator.
However, it means that this method is not suitable for cases where there is only
one feature.
Note that data all and data tuples are generators. data all is a generator of
a list of list of strings. This version assumes that CSV files are simple. The

http://aipython.org Version 0.7.5 June 19, 2018

 7.1. Representations of Data and Predictions 107

standard csv package, that allows quoted arguments, can be used by uncom-
menting the line for data aa and commenting out the following line. data tuples
contains only those lines that contain the delimiter (others lines are assumed to
be empty or comments), and tries to convert the elements to numbers when-
ever possible.
This allows for some of the columns to be included. Note that if include only
is specified, the target index is in the resulting

learnProblem.py — (continued)

124 class Data_from_file(Data_set):

125 def __init__(self, file_name, separator=',', num_train=None, prob_test=0.3,
126 has_header=False, target_index=0, boolean_features=True,
127 categorical=[], include_only=None):
128 """create a dataset from a file
129 separator is the character that separates the attributes
130 num_train is a number n specifying the first n tuples are training, or None
131 prob_test is the probability an example should in the test set (if num_train is None)
132 has_header is True if the first line of file is a header
133 target_index specifies which feature is the target
134 boolean_features specifies whether we want to create Boolean features
135 (if False, is uses the original features).
136 categorical is a set (or list) of features that should be treated as categorical
137 include_only is a list or set of indexes of columns to include
138 """
139 self.boolean_features = boolean_features
140 with open(file_name,'r',newline='') as csvfile:
141 # data_all = csv.reader(csvfile,delimiter=separator) # for more complicted CSV files
142 data_all = (line.strip().split(separator) for line in csvfile)
143 if include_only is not None:
144 data_all = ([v for (i,v) in enumerate(line) if i in include_only] for line in data_a
145 if has_header:
146 header = next(data_all)
147 else:
148 header = None
149 data_tuples = (make_num(d) for d in data_all if len(d)>1)
150 if num_train is not None:
151 # training set is divided into training then text examples
152 # the file is only read once, and the data is placed in appropriate list
153 train = []
154 for i in range(num_train): # will give an error if insufficient examples
155 train.append(next(data_tuples))
156 test = list(data_tuples)
157 Data_set.__init__(self,train, test=test, target_index=target_index,header=header)
158 else: # randomly assign training and test examples
159 Data_set.__init__(self,data_tuples, prob_test=prob_test,
160 target_index=target_index, header=header)
161
162 def __str__(self):
163 if self.train and len(self.train)>0:
164 return ("Data: "+str(len(self.train))+" training examples, "

http://aipython.org Version 0.7.5 June 19, 2018

 108 7. Supervised Machine Learning

165 +str(len(self.test))+" test examples, "

166 +str(len(self.train[0]))+" features.")
167 else:
168 return ("Data: "+str(len(self.train))+" training examples, "
169 +str(len(self.test))+" test examples.")

7.1.4 Creating Binary Features

Some of the algorithms require Boolean features or features with domain {0, 1}.
In order to be able to use these on datasets that allow for arbitrary ranges of
input variables, we construct binary features from the attributes. This method
overrides the method in Data set.
There are 3 cases:

• When the attribute only has two values, we designate one to be the “true”
value.

• When the values are all numeric, we assume they are ordered (as opposed
to just being some classes that happen to be labelled with numbers, but
where the numbers have no meaning) and construct Boolean features for
splits of the data. That is, the feature is e[ind] < cut for some value cut.
We choose a number of cut values, up to a maximum number of cuts,
given by max num cuts.

• When the values are not all numeric, we assume they are unordered, and
create an indicator function for each value. An indicator function for a
value returns true when that value is given and false otherwise. Note
that we can’t create an indicator function for values that appear in the
test set but not in the training set because we haven’t seen the test set.
For the examples in the test set with that value, the indicator functions
return false.

learnProblem.py — (continued)

171 def create_features(self, max_num_cuts=8):

172 """creates boolean features from input features.
173 max_num_cuts is the maximum number of binary variables
174 to split a numerical feature into.
175 """
176 ranges = [set() for i in range(self.num_properties)]
177 for example in self.train:
178 for ind,val in enumerate(example):
179 ranges[ind].add(val)
180 if self.target_index <= self.num_properties:
181 def target(e,index=self.target_index):
182 return e[index]
183 if self.header:
184 target.__doc__ = self.header[ind]

http://aipython.org Version 0.7.5 June 19, 2018

 7.1. Representations of Data and Predictions 109

185 else:
186 target.__doc__ = "e["+str(ind)+"]"
187 target.frange = ranges[self.target_index]
188 self.target = target
189 if self.boolean_features:
190 self.input_features = []
191 for ind,frange in enumerate(ranges):
192 if ind != self.target_index and len(frange)>1:
193 if len(frange) == 2:
194 # two values, the feature is equality to one of them.
195 true_val = list(frange)[1] # choose one as true
196 def feat(e, i=ind, tv=true_val):
197 return e[i]==tv
198 if self.header:
199 feat.__doc__ = self.header[ind]+"=="+str(true_val)
200 else:
201 feat.__doc__ = "e["+str(ind)+"]=="+str(true_val)
202 feat.frange = boolean
203 self.input_features.append(feat)
204 elif all(isinstance(val,(int,float)) for val in frange):
205 # all numeric, create cuts of the data
206 sorted_frange = sorted(frange)
207 num_cuts = min(max_num_cuts,len(frange))
208 cut_positions = [len(frange)*i//num_cuts for i in range(1,num_cuts)]
209 for cut in cut_positions:
210 cutat = sorted_frange[cut]
211 def feat(e, ind_=ind, cutat=cutat):
212 return e[ind_] < cutat
213
214 if self.header:
215 feat.__doc__ = self.header[ind]+"<"+str(cutat)
216 else:
217 feat.__doc__ = "e["+str(ind)+"]<"+str(cutat)
218 feat.frange = boolean
219 self.input_features.append(feat)
220 else:
221 # create an indicator function for every value
222 for val in frange:
223 def feat(e, ind_=ind, val_=val):
224 return e[ind_] == val_
225 if self.header:
226 feat.__doc__ = self.header[ind]+"=="+str(val)
227 else:
228 feat.__doc__= "e["+str(ind)+"]=="+str(val)
229 feat.frange = boolean
230 self.input_features.append(feat)
231 else: # boolean_features is off
232 self.input_features = []
233 for i in range(self.num_properties):
234 def feat(e,index=i):

http://aipython.org Version 0.7.5 June 19, 2018

 110 7. Supervised Machine Learning

235 return e[index]

236 if self.header:
237 feat.__doc__ = self.header[i]
238 else:
239 feat.__doc__ = "e["+str(i)+"]"
240 feat.frange = ranges[i]
241 if i == self.target_index:
242 self.target = feat
243 else:
244 self.input_features.append(feat)

Exercise 7.1 Change the code so that it splits using e[ind] ≤ cut instead of e[ind] <
cut. Check boundary cases, such as 3 elements with 2 cuts, and if there are 30
elements (integers from 100 to 129), and you want 2 cuts, the resulting Boolean
features should be e[ind] ≤ 109 and e[ind] ≤ 119 to make sure that each of the
resulting ranges is equal size.
Exercise 7.2 This splits on whether the feature is less than one of the values in
the training set. Sam suggested it might be better to split between the values in
the training set, and suggested using

cutat = (sorted frange[cut] + sorted frange[cut − 1])/2

Why might Sam have suggested this? Does this work better? (Try it on a few data
sets).
When reading from a file all of the values are strings. This next method
tries to convert each values into a number (an int or a float), if it is possible.
learnProblem.py — (continued)

245 def make_num(str_list):

246 """make the elements of string list str_list numerical if possible.
247 Otherwise remove initial and trailing spaces.
248 """
249 res = []
250 for e in str_list:
251 try:
252 res.append(int(e))
253 except ValueError:
254 try:
255 res.append(float(e))
256 except ValueError:
257 res.append(e.strip())
258 return res

7.1.5 Augmented Features

Sometimes we want to augment the features with new features computed from
the old features (eg. the product of features). Here we allow the creation of a
new dataset from an old dataset but with new features.

http://aipython.org Version 0.7.5 June 19, 2018

 7.1. Representations of Data and Predictions 111

A feature is a function of examples. A unary feature constructor takes a fea-

ture and returns a new feature. A binary feature combiner takes two features
and returns a new feature.
learnProblem.py — (continued)

260 class Data_set_augmented(Data_set):

261 def __init__(self, dataset, unary_functions=[], binary_functions=[], include_orig=True):
262 """creates a dataset like dataset but with new features
263 unary_function is a list of unary feature constructors
264 binary_functions is a list of binary feature combiners.
265 include_orig specifies whether the original features should be included
266 """
267 self.orig_dataset = dataset
268 self.unary_functions = unary_functions
269 self.binary_functions = binary_functions
270 self.include_orig = include_orig
271 self.target = dataset.target
272 Data_set.__init__(self,dataset.train, test=dataset.test,
273 target_index = dataset.target_index)
274
275 def create_features(self):
276 if self.include_orig:
277 self.input_features = self.orig_dataset.input_features.copy()
278 else:
279 self.input_features = []
280 for u in self.unary_functions:
281 for f in self.orig_dataset.input_features:
282 self.input_features.append(u(f))
283 for b in self.binary_functions:
284 for f1 in self.orig_dataset.input_features:
285 for f2 in self.orig_dataset.input_features:
286 if f1 != f2:
287 self.input_features.append(b(f1,f2))
The following are useful unary feature constructors and binary feature com-
biner.
learnProblem.py — (continued)

289 def square(f):

290 """a unary feature constructor to construct the square of a feature
291 """
292 def sq(e):
293 return f(e)**2
294 sq.__doc__ = f.__doc__+"**2"
295 return sq
296
297 def power_feat(n):
298 """given n returns a unary feature constructor to construct the nth power of a feature.
299 e.g., power_feat(2) is the same as square
300 """
301 def fn(f,n=n):

http://aipython.org Version 0.7.5 June 19, 2018

 112 7. Supervised Machine Learning

302 def pow(e,n=n):

303 return f(e)**n
304 pow.__doc__ = f.__doc__+"**"+str(n)
305 return pow
306 return fn
307
308 def prod_feat(f1,f2):
309 """a new feature that is the product of features f1 and f2
310 """
311 def feat(e):
312 return f1(e)*f2(e)
313 feat.__doc__ = f1.__doc__+"*"+f2.__doc__
314 return feat
315
316 def eq_feat(f1,f2):
317 """a new feature that is 1 if f1 and f2 give same value
318 """
319 def feat(e):
320 return 1 if f1(e)==f2(e) else 0
321 feat.__doc__ = f1.__doc__+"=="+f2.__doc__
322 return feat
323
324 def xor_feat(f1,f2):
325 """a new feature that is 1 if f1 and f2 give different values
326 """
327 def feat(e):
328 return 1 if f1(e)!=f2(e) else 0
329 feat.__doc__ = f1.__doc__+"!="+f2.__doc__
330 return feat

Example:

learnProblem.py — (continued)

332 # from learnProblem import Data_set_augmented,prod_feat

333 # data = Data_from_file('data/holiday.csv', num_train=19, target_index=-1)
334 ## data = Data_from_file('data/SPECT.csv', prob_test=0.5, target_index=0)
335 # dataplus = Data_set_augmented(data,[],[prod_feat])
336 # dataplus = Data_set_augmented(data,[],[prod_feat,xor_feat])

Exercise 7.3 For symmetric properties, such as product, we don’t need both
f 1 ∗ f 2 as well as f 2 ∗ f 1 as extra properties. Allow the user to be able to declare
feature constructors as symmetric (by associating a Boolean feature with them).
Change construct features so that it does not create both versions for symmetric
combiners.

7.1.6 Learner
A learner takes a dataset (and possible other arguments specific to the method).
To get it to learn, we call the learn() method. This implements Displayable so

http://aipython.org Version 0.7.5 June 19, 2018

 7.2. Learning With No Input Features 113

that we can display traces at multiple levels of detail (and perhaps with a GUI).

learnProblem.py — (continued)

337 from utilities import Displayable

338
339 class Learner(Displayable):
340 def __init__(self, dataset):
341 raise NotImplementedError("Learner.__init__") # abstract method
342
343 def learn(self):
344 """returns a predictor, a function from a tuple to a value for the target feature
345 """
346 raise NotImplementedError("learn") # abstract method

7.2 Learning With No Input Features

If we make the same prediction for each example, what prediction should we
make?
There are a few alternatives as to what could be allowed in a prediction:

• a point prediction, where we are only allowed to predict one of the values
of the feature. For example, if the values of the feature are {0, 1} we are
only allowed to predict 0 or 1 or of the values are ratings in {1, 2, 3, 4, 5},
we can only predict one of these integers.

• a point prediction, where we are allowed to predict any value. For exam-
ple, if the values of the feature are {0, 1} we may be allowed to predict 0.3,
1, or even 1.7. For all of the criteria we can imagine, there is no point in
predicting a value greater than 1 or less that zero (but that doesn’t mean
we can’t), but it is often useful to predict a value between 0 and 1. If the
values are ratings in {1, 2, 3, 4, 5}, we may want to predict 3.4.

• a probability distribution over the values of the feature. For each value v,
we predict a non-negative number pv , such that the sum over all predic-
tions is 1.

The following code assumes the second of these, where we can make a point
prediction of any value (although median will only predict one of the actual
values for the feature).
The point prediction function takes in a target feature (which is assumed to
be numeric), some training data, and a section of what to return, and returns
a function that takes in an example, and makes a prediction of a value for the
target variable, but makes same prediction for all examples.
This method uses selection, whose value should be “median”, “proportion”,
or “Laplace” determine what prediction should be made.

http://aipython.org Version 0.7.5 June 19, 2018

 114 7. Supervised Machine Learning

learnNoInputs.py — Learning ignoring all input features

11 from learnProblem import Learner, Data_set
12 import math, random
13
14 selections = ["median", "mean", "Laplace"]
15
16 def point_prediction(target, training_data,
17 selection="mean" ):
18 """makes a point prediction for a set of training data.
19 target provides the target
20 training_data provides the training data to use (often a subset of train).
21 selection specifies what statistic of the data to use as the evaluation.
22 to_optimize provides a criteria to optimize (used to guess selection)
23 """
24 assert len(training_data)>0
25 if selection == "median":
26 counts,total = target_counts(target,training_data)
27 middle = total/2
28 cumulative = 0
29 for val,num in sorted(counts.items()):
30 cumulative += num
31 if cumulative > middle:
32 break # exit loop with val as the median
33 elif selection == "mean":
34 val = mean((target(e) for e in training_data))
35 elif selection == "Laplace":
36 val = mean((target(e) for e in training_data),len(target.frange),1)
37 elif selection == "mode":
38 raise NotImplementedError("mode")
39 else:
40 raise RuntimeError("Not valid selection: "+str(selection))
41 fun = lambda x: val
42 fun.__doc__ = str(val)
43 return fun
44
45 def mean(enum,count=0,sum=0):
46 """returns the mean of enumeration enum,
47 count and sum are initial counts and the initial sum.
48 This works for enumerations, even where len() is not defined"""
49 for e in enum:
50 count += 1
51 sum += e
52 return sum/count
53
54 def target_counts(target, data_subset):
55 """returns a value:count dictionary of the count of the number of
56 times target has this value in data_subset, and the number of examples.
57 """
58 counts = {val:0 for val in target.frange}
59 total = 0

http://aipython.org Version 0.7.5 June 19, 2018

 7.2. Learning With No Input Features 115

60 for instance in data_subset:

61 total += 1
62 counts[target(instance)] += 1
63 return counts, total

7.2.1 Testing
To test the point prediction, we will first generate some data from a simple
(Bernoulli) distribution, where there are two possible values, 0 and 1 for the
target feature. Given prob, a number in the range [0, 1], this generate some
training and test data where prob is the probability of each example being 1.
learnNoInputs.py — (continued)

65 class Data_set_random(Data_set):
66 """A data set of a {0,1} feature generated randomly given a probability"""
67 def __init__(self, prob, train_size, test_size=100):
68 """a data set of with train_size training examples,
69 test_size test examples
70 where each examples in generated where prob i the probability of 1
71 """
72 self.max_display_level = 0
73 train = [[1] if random.random()<prob else [0] for i in range(train_size)]
74 test = [[1] if random.random()<prob else [0] for i in range(test_size)]
75 Data_set.__init__(self, train, test, target_index=0)
Let’s try to evaluate the predictions of the possible selections according to
the different evaluation criteria, for various training sizes.
learnNoInputs.py — (continued)

77 def test_no_inputs():
78 num_samples = 1000 #number of runs to average over
79 test_size = 100 # number of test examples for each prediction
80 for train_size in [1,2,3,4,5,10,20,100,1000]:
81 total_error = {(select,crit):0
82 for select in selections
83 for crit in Data_set.evaluation_criteria}
84 for sample in range(num_samples): # average over num_samples
85 p = random.random()
86 data = Data_set_random(p, train_size, test_size)
87 for select in selections:
88 prediction = point_prediction(data.target, data.train, selection=select)
89 for ecrit in Data_set.evaluation_criteria:
90 test_error = data.evaluate_dataset(data.test,prediction,ecrit)
91 total_error[(select,ecrit)] += test_error
92 print("For training size",train_size,":")
93 for ecrit in Data_set.evaluation_criteria:
94 print(" Evaluated according to",ecrit,":")
95 for select in selections:
96 print(" Average error of",select,"is",
97 total_error[(select,ecrit)]/num_samples)

http://aipython.org Version 0.7.5 June 19, 2018

 116 7. Supervised Machine Learning

98
99 if __name__ == "__main__":
100 test_no_inputs()

7.3 Decision Tree Learning

To run the decision tree learning demo, in folder ”aipython”, load
”learnDT.py”, using e.g., ipython -i learnDT.py, and it prints some
test results. To try more examples, copy and paste the commented-
out commands at the bottom of that file. This requires Python 3 with
matplotlib.

The decision tree algorithm does binary splits, and assumes that all input
features are binary functions of the examples. It stops splitting if there are
no input features, the number of examples is less than a specified number of
examples or all of the examples agree on the target feature.
learnDT.py — Learning a binary decision tree
11 from learnProblem import Learner, error_example
12 from learnNoInputs import point_prediction, target_counts, selections
13 import math
14
15 class DT_learner(Learner):
16 def __init__(self,
17 dataset,
18 to_optimize="sum-of-squares",
19 leaf_selection="mean", # what to use for point prediction at leaves
20 train=None, # used for cross validation
21 min_number_examples=10):
22 self.dataset = dataset
23 self.target = dataset.target
24 self.to_optimize = to_optimize
25 self.leaf_selection = leaf_selection
26 self.min_number_examples = min_number_examples
27 if train is None:
28 self.train = self.dataset.train
29 else:
30 self.train = train
31
32 def learn(self):
33 return self.learn_tree(self.dataset.input_features, self.train)
The main recursive algorithm, takes in a set of input features and a set of
training data. It first decides whether to split. If it doesn’t split, it makes a point
prediction, ignoring the input features.
It doesn’t split if there are no more input features, if there are fewer exam-
ples than min number examples, if all the examples agree on the value of the
target or if the best split makes all examples in the same partition

http://aipython.org Version 0.7.5 June 19, 2018

 7.3. Decision Tree Learning 117

If it decides to split, it selects the best split and returns the condition to split
on (in the variable split) and the corresponding partition of the examples.

learnDT.py — (continued)

35 def learn_tree(self, input_features, data_subset):

36 """returns a decision tree
37 for input_features is a set of possible conditions
38 data_subset is a subset of the data used to build this (sub)tree
39
40 where a decision tree is a function that takes an example and
41 makes a prediction on the target feature
42 """
43 if (input_features and len(data_subset) >= self.min_number_examples):
44 first_target_val = self.target(data_subset[0])
45 allagree = all(self.target(inst)==first_target_val for inst in data_subset)
46 if not allagree:
47 split, partn = self.select_split(input_features, data_subset)
48 if split: # the split succeeded in splitting the data
49 false_examples, true_examples = partn
50 rem_features = [fe for fe in input_features if fe != split]
51 self.display(2,"Splitting on",split.__doc__,"with examples split",
52 len(true_examples),":",len(false_examples))
53 true_tree = self.learn_tree(rem_features,true_examples)
54 false_tree = self.learn_tree(rem_features,false_examples)
55 def fun(e):
56 if split(e):
57 return true_tree(e)
58 else:
59 return false_tree(e)
60 #fun = lambda e: true_tree(e) if split(e) else false_tree(e)
61 fun.__doc__ = ("if "+split.__doc__+" then ("+true_tree.__doc__+
62 ") else ("+false_tree.__doc__+")")
63 return fun
64 # don't expand the trees but return a point prediction
65 return point_prediction(self.target, data_subset, selection=self.leaf_selection)

learnDT.py — (continued)

67 def select_split(self, input_features, data_subset):

68 """finds best feature to split on.
69
70 input_features is a non-empty list of features.
71 returns feature, partition
72 where feature is an input feature with the smallest error as
73 judged by to_optimize or
74 feature==None if there are no splits that improve the error
75 partition is a pair (false_examples, true_examples) if feature is not None
76 """
77 best_feat = None # best feature
78 # best_error = float("inf") # infinity - more than any error
79 best_error = training_error(self.dataset, data_subset, self.to_optimize)

http://aipython.org Version 0.7.5 June 19, 2018

 118 7. Supervised Machine Learning

80 best_partition = None
81 for feat in input_features:
82 false_examples, true_examples = partition(data_subset,feat)
83 if false_examples and true_examples: #both partitons are non-empty
84 err = (training_error(self.dataset,false_examples,self.to_optimize)
85 + training_error(self.dataset,true_examples,self.to_optimize))
86 self.display(3," split on",feat.__doc__,"has err=",err,
87 "splits into",len(true_examples),":",len(false_examples))
88 if err < best_error:
89 best_feat = feat
90 best_error=err
91 best_partition = false_examples, true_examples
92 self.display(3,"best split is on",best_feat.__doc__,
93 "with err=",best_error)
94 return best_feat, best_partition
95
96 def partition(data_subset,feature):
97 """partitions the data_subset by the feature"""
98 true_examples = []
99 false_examples = []
100 for example in data_subset:
101 if feature(example):
102 true_examples.append(example)
103 else:
104 false_examples.append(example)
105 return false_examples, true_examples
106
107
108 def training_error(dataset, data_subset, to_optimize):
109 """returns training error for dataset on to_optimize.
110 This assumes that we choose the best value for the optimization
111 criteria for dataset according to point_prediction
112 """
113 select_dict = {"sum-of-squares":"mean", "sum_absolute":"median",
114 "logloss":"Laplace"} # arbitrary mapping. Perhaps wrong.
115 selection = select_dict[to_optimize]
116 predictor = point_prediction(dataset.target, data_subset, selection=selection)
117 error = sum(error_example(predictor(example),
118 dataset.target(example),
119 to_optimize)
120 for example in data_subset)
121 return error

Test cases:

learnDT.py — (continued)

123 from learnProblem import Data_set, Data_from_file

124
125 def test(data):
126 """Prints errors and the trees for various evaluation criteria and ways to select leaves.
127 """

http://aipython.org Version 0.7.5 June 19, 2018

 7.3. Decision Tree Learning 119

128 for crit in Data_set.evaluation_criteria:

129 for leaf in selections:
130 tree = DT_learner(data, to_optimize=crit, leaf_selection=leaf).learn()
131 print("For",crit,"using",leaf,"at leaves, tree built is:",tree.__doc__)
132 if data.test:
133 for ecrit in Data_set.evaluation_criteria:
134 test_error = data.evaluate_dataset(data.test, tree, ecrit)
135 print(" Average error for", ecrit,"using",leaf, "at leaves is", test_error)
136
137 if __name__ == "__main__":
138 #print("carbool.csv"); test(data = Data_from_file('data/carbool.csv', target_index=-1))
139 # print("SPECT.csv"); test(data = Data_from_file('data/SPECT.csv', target_index=0))
140 print("mail_reading.csv"); test(data = Data_from_file('data/mail_reading.csv', target_index=-1)
141 # print("holiday.csv"); test(data = Data_from_file('data/holiday.csv', num_train=19, target_ind

Exercise 7.4 The current algorithm does not have a very sophisticated stopping
criterion. What is the current stopping criterion? (Hint: you need to look at both
learn tree and select split.)

Exercise 7.5 Extend the current algorithm to include in the stopping criterion

(a) A minimum child size; don’t use a split if one of the children has fewer
elements that this.

(b) A depth-bound on the depth of the tree.

(c) An improvement bound such that a split is only carried out if error with the
split is better than the error without the split by at least the improvement
bound.

Which values for these parameters make the prediction errors on the test set the
smallest? Try it on more than one dataset.

Exercise 7.6 Without any input features, it is often better to include a pseudo-
count that is added to the counts from the training data. Modify the code so that
it includes a pseudo-count for the predictions. When evaluating a split, including
pseudo counts can make the split worse than no split. Does pruning with an im-
provement bound and pseudo-counts make the algorithm work better than with
an improvement bound by itself?

Exercise 7.7 Some people have suggested using information gain (which is equiv-
alent to greedy optimization of logloss) as the measure of improvement when
building the tree, even in they want to have non-probabilistic predictions in the
final tree. Does this work better than myopically choosing the split that is best for
the evaluation criteria we will use to judge the final prediction?

http://aipython.org Version 0.7.5 June 19, 2018

 120 7. Supervised Machine Learning

7.4 Cross Validation and Parameter Tuning

To run the cross validation demo, in folder ”aipython”,
load ”learnCrossValidation.py”, using e.g., ipython -i
learnCrossValidation.py. Run plot fig 7 15() to produce a
graph like Figure 7.15. Note that different runs will produce different
graphs, so your graph will not look like the one in the textbook. To try
more examples, copy and paste the commented-out commands at the
bottom of that file. This requires Python 3 with matplotlib.

The above decision tree overfits the data. One way to determine whether
the prediction is overfitting is by cross validation. The code below implements
k-fold cross validation, which can be used to choose the value of parameters
to best fit the training data. If we want to use parameter tuning to improve
predictions on a particular data set, we can only use the training data (and not
the test data) to tune the parameter.
In k-fold cross validation, we partition the training set into k approximately
equal-sized folds (each fold is an enumeration of examples). For each fold, we
train on the other examples, and determine the error of the prediction on that
fold. For example, if there are 10 folds, we train on 90% of the data, and then
test on remaining 10% of the data. We do this 10 times, so that each example
gets used as a test set once, and in the training set 9 times.
The code below creates one copy of the data, and multiple views of the data.
For each fold, fold enumerates the examples in the fold, and fold complement
enumerates the examples not in the fold.

learnCrossValidation.py — Cross Validation for Parameter Tuning

11 from learnProblem import Data_set, Data_from_file, error_example
12 from learnDT import DT_learner
13 import matplotlib.pyplot as plt
14 import random
15
16 class K_fold_dataset(object):
17 def __init__(self, training_set, num_folds):
18 self.data = training_set.train.copy()
19 self.target = training_set.target
20 self.input_features = training_set.input_features
21 self.num_folds = num_folds
22 random.shuffle(self.data)
23 self.fold_boundaries = [(len(self.data)*i)//num_folds
24 for i in range(0,num_folds+1)]
25
26 def fold(self, fold_num):
27 for i in range(self.fold_boundaries[fold_num],
28 self.fold_boundaries[fold_num+1]):
29 yield self.data[i]
30

http://aipython.org Version 0.7.5 June 19, 2018

 7.4. Cross Validation and Parameter Tuning 121

31 def fold_complement(self, fold_num):

32 for i in range(0,self.fold_boundaries[fold_num]):
33 yield self.data[i]
34 for i in range(self.fold_boundaries[fold_num+1],len(self.data)):
35 yield self.data[i]
The validation error is the average error for each example, where we test on
each fold, and learn on the other folds.
learnCrossValidation.py — (continued)

37 def validation_error(self, learner, criterion, **other_params):

38 error = 0
39 try:
40 for i in range(self.num_folds):
41 predictor = learner(self, train=list(self.fold_complement(i)),
42 **other_params).learn()
43 error += sum( error_example(predictor(example),
44 self.target(example),
45 criterion)
46 for example in self.fold(i))
47 except ValueError:
48 return float("inf") #infinity
49 return error/len(self.data)
The plot error method plots the average error as a function of a the minimun
number of examples in decision-tree search, both for the validation set and for
the test set. The error on the validation set can be used to tune the parameter
— choose the value of the parameter that minimizes the error. The error on the
test set cannot be used to tune the parameters; if is were to be used this way
then it cannot be used to test.
learnCrossValidation.py — (continued)

51 def plot_error(data,criterion="sum-of-squares", num_folds=5, xscale='log'):

52 """Plots the error on the validation set and the test set
53 with respect to settings of the minimum number of examples.
54 xscale should be 'log' or 'linear'
55 """
56 plt.ion()
57 plt.xscale('linear') # change between log and linear scale
58 plt.xlabel("minimum number of examples")
59 plt.ylabel("average "+criterion+" error")
60 folded_data = K_fold_dataset(data, num_folds)
61 verrors = [] # validation errors
62 terrors = [] # test set errors
63 for mne in range(1,len(data.train)+2):
64 verrors.append(folded_data.validation_error(DT_learner,criterion,
65 min_number_examples=mne))
66 tree = DT_learner(data, criterion, min_number_examples=mne).learn()
67 terrors.append(data.evaluate_dataset(data.test,tree,criterion))
68 plt.plot(range(1,len(data.train)+2), verrors, ls='-',color='k', label="validation for "+criteri
69 plt.plot(range(1,len(data.train)+2), terrors, ls='--',color='k', label="test set for "+criterio

http://aipython.org Version 0.7.5 June 19, 2018

 122 7. Supervised Machine Learning

70 plt.legend()
71 plt.draw()
72
73 # Try
74 # data = Data_from_file('data/mail_reading.csv', target_index=-1)
75 # data = Data_from_file('data/SPECT.csv',target_index=0)
76 # data = Data_from_file('data/carbool.csv', target_index=-1)
77 # plot_error(data) # warning, may take a long time depending on the dataset
78
79 def plot_fig_7_15(): # different runs produce different plots
80 data = Data_from_file('data/SPECT.csv',target_index=0)
81 # data = Data_from_file('data/carbool.csv', target_index=-1)
82 plot_error(data)
83 # plot_fig_7_15() # warning takes a long time!

7.5 Linear Regression and Classification

Here we give a gradient descent searcher for linear regression and classifica-
tion.
learnLinear.py — Linear Regression and Classification
11 from learnProblem import Learner
12 import random, math
13
14 class Linear_learner(Learner):
15 def __init__(self, dataset, train=None,
16 learning_rate=0.1, max_init = 0.2,
17 squashed=True):
18 """Creates a gradient descent searcher for a linear classifier.
19 The main learning is carried out by learn()
20
21 dataset provides the target and the input features
22 train provides a subset of the training data to use
23 number_iterations is the default number of steps of gradient descent
24 learning_rate is the gradient descent step size
25 max_init is the maximum absolute value of the initial weights
26 squashed specifies whether the output is a squashed linear function
27 """
28 self.dataset = dataset
29 self.target = dataset.target
30 if train==None:
31 self.train = self.dataset.train
32 else:
33 self.train = train
34 self.learning_rate = learning_rate
35 self.squashed = squashed
36 self.input_features = dataset.input_features+[one] # one is defined below
37 self.weights = {feat:random.uniform(-max_init,max_init)
38 for feat in self.input_features}

http://aipython.org Version 0.7.5 June 19, 2018

 7.5. Linear Regression and Classification 123

predictor predicts the value of an example from the current parameter settings.
predictor string gives a string representation of the predictor.
learnLinear.py — (continued)

40
41 def predictor(self,e):
42 """returns the prediction of the learner on example e"""
43 linpred = sum(w*f(e) for f,w in self.weights.items())
44 if self.squashed:
45 return sigmoid(linpred)
46 else:
47 return linpred
48
49 def predictor_string(self, sig_dig=3):
50 """returns the doc string for the current prediction function
51 sig_dig is the number of significant digits in the numbers"""
52 doc = "+".join(str(round(val,sig_dig))+"*"+feat.__doc__
53 for feat,val in self.weights.items())
54 if self.squashed:
55 return "sigmoid("+ doc+")"
56 else:
57 return doc
learn is the main algorithm of the learner. It does num iter steps of gradient
descent. The other parameters it gets from the class.
learnLinear.py — (continued)

59 def learn(self,num_iter=100):
60 for it in range(num_iter):
61 self.display(2,"prediction=",self.predictor_string())
62 for e in self.train:
63 predicted = self.predictor(e)
64 error = self.target(e) - predicted
65 update = self.learning_rate*error
66 for feat in self.weights:
67 self.weights[feat] += update*feat(e)
68 #self.predictor.__doc__ = self.predictor_string()
69 #return self.predictor
one is a function that always returns 1. This is used for one of the input prop-
erties.
learnLinear.py — (continued)

71 def one(e):
72 "1"
73 return 1
sigmoid(x) is the function
1
1 + e−x

http://aipython.org Version 0.7.5 June 19, 2018

 124 7. Supervised Machine Learning

learnLinear.py — (continued)

75 def sigmoid(x):
76 return 1/(1+math.exp(-x))

The following tests the learner on a data sets. Uncomment the other data
sets for different examples.
learnLinear.py — (continued)

78 from learnProblem import Data_set, Data_from_file

79 import matplotlib.pyplot as plt
80 def test(**args):
81 data = Data_from_file('data/SPECT.csv', target_index=0)
82 # data = Data_from_file('data/mail_reading.csv', target_index=-1)
83 # data = Data_from_file('data/carbool.csv', target_index=-1)
84 learner = Linear_learner(data,**args)
85 learner.learn()
86 print("function learned is", learner.predictor_string())
87 for ecrit in Data_set.evaluation_criteria:
88 test_error = data.evaluate_dataset(data.test, learner.predictor, ecrit)
89 print(" Average", ecrit, "error is", test_error)

The following plots the errors on the training and test sets as a function of
the number of steps of gradient descent.
learnLinear.py — (continued)

91 def plot_steps(learner=None,
92 data = None,
93 criterion="sum-of-squares",
94 step=1,
95 num_steps=1000,
96 log_scale=True,
97 label=""):
98 """
99 plots the training and test error for a learner.
100 data is the
101 learner_class is the class of the learning algorithm
102 criterion gives the evaluation criterion plotted on the y-axis
103 step specifies how many steps are run for each point on the plot
104 num_steps is the number of points to plot
105
106 """
107 plt.ion()
108 plt.xlabel("step")
109 plt.ylabel("Average "+criterion+" error")
110 if log_scale:
111 plt.xscale('log') #plt.semilogx() #Makes a log scale
112 else:
113 plt.xscale('linear')
114 if data is None:
115 data = Data_from_file('data/holiday.csv', num_train=19, target_index=-1)
116 #data = Data_from_file('data/SPECT.csv', target_index=0)

http://aipython.org Version 0.7.5 June 19, 2018

 7.5. Linear Regression and Classification 125

117 # data = Data_from_file('data/mail_reading.csv', target_index=-1)

118 # data = Data_from_file('data/carbool.csv', target_index=-1)
119 random.seed(None) # reset seed
120 if learner is None:
121 learner = Linear_learner(data)
122 train_errors = []
123 test_errors = []
124 for i in range(1,num_steps+1,step):
125 test_errors.append(data.evaluate_dataset(data.test, learner.predictor, criterion))
126 train_errors.append(data.evaluate_dataset(data.train, learner.predictor, criterion))
127 learner.display(2, "Train error:",train_errors[-1],
128 "Test error:",test_errors[-1])
129 learner.learn(num_iter=step)
130 plt.plot(range(1,num_steps+1,step),train_errors,ls='-',c='k',label="training errors")
131 plt.plot(range(1,num_steps+1,step),test_errors,ls='--',c='k',label="test errors")
132 plt.legend()
133 plt.draw()
134 learner.display(1, "Train error:",train_errors[-1],
135 "Test error:",test_errors[-1])
136
137 if __name__ == "__main__":
138 test()
139
140 # This generates the figure
141 # from learnProblem import Data_set_augmented,prod_feat
142 # data = Data_from_file('data/SPECT.csv', prob_test=0.5, target_index=0)
143 # dataplus = Data_set_augmented(data,[],[prod_feat])
144 # plot_steps(data=data,num_steps=10000)
145 # plot_steps(data=dataplus,num_steps=10000) # warning very slow

Exercise 7.8 The squashed learner only makes predictions in the range (0, 1). If
the output values are {1, 2, 3, 4} there is no use prediction less than 1 or greater
than 4. Change the squashed learner so that it can learn values in the range (1, 4).
Test it on the file 'data/car.csv'.
The following plots the prediction as a function of the function of the num-
ber of steps of gradient descent. We first define a version of range that allows
for real numbers (integers and floats).
learnLinear.py — (continued)

146 def arange(start,stop,step):

147 """returns enumeration of values in the range [start,stop) separated by step.
148 like the built-in range(start,stop,step) but allows for integers and floats.
149 Note that rounding errors are expected with real numbers.
150 """
151 while start<stop:
152 yield start
153 start += step
154
155 def plot_prediction(learner=None,
156 data = None,

http://aipython.org Version 0.7.5 June 19, 2018

 126 7. Supervised Machine Learning

157 minx = 0,
158 maxx = 5,
159 step_size = 0.01, # for plotting
160 label="function"):
161 plt.ion()
162 plt.xlabel("x")
163 plt.ylabel("y")
164 if data is None:
165 data = Data_from_file('data/simp_regr.csv', prob_test=0,
166 boolean_features=False, target_index=-1)
167 if learner is None:
168 learner = Linear_learner(data,squashed=False)
169 learner.learning_rate=0.001
170 learner.learn(100)
171 learner.learning_rate=0.0001
172 learner.learn(1000)
173 learner.learning_rate=0.00001
174 learner.learn(10000)
175 learner.display(1,"function learned is", learner.predictor_string(),
176 "error=",data.evaluate_dataset(data.train, learner.predictor, "sum-of-squares"))
177 plt.plot([e[0] for e in data.train],[e[-1] for e in data.train],"bo",label="data")
178 plt.plot(list(arange(minx,maxx,step_size)),[learner.predictor([x])
179 for x in arange(minx,maxx,step_size)],
180 label=label)
181 plt.legend()
182 plt.draw()

learnLinear.py — (continued)

184 from learnProblem import Data_set_augmented, power_feat

185 def plot_polynomials(data=None,
186 learner_class = Linear_learner,
187 max_degree=5,
188 minx = 0,
189 maxx = 5,
190 num_iter = 100000,
191 learning_rate = 0.0001,
192 step_size = 0.01, # for plotting
193 ):
194 plt.ion()
195 plt.xlabel("x")
196 plt.ylabel("y")
197 if data is None:
198 data = Data_from_file('data/simp_regr.csv', prob_test=0,
199 boolean_features=False, target_index=-1)
200 plt.plot([e[0] for e in data.train],[e[-1] for e in data.train],"ko",label="data")
201 x_values = list(arange(minx,maxx,step_size))
202 line_styles = ['-','--','-.',':']
203 colors = ['0.5','k','k','k','k']
204 for degree in range(max_degree):
205 data_aug = Data_set_augmented(data,[power_feat(n) for n in range(1,degree+1)],

http://aipython.org Version 0.7.5 June 19, 2018

 7.5. Linear Regression and Classification 127

206 include_orig=False)
207 learner = learner_class(data_aug,squashed=False)
208 learner.learning_rate=learning_rate
209 learner.learn(num_iter)
210 learner.display(1,"For degree",degree,
211 "function learned is", learner.predictor_string(),
212 "error=",data.evaluate_dataset(data.train, learner.predictor, "sum-of-squares")
213 ls = line_styles[degree % len(line_styles)]
214 col = colors[degree % len(colors)]
215 plt.plot(x_values,[learner.predictor([x]) for x in x_values], linestyle=ls, color=col,
216 label="degree="+str(degree))
217 plt.legend(loc='upper left')
218 plt.draw()
219
220 # Try:
221 # plot_prediction()
222 # plot_polynomials()
223 #data = Data_from_file('data/mail_reading.csv', target_index=-1)
224 #plot_prediction(data=data)

7.5.1 Batched Stochastic Gradient Descent

This implements batched stochastic gradient descent. If the batch size is 1, it
can be simplified by not storing the differences in d, but applying them directly;
this would the be equivalent to the original code!
This overrides the learner Linear learner. Note that the comparison with
regular gradient descent is unfair as the number of updates per step is not the
same. (How could it me made more fair?)
learnLinearBSGD.py — Linear Learner with Batched Stochastic Gradient Descent
11 from learnLinear import Linear_learner
12 import random, math
13
14 class Linear_learner_bsgd(Linear_learner):
15 def __init__(self, *args, batch_size=10, **kargs):
16 Linear_learner.__init__(self, *args, **kargs)
17 self.batch_size = batch_size
18
19 def learn(self,num_iter=None):
20 if num_iter is None:
21 num_iter = self.number_iterations
22 batch_size = min(self.batch_size, len(self.train))
23 d = {feat:0 for feat in self.weights}
24 for it in range(num_iter):
25 self.display(2,"prediction=",self.predictor_string())
26 for e in random.sample(self.train, batch_size):
27 predicted = self.predictor(e)
28 error = self.target(e) - predicted
29 update = self.learning_rate*error

http://aipython.org Version 0.7.5 June 19, 2018

 128 7. Supervised Machine Learning

30 for feat in self.weights:

31 d[feat] += update*feat(e)
32 for feat in self.weights:
33 self.weights[feat] += d[feat]
34 d[feat]=0
35
36 # from learnLinear import plot_steps
37 # from learnProblem import Data_from_file
38 # data = Data_from_file('data/holiday.csv', target_index=-1)
39 # learner = Linear_learner_bsgd(data)
40 # plot_steps(learner = learner, data=data)
41
42 # to plot polynomials with batching (compare to SGD)
43 # from learnLinear import plot_polynomials
44 # plot_polynomials(learner_class = Linear_learner_bsgd)

7.6 Deep Neural Network Learning

This provides a modular implementation that implements the layers modu-
larly. Layers can easily be configured in many configurations. A layer needs to
implement a function to compute the output values from the inputs and a way
to back-propagate the error.
learnNN.py — Neural Network Learning
11 from learnProblem import Learner, Data_set, Data_from_file
12 from learnLinear import sigmoid, one
13 import random, math
14
15 class Layer(object):
16 def __init__(self,nn,num_outputs=None):
17 """Given a list of inputs, outputs will produce a list of length num_outputs.
18 nn is the neural network this is part of
19 num outputs is the number of outputs for this layer.
20 """
21 self.nn = nn
22 self.num_inputs = nn.num_outputs # output of nn is the input to this layer
23 if num_outputs:
24 self.num_outputs = num_outputs
25 else:
26 self.num_outputs = nn.num_outputs # same as the inputs
27
28 def output_values(self,input_values):
29 """Return the outputs for this layer for the given input values.
30 input_values is a list of the inputs to this layer (of length num_inputs)
31 returns a list of length self.num_outputs
32 """
33 raise NotImplementedError("output_values") # abstract method
34
35 def backprop(self,errors):

http://aipython.org Version 0.7.5 June 19, 2018

 7.6. Deep Neural Network Learning 129

36 """Backpropagate the errors on the outputs, return the errors on the inputs.
37 errors is a list of errors for the outputs (of length self.num_outputs).
38 Return the errors for the inputs to this layer (of length self.num_inputs).
39 You can assume that this is only called after corresponding output_values,
40 and it can remember information information required for the backpropagation.
41 """
42 raise NotImplementedError("backprop") # abstract method
A linear layer maintains an array of weights. self .weights[o][i] is the weight
between input i and output o. A 1 is added to the inputs.
learnNN.py — (continued)

44 class Linear_complete_layer(Layer):
45 """a completely connected layer"""
46 def __init__(self, nn, num_outputs, max_init=0.2):
47 """A completely connected linear layer.
48 nn is a neural network that the inputs come from
49 num_outputs is the number of outputs
50 max_init is the maximum value for random initialization of parameters
51 """
52 Layer.__init__(self, nn, num_outputs)
53 # self.weights[o][i] is the weight between input i and output o
54 self.weights = [[random.uniform(-max_init, max_init)
55 for inf in range(self.num_inputs+1)]
56 for outf in range(self.num_outputs)]
57
58 def output_values(self,input_values):
59 """Returns the outputs for the input values.
60 It remembers the values for the backprop.
61
62 Note in self.weights there is a weight list for every output,
63 so wts in self.weights effectively loops over the outputs.
64 """
65 self.inputs = input_values + [1]
66 return [sum(w*val for (w,val) in zip(wts,self.inputs))
67 for wts in self.weights]
68
69 def backprop(self,errors):
70 """Backpropagate the errors, updating the weights and returning the error in its inputs.
71 """
72 input_errors = [0]*(self.num_inputs+1)
73 for out in range(self.num_outputs):
74 for inp in range(self.num_inputs+1):
75 input_errors[inp] += self.weights[out][inp] * errors[out]
76 self.weights[out][inp] += self.nn.learning_rate * self.inputs[inp] * errors[out]
77 return input_errors[:-1] # remove the error for the "1"

learnNN.py — (continued)

79 class Sigmoid_layer(Layer):
80 """sigmoids of the inputs.
81 The number of outputs is equal to the number of inputs.

http://aipython.org Version 0.7.5 June 19, 2018

 130 7. Supervised Machine Learning

82 Each output is the sigmoid of its corresponding input.

83 """
84 def __init__(self, nn):
85 Layer.__init__(self, nn)
86
87 def output_values(self,input_values):
88 """Returns the outputs for the input values.
89 It remembers the output values for the backprop.
90 """
91 self.outputs= [sigmoid(inp) for inp in input_values]
92 return self.outputs
93
94 def backprop(self,errors):
95 """Returns the derivative of the errors"""
96 return [e*out*(1-out) for e,out in zip(errors, self.outputs)]

learnNN.py — (continued)

98 class ReLU_layer(Layer):
99 """Rectified linear unit (ReLU) f(z) = max(0, z).
100 The number of outputs is equal to the number of inputs.
101 """
102 def __init__(self, nn):
103 Layer.__init__(self, nn)
104
105 def output_values(self,input_values):
106 """Returns the outputs for the input values.
107 It remembers the input values for the backprop.
108 """
109 self.input_values = input_values
110 self.outputs= [max(0,inp) for inp in input_values]
111 return self.outputs
112
113 def backprop(self,errors):
114 """Returns the derivative of the errors"""
115 return [e if inp>0 else 0 for e,inp in zip(errors, self.input_values)]

learnNN.py — (continued)

117 class NN(Learner):

118 def __init__(self, dataset, learning_rate=0.1):
119 self.dataset = dataset
120 self.learning_rate = learning_rate
121 self.input_features = dataset.input_features
122 self.num_outputs = len(self.input_features)
123 self.layers = []
124
125 def add_layer(self,layer):
126 """add a layer to the network.
127 Each layer gets values from the previous layer.
128 """
129 self.layers.append(layer)

http://aipython.org Version 0.7.5 June 19, 2018

 7.6. Deep Neural Network Learning 131

130 self.num_outputs = layer.num_outputs

131
132 def predictor(self,ex):
133 """Predicts the value of the first output feature for example ex.
134 """
135 values = [f(ex) for f in self.input_features]
136 for layer in self.layers:
137 values = layer.output_values(values)
138 return values[0]
139
140 def predictor_string(self):
141 return "not implemented"

The test method learns a network and evaluates it according to various criteria.

learnNN.py — (continued)

143
144 def learn(self,num_iter):
145 """Learns parameters for a neural network using stochastic gradient decent.
146 num_iter is the number of iterations
147 """
148 for i in range(num_iter):
149 for e in random.sample(self.dataset.train,len(self.dataset.train)):
150 # compute all outputs
151 values = [f(e) for f in self.input_features]
152 for layer in self.layers:
153 values = layer.output_values(values)
154 # backpropagate
155 errors = self.sum_squares_error([self.dataset.target(e)],values)
156 for layer in reversed(self.layers):
157 errors = layer.backprop(errors)
158
159 def sum_squares_error(self,observed,predicted):
160 """Returns the errors for each of the target features.
161 """
162 return [obsd-pred for obsd,pred in zip(observed,predicted)]

This constructs a neural network consisting of neural network with one

hidden layer. The hidden using used a ReLU activation function. The output
layer used a sigmoid.

learnNN.py — (continued)

165 data = Data_from_file('data/mail_reading.csv', target_index=-1)

166 #data = Data_from_file('data/mail_reading_consis.csv', target_index=-1)
167 #data = Data_from_file('data/SPECT.csv', prob_test=0.5, target_index=0)
168 #data = Data_from_file('data/holiday.csv', target_index=-1) #, num_train=19)
169 nn1 = NN(data)
170 nn1.add_layer(Linear_complete_layer(nn1,3))
171 nn1.add_layer(Sigmoid_layer(nn1)) # comment this or the next
172 # nn1.add_layer(ReLU_layer(nn1))
173 nn1.add_layer(Linear_complete_layer(nn1,1))

http://aipython.org Version 0.7.5 June 19, 2018

 132 7. Supervised Machine Learning

174 nn1.add_layer(Sigmoid_layer(nn1))
175 nn1.learning_rate=0.1
176 #nn1.learn(100)
177
178 from learnLinear import plot_steps
179 import time
180 start_time = time.perf_counter()
181 plot_steps(learner = nn1, data = data, num_steps=10000)
182 for eg in data.train:
183 print(eg,nn1.predictor(eg))
184 end_time = time.perf_counter()
185 print("Time:", end_time - start_time)

Exercise 7.9 In the definition of nn1 above, for each of the following, first hy-
pothesize what will happen, then test your hypothesis, then explain whether you
testing confirms your hypothesis or not. Test it for more than one data set, and use
more than one run for each data set.
(a) Which fits the data better, having a sigmoid layer or a ReLU layer after the
first linear layer?
(b) Which is faster, having a sigmoid layer or a ReLU layer after the first linear
layer?
(c) What happens if you have both the sigmoid layer and then a ReLU layer
after the first linear layer and before the second linear layer?
(d) What happens if you have neither the sigmoid layer nor a ReLU layer after
the first linear layer?
(e) What happens if you have a ReLU layer then a sigmoid layer after the first
linear layer and before the second linear layer?
Exercise 7.10 Do some
It is even possible to define a perceptron layer. Warning: you may need to
change the learning rate to make this work. Should I add it into the code? It
doesn’t follow the official line.
class PerceptronLayer(Layer):
def __init__(self, nn):
Layer.__init__(self, nn)

def output_values(self,input_values):
"""Returns the outputs for the input values.
"""
self.outputs= [1 if inp>0 else -1 for inp in input_values]
return self.outputs

def backprop(self,errors):
"""Pass the errors through"""
return errors

http://aipython.org Version 0.7.5 June 19, 2018

 7.7. Boosting 133

7.7 Boosting
The following code implements functional gradient boosting for regression.
A Boosted dataset is created from a base dataset by subtracting the pre-
diction of the offset function from each example. This does not save the new
dataset, but generates it as needed. The amount of space used is constant, in-
dependent on the size of the data set.
learnBoosting.py — Functional Gradient Boosting
11 from learnProblem import Data_set, Learner
12
13 class Boosted_dataset(Data_set):
14 def __init__(self, base_dataset, offset_fun):
15 """new dataset which is like base_dataset,
16 but offset_fun(e) is subtracted from the target of each example e
17 """
18 self.base_dataset = base_dataset
19 self.offset_fun = offset_fun
20 Data_set.__init__(self, base_dataset.train, base_dataset.test,
21 base_dataset.prob_test, base_dataset.target_index)
22
23 def create_features(self):
24 self.input_features = self.base_dataset.input_features
25 def newout(e):
26 return self.base_dataset.target(e) - self.offset_fun(e)
27 newout.frange = self.base_dataset.target.frange
28 self.target = newout
A boosting learner takes in a dataset and a base learner, and returns a new
predictor. The base learner, takes a dataset, and returns a Learner object.
learnBoosting.py — (continued)

30 class Boosting_learner(Learner):
31 def __init__(self, dataset, base_learner_class):
32 self.dataset = dataset
33 self.base_learner_class = base_learner_class
34 mean = sum(self.dataset.target(e)
35 for e in self.dataset.train)/len(self.dataset.train)
36 self.predictor = lambda e:mean # function that returns mean for each example
37 self.predictor.__doc__ = "lambda e:"+str(mean)
38 self.offsets = [self.predictor]
39 self.errors = [data.evaluate_dataset(data.test, self.predictor, "sum-of-squares")]
40 self.display(1,"Predict mean test set error=", self.errors[0] )
41
42
43 def learn(self, num_ensemble=10):
44 """adds num_ensemble learners to the ensemble.
45 returns a new predictor.
46 """
47 for i in range(num_ensemble):
48 train_subset = Boosted_dataset(self.dataset, self.predictor)

http://aipython.org Version 0.7.5 June 19, 2018

 134 7. Supervised Machine Learning

49 learner = self.base_learner_class(train_subset)
50 new_offset = learner.learn()
51 self.offsets.append(new_offset)
52 def new_pred(e, old_pred=self.predictor, off=new_offset):
53 return old_pred(e)+off(e)
54 self.predictor = new_pred
55 self.errors.append(data.evaluate_dataset(data.test, self.predictor,"sum-of-squares"))
56 self.display(1,"After Iteration",len(self.offsets)-1,"test set error=", self.errors[-1])
57 return self.predictor
For testing, sp DT learner returns a function that constructs a decision tree learner
where the minimum number of examples is a proportion of the number of
training examples. The value of 0.9 tends to have one split, and a value of 0.5
tends to have two splits (but test it). Thus this can be used to construct small
decision trees that can be used as weak learners.
learnBoosting.py — (continued)

59 # Testing
60
61 from learnDT import DT_learner
62 from learnProblem import Data_set, Data_from_file
63
64 def sp_DT_learner(min_prop=0.9):
65 def make_learner(dataset):
66 mne = len(dataset.train)*min_prop
67 return DT_learner(dataset,min_number_examples=mne)
68 return make_learner
69
70 data = Data_from_file('data/carbool.csv', target_index=-1)
71 #data = Data_from_file('data/SPECT.csv', target_index=0)
72 #data = Data_from_file('data/mail_reading.csv', target_index=-1)
73 #data = Data_from_file('data/holiday.csv', num_train=19, target_index=-1)
74 learner9 = Boosting_learner(data, sp_DT_learner(0.9))
75 #learner7 = Boosting_learner(data, sp_DT_learner(0.7))
76 #learner5 = Boosting_learner(data, sp_DT_learner(0.5))
77 predictor9 =learner9.learn(10)
78 for i in learner9.offsets: print(i.__doc__)
79 import matplotlib.pyplot as plt
80
81 def plot_boosting(data,steps=10, thresholds=[0.5,0.1,0.01,0.001], markers=['-','--','-.',':'] ):
82 learners = [Boosting_learner(data, sp_DT_learner(th)) for th in thresholds]
83 predictors = [learner.learn(steps) for learner in learners]
84 plt.ion()
85 plt.xscale('linear') # change between log and linear scale
86 plt.xlabel("number of trees")
87 plt.ylabel(" error")
88 for (learner,(threshold,marker)) in zip(learners,zip(thresholds,markers)):
89 plt.plot(range(len(learner.errors)), learner.errors, ls=marker,c='k',
90 label=str(round(threshold*100))+"% min example threshold")
91 plt.legend()
92 plt.draw()

http://aipython.org Version 0.7.5 June 19, 2018

 7.7. Boosting 135

93
94 # plot_boosting(data)

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 8

Reasoning Under Uncertainty

8.1 Representing Probabilistic Models

In the implementation of probabilistic models we will assume that the variables
are objects, rather than the strings we used for CSPs. (Note that in the CSP code
variables could be anything; we just used strings for the examples.) We use a
class here because it is more amenable to extend to richer models, such as when
we introduce time.
A variable consists of a name and a domain. The domain of a variable is
a list or a tuple, as the ordering will matter in the representation of factors.
The code below internally uses the index of each value. We define a function
val to index that maps from the value to the index.
probVariables.py — Probabilistic Variables
11 class Variable(object):
12 """A random variable.
13 name (string) - name of the variable
14 domain (list) - a list of the values for the variable.
15 Variables are ordered according to their name.
16 """
17
18 def __init__(self,name,domain):
19 self.name = name
20 self.size = len(domain)
21 self.domain = domain
22 self.val_to_index = {} # map from domain to index
23 for i,val in enumerate(domain):
24 self.val_to_index[val]=i
25
26 def __str__(self):
27 return self.name

137
 138 8. Reasoning Under Uncertainty

A B C Value
0 a s v0
0 a t v1
0 b s v2
0 b t v3
0 c s v4
0 c t v5
1 a s v6
1 a t v7
1 b s v8
1 b t v9
1 c s v10
1 c t v11

Figure 8.1: A representation for a factor for the variable ordering A, B, C

28
29 def __repr__(self):
30 return "Variable('"+self.name+"')"

8.2 Factors
Factors are functions from variables into values. The main problem with vari-
able elimination is the amount of space used, because it saves the intermedi-
ate factors. (If instead it recomputed factors rather than saving the factors, it
would be effectively enumerating the worlds, and so would be exponential in
the number of variables). We only want to store the list of numbers, with as
little bookkeeping as possible.
A total ordering of the variables, and a total ordering of the values in the
domains of the variables induces a total ordering of the values of the factor
according to the lexicographic ordering. E.g., suppose the domain of A is [0, 1],
domain of B is [0 a0 ,0 b0 ,0 c0 ], and the domain of C is [0 s0 ,0 t0 ], the ordering [A, B, C]
of variables induces an ordering on the values of the factor, as in Figure 8.1.
We just need to store the list of variables and the vi s. For any assignment to A,
B and C, we can compute the index of the value for that assignment. A = a, B =
b, C = c is stored at location a0 ∗ 6 + b0 ∗ 2 + c0 , where a0 is A.val to index[a], and
similarly for b0 and c0 .
probFactors.py — Factor manipulation for graphical models
11 from functools import reduce
12 #from probVariables import Variable
13
14 class Factor(object):
15 nextid=0 # each factor has a unique identifier; for printing

http://aipython.org Version 0.7.5 June 19, 2018

 8.2. Factors 139

16
17 def __init__(self,variables):
18 """variables is the ordered list of variables
19 """
20 self.variables = variables # ordered list of variables
21 # Compute the size and the offsets for the variables
22 self.var_offsets = {}
23 self.size = 1
24 for i in range(len(variables)-1,-1,-1):
25 self.var_offsets[variables[i]]=self.size
26 self.size *= variables[i].size
27 self.id = Factor.nextid
28 self.name = "f"+str(self.id)
29 Factor.nextid += 1

For each factor, get value returns the value of the factor for an assignment. An
assignment is a variable:value dictionary. The assignment must include all of
the variables involved in the factor, and can include variables not in the factor.
This needs to be defined for every subclass.
probFactors.py — (continued)

31 def get_value(self,assignment):
32 raise NotImplementedError("get_value") # abstract method

The methods str and brief return string representations of the factor, as a table
or just as a name with the variables it is a factor on.
probFactors.py — (continued)

34 def __str__(self, variables=None):

35 """returns a string representation of the factor.
36 Allows for an arbitrary variable ordering.
37 variables is a list of the variables in the factor
38 (can contain other variables)"""
39 if variables==None:
40 variables = self.variables
41 else:
42 variables = [v for v in variables if v in self.variables]
43 res = ""
44 for v in variables:
45 res += str(v) + "\t"
46 res += self.name+"\n"
47 for i in range(self.size):
48 asst = self.index_to_assignment(i)
49 for v in variables:
50 res += str(asst[v])+"\t"
51 res += str(self.get_value(asst))
52 res += "\n"
53 return res
54
55 def brief(self):
56 """returns a string representing a summary of the factor"""

http://aipython.org Version 0.7.5 June 19, 2018

 140 8. Reasoning Under Uncertainty

57 res = self.name+"("
58 for i in range(0,len(self.variables)-1):
59 res += str(self.variables[i])+","
60 if len(self.variables)>0:
61 res += str(self.variables[len(self.variables)-1])
62 res += ")"
63 return res

The methods assignment to index and index to assignment map between the as-
signments of values to variables and the index of where that assignment would
be stored.

probFactors.py — (continued)

65 def assignment_to_index(self,assignment):
66 """returns the index where the variable:value assignment is stored"""
67 index = 0
68 for var in self.variables:
69 index += var.val_to_index[assignment[var]]*self.var_offsets[var]
70 return index
71
72 def index_to_assignment(self,index):
73 """gives a dict representation of the variable assignment for index
74 """
75 asst = {}
76 for i in range(len(self.variables)-1,-1,-1):
77 asst[self.variables[i]] = self.variables[i].domain[index % self.variables[i].size]
78 index = index // self.variables[i].size
79 return asst

A Factor stored is a factor that has the values stored in a list.

probFactors.py — (continued)

81 class Factor_stored(Factor):
82 def __init__(self,variables,values):
83 Factor.__init__(self, variables)
84 self.values = values
85
86 def get_value(self,assignment):
87 return self.values[self.assignment_to_index(assignment)]

A Factor observed is a factor that is the result of some observations on an-

other factor. We don’t store the values in a list; we just look them up as needed.
The observations can include variables that are not in the list, but should have
some intersection with the variables in the factor.

probFactors.py — (continued)

89 class Factor_observed(Factor):
90 def __init__(self,factor,obs):
91 Factor.__init__(self, [v for v in factor.variables if v not in obs])
92 self.observed = obs
93 self.orig_factor = factor

http://aipython.org Version 0.7.5 June 19, 2018

 8.2. Factors 141

94
95 def get_value(self,assignment):
96 ass = assignment.copy()
97 for ob in self.observed:
98 ass[ob]=self.observed[ob]
99 return self.orig_factor.get_value(ass)

A Factor sum is a factor that is the result of summing out a variable from the
product of other factors. Ie., it constructs a representation of:

∑ ∏ f.
var f ∈factors

We store the values in a list in a lazy manner; if they are already computed, we
used the stored values. If they are not already computed we can compute and
store them.
probFactors.py — (continued)

101 class Factor_sum(Factor_stored):

102 def __init__(self,var,factors):
103 self.var_summed_out = var
104 self.factors = factors
105 vars = []
106 for fac in factors:
107 for v in fac.variables:
108 if v is not var and v not in vars:
109 vars.append(v)
110 Factor_stored.__init__(self,vars,None)
111 self.values = [None]*self.size
112
113 def get_value(self,assignment):
114 """lazy implementation: if not saved, compute it. Return saved value"""
115 index = self.assignment_to_index(assignment)
116 if self.values[index]:
117 return self.values[index]
118 else:
119 total = 0
120 new_asst = assignment.copy()
121 for val in self.var_summed_out.domain:
122 new_asst[self.var_summed_out] = val
123 prod = 1
124 for fac in self.factors:
125 prod *= fac.get_value(new_asst)
126 total += prod
127 self.values[index] = total
128 return total

The method factor times multiples a set of factors that are all factors on the same
variable (or on no variables). This is the last step in variable elimination before
normalizing. It returns an array giving the product for each value of variable.

http://aipython.org Version 0.7.5 June 19, 2018

 142 8. Reasoning Under Uncertainty

probFactors.py — (continued)

130 def factor_times(variable,factors):

131 """when factors are factors just on variable (or on no variables)"""
132 prods= []
133 facs = [f for f in factors if variable in f.variables]
134 for val in variable.domain:
135 prod = 1
136 ast = {variable:val}
137 for f in facs:
138 prod *= f.get_value(ast)
139 prods.append(prod)
140 return prods
Prob is a factor that represents a conditional probability.
probFactors.py — (continued)

142 class Prob(Factor_stored):

143 """A factor defined by a conditional probability table"""
144 def __init__(self,var,pars,cpt):
145 """Creates a factor from a conditional probability table, cptf.
146 The cpt values are assumed to be for the ordering par+[var]
147 """
148 Factor_stored.__init__(self,pars+[var],cpt)
149 self.child = var
150 self.parents = pars
151 assert self.size==len(cpt),"Table size incorrect "+str(self)
cond dist returns the probability distribution of the child given values from the
parent. This code is based on assignment to index. Similarly, cont prob returns
the probability that the child has a particular value given an assignment of
values to the parents. In both of these par assignment is a dict that assigns all of
the parents (and can also assign other variables, but these are ignored).
probFactors.py — (continued)

153 def cond_dist(self,par_assignment):

154 """returns the distribution (a val:prob dictionary) over the child given
155 assignment to the parents
156
157 par_assignment is a variable:value dictionary that assigns values to parents
158 """
159 index = 0
160 for var in self.parents:
161 index += var.val_to_index[par_assignment[var]]*self.var_offsets[var]
162 # index is the position where the disgribution starts
163 return {self.child.domain[i]:self.values[index+i] for i in range(len(self.child.domain))}
164
165 def cond_prob(self,par_assignment,child_value):
166 """returns the probability child has child_value given
167 assignment to the parents
168
169 par_assignment is a variable:value dictionary that assigns values to parents

http://aipython.org Version 0.7.5 June 19, 2018

 8.3. Graphical Models 143

170 child_value is a value to the child

171 """
172 index = self.child.val_to_index[child_value]
173 for var in self.parents:
174 index += var.val_to_index[par_assignment[var]]*self.var_offsets[var]
175 return self.values[index]
A Factor rename is a factor that is the result renaming the variables in the factor.
It takes a factor, fac, and a new : old dictionary, where new is the name of a
variable in the resulting factor and old is the corresponding name in fac. This
assumes that the all variables are renamed.
probFactors.py — (continued)

177 class Factor_rename(Factor):

178 def __init__(self,fac,renaming):
179 Factor.__init__(self,list(renaming.keys()))
180 self.orig_fac = fac
181 self.renaming = renaming
182
183 def get_value(self,assignment):
184 return self.orig_fac.get_value({self.renaming[var]:val
185 for (var,val) in assignment.items()
186 if var in self.variables})

8.3 Graphical Models

A graphical model consists of a set of variables and a set of factors. A be-
lief network is a graphical model where all of the factors represent conditional
probabilities. There are some operations (such as pruning variables) which are
applicable to belief networks, but are not applicable to more general models.
At the moment, we will treat them as the same.
probGraphicalModels.py — Graphical Models and Belief Networks
11 class Graphical_model(object):
12 """The class of graphical models.
13 A graphical model consists of a set of variables and a set of factors.
14
15 List vars is a list of variables
16 List factors is a list of factors
17 """
18 def __init__(self,vars=None,factors=None):
19 self.variables = vars
20 self.factors = factors
A belief network is a graphical model where all of the factors are condi-
tional probabilities, and every variable has a conditional probability. This only
checks the first condition:
probGraphicalModels.py — (continued)

http://aipython.org Version 0.7.5 June 19, 2018

 144 8. Reasoning Under Uncertainty

22 class Belief_network(Graphical_model):
23 """The class of belief networks."""
24
25 def __init__(self,vars=None,factors=None):
26 """vars is a list of variables
27 factors is a list of factors. Here we assume that all of the factors are instances of Prob.
28 """
29 Graphical_model.__init__(self,vars,factors)
30 assert all(isinstance(f,Prob) for f in factors) if factors else True

Each of the inference methods implements the query method that com-
putes the posterior probability of a variable given a dictionary of variable:value
observations. These are all Displayable because they implement the display
method which is currently text-based.

probGraphicalModels.py — (continued)

32 from utilities import Displayable

33
34 class Inference_method(Displayable):
35 """The abstract class of graphical model inference methods"""
36 def query(self,qvar,obs={}):
37 raise NotImplementedError("Inference_method query") # abstract method

The first example belief network is a simple chain A −→ B −→ C.

probGraphicalModels.py — (continued)

39 from probVariables import Variable

40 from probFactors import Prob
41
42 boolean = [False, True]
43 A = Variable("A", boolean)
44 B = Variable("B", boolean)
45 C = Variable("C", boolean)
46
47 f_a = Prob(A,[],[0.4,0.6])
48 f_b = Prob(B,[A],[0.9,0.1,0.2,0.8])
49 f_c = Prob(C,[B],[0.5,0.5,0.3,0.7])
50
51 bn1 = Belief_network([A,B,C],[f_a,f_b,f_c])

The second Bayesian network is the report-of-leaving example from Poole and
Mackworth, Artificial Intelligence, 2010 http://artint.info. This is Example
6.10 (page 236) shown in Figure 6.1.

probGraphicalModels.py — (continued)

53 # Bayesian network report of leaving example from

54 # Poole and Mackworth, Artificial Intelligence, 2010 http://artint.info
55 # This is Example 6.10 (page 236) shown in Figure 6.1
56
57 Al = Variable("Alarm", boolean)
58 Fi = Variable("Fire", boolean)

http://aipython.org Version 0.7.5 June 19, 2018

 8.4. Variable Elimination 145

59 Le = Variable("Leaving", boolean)
60 Re = Variable("Report", boolean)
61 Sm = Variable("Smoke", boolean)
62 Ta = Variable("Tamper", boolean)
63
64 f_ta = Prob(Ta,[],[0.98,0.02])
65 f_fi = Prob(Fi,[],[0.99,0.01])
66 f_sm = Prob(Sm,[Fi],[0.99,0.01,0.1,0.9])
67 f_al = Prob(Al,[Fi,Ta],[0.9999, 0.0001, 0.15, 0.85, 0.01, 0.99, 0.5, 0.5])
68 f_lv = Prob(Le,[Al],[0.999, 0.001, 0.12, 0.88])
69 f_re = Prob(Re,[Le],[0.99, 0.01, 0.25, 0.75])
70
71 bn2 = Belief_network([Al,Fi,Le,Re,Sm,Ta],[f_ta,f_fi,f_sm,f_al,f_lv,f_re])

The third Bayesian network is the sprinkler example from Pearl.

probGraphicalModels.py — (continued)

73
74 Season = Variable("Season",["summer","winter"])
75 Sprinkler = Variable("Sprinkler",["on","off"])
76 Rained = Variable("Rained",boolean)
77 Grass_wet = Variable("Grass wet",boolean)
78 Grass_shiny = Variable("Grass shiny",boolean)
79 Shoes_wet = Variable("Shoes wet",boolean)
80
81 f_season = Prob(Season,[],[0.5,0.5])
82 f_sprinkler = Prob(Sprinkler,[Season],[0.9,0.1,0.05,0.95])
83 f_rained = Prob(Rained,[Season],[0.7,0.3,0.2,0.8])
84 f_wet = Prob(Grass_wet,[Sprinkler,Rained], [1,0,0.1,0.9,0.2,0.8,0.02,0.98])
85 f_shiny = Prob(Grass_shiny, [Grass_wet], [0.95,0.05,0.3,0.7])
86 f_shoes = Prob(Shoes_wet, [Grass_wet], [0.92,0.08,0.35,0.65])
87
88 bn3 = Belief_network([Season, Sprinkler, Rained, Grass_wet, Grass_shiny, Shoes_wet],
89 [f_season, f_sprinkler, f_rained, f_wet, f_shiny, f_shoes])

8.4 Variable Elimination

An instance of a VE object takes in a graphical model. The query method uses
variable elimination to compute the probability of a variable given observa-
tions on some other variables.
probVE.py — Variable Elimination for Graphical Models
11 from probFactors import Factor, Factor_observed, Factor_sum, factor_times
12 from probGraphicalModels import Graphical_model, Inference_method
13
14 class VE(Inference_method):
15 """The class that queries Graphical Models using variable elimination.
16
17 gm is graphical model to query

http://aipython.org Version 0.7.5 June 19, 2018

 146 8. Reasoning Under Uncertainty

18 """
19 def __init__(self,gm=None):
20 self.gm = gm
21
22 def query(self,var,obs={},elim_order=None):
23 """computes P(var|obs) where
24 var is a variable
25 obs is a variable:value dictionary"""
26 if var in obs:
27 return [1 if val == obs[var] else 0 for val in var.domain]
28 else:
29 if elim_order == None:
30 elim_order = self.gm.variables
31 projFactors = [self.project_observations(fact,obs)
32 for fact in self.gm.factors]
33 for v in elim_order:
34 if v != var and v not in obs:
35 projFactors = self.eliminate_var(projFactors,v)
36 unnorm = factor_times(var,projFactors)
37 p_obs=sum(unnorm)
38 self.display(1,"Unnormalized probs:",unnorm,"Prob obs:",p_obs)
39 return {val:pr/p_obs for val,pr in zip(var.domain, unnorm)}
To project observations onto a factor, for each variable that is observed in the
factor, we construct a new factor that is the factor projected onto that variable.
Factor observed creates a new factor that is the result is assigning a value to a
single variable.
probVE.py — (continued)

41 def project_observations(self,factor,obs):
42 """Returns the resulting factor after observing obs
43
44 obs is a dictionary of variable:value pairs.
45 """
46 if any((var in obs) for var in factor.variables):
47 # a variable in factor is observed
48 return Factor_observed(factor,obs)
49 else:
50 return factor
51
52 def eliminate_var(self,factors,var):
53 """Eliminate a variable var from a list of factors.
54 Returns a new set of factors that has var summed out.
55 """
56 self.display(2,"eliminating ",str(var))
57 contains_var = []
58 not_contains_var = []
59 for fac in factors:
60 if var in fac.variables:
61 contains_var.append(fac)
62 else:

http://aipython.org Version 0.7.5 June 19, 2018

 8.5. Stochastic Simulation 147

63 not_contains_var.append(fac)
64 if contains_var == []:
65 return factors
66 else:
67 newFactor = Factor_sum(var,contains_var)
68 self.display(2,"Multiplying:",[f.brief() for f in contains_var])
69 self.display(2,"Creating factor:", newFactor.brief())
70 self.display(3, newFactor) # factor in detail
71 not_contains_var.append(newFactor)
72 return not_contains_var
73
74 from probGraphicalModels import bn1, A,B,C
75 bn1v = VE(bn1)
76 ## bn1v.query(A,{})
77 ## bn1v.query(C,{})
78 ## Inference_method.max_display_level = 3 # show more detail in displaying
79 ## Inference_method.max_display_level = 1 # show less detail in displaying
80 ## bn1v.query(A,{C:True})
81 ## bn1v.query(B,{A:True,C:False})
82
83 from probGraphicalModels import bn2,Al,Fi,Le,Re,Sm,Ta
84 bn2v = VE(bn2) # answers queries using variable elimination
85 ## bn2v.query(Ta,{})
86 ## Inference_method.max_display_level = 0 # show no detail in displaying
87 ## bn2v.query(Le,{})
88 ## bn2v.query(Ta,{},elim_order=[Sm,Re,Le,Al,Fi])
89 ## bn2v.query(Ta,{Re:True})
90 ## bn2v.query(Ta,{Re:True,Sm:False})
91
92 from probGraphicalModels import bn3, Season, Sprinkler, Rained, Grass_wet, Grass_shiny, Shoes_wet
93 bn3v = VE(bn3)
94 ## bn3v.query(Shoes_wet,{})
95 ## bn3v.query(Shoes_wet,{Rained:True})
96 ## bn3v.query(Shoes_wet,{Grass_shiny:True})
97 ## bn3v.query(Shoes_wet,{Grass_shiny:False,Rained:True})

8.5 Stochastic Simulation

8.5.1 Sampling from a discrete distribution
The method sample one generates a single sample from a (possible unnormal-
ized) distribution. dist is a value : weight dictionary, where weight ≥ 0. This
returns a value with probability in proportion to its weight.
probStochSim.py — Probabilistic inference using stochastic simulation
11 import random
12 from probGraphicalModels import Inference_method
13
14 def sample_one(dist):

http://aipython.org Version 0.7.5 June 19, 2018

 148 8. Reasoning Under Uncertainty

15 """returns the index of a single sample from normalized distribution dist."""

16 rand = random.random()*sum(dist.values())
17 cum = 0 # cumulative weights
18 for v in dist:
19 cum += dist[v]
20 if cum > rand:
21 return v
If we want to generate multiple samples, repeatedly calling sample one may
not be efficient. If we want to generate n samples, and the distribution is over
m values, sample one takes time O(mn). If m and n are of the same order of
magnitude, we can do better.
The method sample multiple generates multiple samples from a distribution
defined by dist, where dist is a value : weight dictionary, where weight ≥ 0
and the weights cannot all be zero. This returns a list of values, of length
num samples, where each sample is selected with a probability proportional to
its weight.
The method generates all of the random numbers, sorts them, and then
goes through the distribution once, saving the selected samples.
probStochSim.py — (continued)

23 def sample_multiple(dist, num_samples):

24 """returns a list of num_samples values selected using distribution dist.
25 dist is a value:weight dictionary that does not need to be normalized
26 """
27 total = sum(dist.values())
28 rands = sorted(random.random()*total for i in range(num_samples))
29 result = []
30 dist_items = list(dist.items())
31 cum = dist_items[0][1] # cumulative sum
32 index = 0
33 for r in rands:
34 while r>cum:
35 index += 1
36 cum += dist_items[index][1]
37 result.append(dist_items[index][0])
38 return result

Exercise 8.1
What is the time and space complexity the following 4 methods to generate n
samples, where m is the length of dist:
(a) n calls to sample one
(b) sample multiple
(c) Create the cumulative distribution (choose how this is represented) and, for
each random number, do a binary search to determine the sample associated
with the random number.
(d) Choose a random number in the range [i/n, (i + 1)/n) for each i ∈ range(n),
where n is the number of samples. Use these as the random numbers to
select the particles. (Does this give random samples?)

http://aipython.org Version 0.7.5 June 19, 2018

 8.5. Stochastic Simulation 149

For each method suggest when it might be the best method.

The test sampling method can be used to generate the statistics from a num-
ber of samples. It is useful to see the variability as a function of the number of
samples. Try it for few samples and also for many samples.
probStochSim.py — (continued)

40 def test_sampling(dist, num_samples):

41 """Given a distribution, dist, draw num_samples samples
42 and return the resulting counts
43 """
44 result = {v:0 for v in dist}
45 for v in sample_multiple(dist, num_samples):
46 result[v] += 1
47 return result
48
49 # try the following queries a number of times each:
50 # test_sampling({1:1,2:2,3:3,4:4}, 100)
51 # test_sampling({1:1,2:2,3:3,4:4}, 100000)

8.5.2 Sampling Methods for Belief Network Inference

A Sampling inference method is an Inference method, but the query method also
takes arguments for the number of samples and the sample-order (which is an
ordering of factors). The first methods assume a Bayesian network (and not an
undirected graphical model).
probStochSim.py — (continued)

53 class Sampling_inference_method(Inference_method):
54 """The abstract class of sampling-based belief network inference methods"""
55 def query(self,qvar,obs={},number_samples=1000,sample_order=None):
56 raise NotImplementedError("Sampling_inference_method query") # abstract
Some of the sampling methods require a sample order of factors represent-
ing conditional probabilities, where the parents of a node must come before
the node in the sample order. The following method computes such a sample
ordering, and is used when the sample order argument is None.
probStochSim.py — (continued)

58 def select_sample_ordering(bn):
59 """creates a sample ordering of factors such that the parents of a node
60 are before the node.
61 raises StopIteration if there is no such ordering. This would occur in next(.).
62 """
63 sample_order=[]
64 defined = set() # set of variables whose probability is defined
65 factors_to_sample = bn.factors.copy()
66 while factors_to_sample:
67 fac = next(f for f in factors_to_sample
68 if all(par in defined for par in f.parents))

http://aipython.org Version 0.7.5 June 19, 2018

 150 8. Reasoning Under Uncertainty

69 factors_to_sample.remove(fac)
70 sample_order.append(fac)
71 defined.add(fac.child)
72 return sample_order

8.5.3 Rejection Sampling

probStochSim.py — (continued)

74 class Rejection_sampling(Sampling_inference_method):
75 """The class that queries Graphical Models using Rejection Sampling.
76
77 bn is a belief network to query
78 """
79 def __init__(self,bn=None):
80 self.bn = bn
81 self.label = "Rejection Sampling"
82
83 def query(self,qvar,obs={},number_samples=1000,sample_order=None):
84 """computes P(qvar|obs) where
85 qvar is a variable.
86 obs is a variable:value dictionary.
87 sample_order is a list of factors where factors defining the parents
88 come before the factors for the child.
89 """
90 if sample_order is None:
91 sample_order = select_sample_ordering(self.bn)
92 self.display(2,*[f.child for f in sample_order],sep="\t")
93 counts = {val:0 for val in qvar.domain}
94 for i in range(number_samples):
95 rejected = False
96 sample = {}
97 for fac in sample_order:
98 nvar = fac.child #next variable
99 val = sample_one(fac.cond_dist(sample))
100 self.display(2,val,end="\t")
101 if nvar in obs and obs[nvar] != val:
102 rejected = True
103 self.display(2,"Rejected")
104 break
105 sample[nvar] = val
106 if not rejected:
107 counts[sample[qvar]] += 1
108 self.display(2,"Accepted")
109 tot = sum(counts.values())
110 return counts, {c:divide(v,tot) for (c,v) in counts.items()}

It is possible that all samples get rejected. In that case, Python would give
as a arithmetic error. Instead, we implement the convention that 0/0 = 1. You
need to be careful is using these numbers as probabilities.

http://aipython.org Version 0.7.5 June 19, 2018

 8.5. Stochastic Simulation 151

probStochSim.py — (continued)

112 def divide(num,denom):

113 """returns num/denom without divide-by-zero errors.
114 defines 0/0 to be 1."""
115 if denom == 0:
116 return 1.0
117 else:
118 return num/denom

8.5.4 Likelihood Weighting

Likelihood weighting includes a weight for each sample. Instead of rejecting
samples based on observations, likelihood weighting changes the weights of
the sample in proportion with the probability of the observation. The weight
then becomes the probability that the variable would have been rejected.
probStochSim.py — (continued)

120 class Likelihood_weighting(Sampling_inference_method):

121 """The class that queries Graphical Models using Likelihood weighting.
122
123 bn is a belief network to query
124 """
125 def __init__(self,bn=None):
126 self.bn = bn
127 self.label = "Likelihood weighting"
128
129 def query(self,qvar,obs={},number_samples=1000,sample_order=None):
130 """computes P(qvar|obs) where
131 qvar is a variable.
132 obs is a variable:value dictionary.
133 sample_order is a list of factors where factors defining the parents
134 come before the factors for the child.
135 """
136 if sample_order is None:
137 sample_order = select_sample_ordering(self.bn)
138 self.display(2,*[f.child for f in sample_order
139 if f.child not in obs],sep="\t")
140 counts = [0 for val in qvar.domain]
141 for i in range(number_samples):
142 sample = {}
143 weight = 1.0
144 for fac in sample_order:
145 nvar = fac.child # next variable sampled
146 if nvar in obs:
147 sample[nvar] = obs[nvar]
148 weight *= fac.get_value(sample)
149 else:
150 val = sample_one(fac.cond_dist(sample))
151 self.display(2,val,end="\t")
152 sample[nvar] = val
153 counts[sample[qvar]] += weight

http://aipython.org Version 0.7.5 June 19, 2018

 152 8. Reasoning Under Uncertainty

154 self.display(2,weight)
155 tot = sum(counts)
156 return counts, {c:v/tot for (c,v) in counts.items()}

Exercise 8.2 Change this algorithm so that it does importance sampling using a
proposal distribution. It needs sample one using a different distribution and then
update the weight of the current sample. For testing, use a proposal distribution
that only specifies probabilities for some of the variables (and the algorithm uses
the probabilities for the network in other cases).

8.5.5 Particle Filtering

In this implementation, a particle is a variable : value dictionary. Because adding
a new value to dictionary involves a side effect, the dictionaries need to be
copied during resampling.
probStochSim.py — (continued)

158 class Particle_filtering(Sampling_inference_method):

159 """The class that queries Graphical Models using Particle Filtering.
160
161 bn is a belief network to query
162 """
163 def __init__(self,bn=None):
164 self.bn = bn
165 self.label = "Particle Filtering"
166
167 def query(self, qvar, obs={}, number_samples=1000, sample_order=None):
168 """computes P(qvar|obs) where
169 qvar is a variable.
170 obs is a variable:value dictionary.
171 sample_order is a list of factors where factors defining the parents
172 come before the factors for the child.
173 """
174 if sample_order is None:
175 sample_order = select_sample_ordering(self.bn)
176 self.display(2,*[f.child for f in sample_order
177 if f.child not in obs],sep="\t")
178 particles = [{} for i in range(number_samples)]
179 for fac in sample_order:
180 nvar = fac.child # the variable sampled
181 if nvar in obs:
182 weights = {part:fac.cond_prob(part,obs[nvar]) for part in particles}
183 particles = [p.copy for p in resample(particles, weights, number_samples)]
184 else:
185 for part in particles:
186 part[nvar] = sample_one(fac.cond_dist(part))
187 self.display(2,part[nvar],end="\t")
188 counts = [0 for val in qvar.domain]
189 for part in particles:
190 counts[part[qvar]] += 1

http://aipython.org Version 0.7.5 June 19, 2018

 8.5. Stochastic Simulation 153

191 self.display(2,weight)
192 return counts

Resampling
Resample is based on sample multiple but works with an array of particles.
(Aside: Python doesn’t let us use sample multiple directly as it uses a dictionary,
and particles, represented as dictionaries can’t be the key of dictionaries).
probStochSim.py — (continued)

194 def resample(particles, weights, num_samples):

195 """returns num_samples copies of particles resampled according to weights.
196 particles is a list of particles
197 weights is a list of positive numbers, of same length as particles
198 num_samples is n integer
199 """
200 total = sum(weights)
201 rands = sorted(random.random()*total for i in range(num_samples))
202 result = []
203 cum = weights[0] # cumulative sum
204 index = 0
205 for r in rands:
206 while r>cum:
207 index += 1
208 cum += weights[index]
209 result.append(particles[index])
210 return result

8.5.6 Examples
probStochSim.py — (continued)

212 from probGraphicalModels import bn1, A,B,C

213 bn1r = Rejection_sampling(bn1)
214 bn1L = Likelihood_weighting(bn1)
215 ## Inference_method.max_display_level = 2 # detailed tracing for all inference methods
216 ## bn1r.query(A,{})
217 ## bn1r.query(C,{})
218 ## bn1r.query(A,{C:True})
219 ## bn1r.query(B,{A:True,C:False})
220
221 from probGraphicalModels import bn2,Al,Fi,Le,Re,Sm,Ta
222 bn2r = Rejection_sampling(bn2) # answers queries using rejection sampling
223 bn2L = Likelihood_weighting(bn2) # answers queries using rejection sampling
224 bn2p = Particle_filtering(bn2) # answers queries using particle filtering
225 ## bn2r.query(Ta,{})
226 ## bn2r.query(Ta,{})
227 ## bn2r.query(Ta,{Re:True})
228 ## Inference_method.max_display_level = 0 # no detailed tracing for all inference methods

http://aipython.org Version 0.7.5 June 19, 2018

 154 8. Reasoning Under Uncertainty

229 ## bn2r.query(Ta,{Re:True},number_samples=100000)
230 ## bn2r.query(Ta,{Re:True,Sm:False})
231 ## bn2r.query(Ta,{Re:True,Sm:False},number_samples=100)
232
233 ## bn2L.query(Ta,{Re:True,Sm:False},number_samples=100)
234 ## bn2L.query(Ta,{Re:True,Sm:False},number_samples=100)
235
236
237 from probGraphicalModels import bn3,Season, Sprinkler
238 from probGraphicalModels import Rained, Grass_wet, Grass_shiny, Shoes_wet
239 bn3r = Rejection_sampling(bn3) # answers queries using rejection sampling
240 bn3L = Likelihood_weighting(bn3) # answers queries using rejection sampling
241 bn3p = Particle_filtering(bn3) # answers queries using particle filtering
242 #bn3r.query(Shoes_wet,{Grass_shiny:True,Rained:True})
243 #bn3L.query(Shoes_wet,{Grass_shiny:True,Rained:True})
244 #bn3p.query(Shoes_wet,{Grass_shiny:True,Rained:True})
Exercise 8.3 This code keeps regenerating the distribution of a variable given
its parents. Implement one or both of the following, and compare them to the
original. Make cond dist return a slice that corresponds to the distribution, and
then use the slice instead of the dictionary (a list slice does not generate new data
structures). Make cond dist remember values it has already computed, and only
return these.

8.5.7 Plotting Behaviour of Stochastic Simulators

The stochastic simulation runs can give different answers each time they are
run. For the algorithms that give the same answer in the limit as the number of
samples approaches infinity (as do all of these algorithms), the algorithms can
be compared by comparing the accuracy for multiple runs. Summary statistics
like the variance may provide some information, but the assumptions behind
the variance being appropriate (namely that the distribution is approximately
Gaussian) may not hold for cases where the predictions are bounded and often
skewed.
It is more appropriate to plot the distribution of predictions over multiple
runs. The plot stats method plots the prediction of a particular variable (or for
the partition function) for a number of runs of the same algorithm. On the x-
axis, is the prediction of the algorithm. On the y-axis is the number of runs
with prediction less than or equal to the x value. Thus this is like a cumulative
distribution over the predictions, but with counts on the y-axis.
Note that for runs where there are no samples that are consistent with the
observations (as can happen with rejection sampling), the prediction of proba-
bility is 1.0 (as a convention for 0/0).
That variable what contains the query variable, or what is “prob ev”, the
probability of evidence.
probStochSim.py — (continued)

246 import matplotlib.pyplot as plt

http://aipython.org Version 0.7.5 June 19, 2018

 8.6. Markov Chain Monte Carlo 155

247
248 def plot_stats(method, what, qvar, obs, number_samples=100, number_runs=1000):
249 """Plots a cumulative distribution of the prediction of the model.
250 method is a Sampling_inference_method (that implements appropriate query(.))
251 what is either "prob_ev" or the value of qvar to plot
252 qvar is the query variable
253 obs is the variable:value dictionary representing the observations
254 number_samples is the number of samples for each run
255 number_iterations is the number of runs that are plotted
256 """
257 plt.ion()
258 plt.xlabel("value")
259 plt.ylabel("Cumulative Number")
260 Inference_method.max_display_level, prev_max_display_level = 0, Inference_method.max_display_le
261 answers = [method.query(qvar,obs,number_samples=number_samples)
262 for i in range(number_runs)]
263 if what == "prob_ev":
264 values = [sum(ans)/number_samples for ans in answers]
265 label = method.label+"(prob of evidence)"
266 else:
267 values = [divide(ans[qvar.val_to_index[what]],sum(ans)) for ans in answers]
268 label = method.label+" ("+str(qvar)+"="+str(what)+")"
269 values.sort()
270 plt.plot(values,range(number_runs),label=label)
271 plt.legend(loc="upper left")
272 plt.draw()
273 Inference_method.max_display_level = prev_max_display_level # restore display level
274
275
276 # plot_stats(bn2r,False,Ta,{Re:True,Sm:False},number_samples=1000, number_runs=1000)
277 # plot_stats(bn2L,False,Ta,{Re:True,Sm:False},number_samples=1000, number_runs=1000)
278 # plot_stats(bn2r,False,Ta,{Re:True,Sm:False},number_samples=100, number_runs=1000)
279 # plot_stats(bn2L,False,Ta,{Re:True,Sm:False},number_samples=100, number_runs=1000)
280 # plot_stats(bn3r,True,Shoes_wet,{Grass_shiny:True,Rained:True},number_samples=1000)
281 # plot_stats(bn3L,True,Shoes_wet,{Grass_shiny:True,Rained:True},number_samples=1000)
282 # plot_stats(bn2r,"prob_ev",Ta,{Re:True,Sm:False},number_samples=1000, number_runs=1000)
283 # plot_stats(bn2L,"prob_ev",Ta,{Re:True,Sm:False},number_samples=1000, number_runs=1000)

8.6 Markov Chain Monte Carlo

The following implements Gibbs sampling, a form of Markov Chain Monte
Carlo MCMC.
probMCMC.py — Markov Chain Monte Carlo (Gibbs sampling)

11 import random
12 from probGraphicalModels import Inference_method
13
14 from probStochSim import sample_one, Sampling_inference_method
15

http://aipython.org Version 0.7.5 June 19, 2018

 156 8. Reasoning Under Uncertainty

16 class Gibbs_sampling(Sampling_inference_method):
17 """The class that queries Graphical Models using Gibbs Sampling.
18
19 bn is a graphical model (e.g., a belief network) to query
20 """
21 def __init__(self,bn=None):
22 self.bn = bn
23 self.label = "Gibbs Sampling"
24
25 def query(self, qvar, obs={}, number_samples=1000, burn_in=100, sample_order=None):
26 """computes P(qvar|obs) where
27 qvar is a variable.
28 obs is a variable:value dictionary.
29 sample_order is a list of non-observed variables in order.
30 """
31 counts = {val:0 for val in qvar.domain}
32 if sample_order is not None:
33 variables = sample_order
34 else:
35 variables = [v for v in self.bn.variables if v not in obs]
36 var_to_factors = {v:set() for v in self.bn.variables}
37 for fac in self.bn.factors:
38 for var in fac.variables:
39 var_to_factors[var].add(fac)
40 sample = {var:random.choice(var.domain) for var in variables}
41 self.display(2,"Sample:",sample)
42 sample.update(obs)
43 for i in range(burn_in + number_samples):
44 if sample_order == None:
45 random.shuffle(variables)
46 for var in variables:
47 # get probability distribution of var given its neighbours
48 vardist = {val:1 for val in var.domain}
49 for val in var.domain:
50 sample[var] = val
51 for fac in var_to_factors[var]: # Markov blanket
52 vardist[val] *= fac.get_value(sample)
53 sample[var] = sample_one(vardist)
54 if i >= burn_in:
55 counts[sample[qvar]] +=1
56 tot = sum(counts.values())
57 return counts, {c:v/tot for (c,v) in counts.items()}
58
59 from probGraphicalModels import bn1, A,B,C
60 bn1g = Gibbs_sampling(bn1)
61 ## Inference_method.max_display_level = 2 # detailed tracing for all inference methods
62 bn1g.query(A,{})
63 ## bn1g.query(C,{})
64 ## bn1g.query(A,{C:True})
65 ## bn1g.query(B,{A:True,C:False})

http://aipython.org Version 0.7.5 June 19, 2018

 8.7. Hidden Markov Models 157

66
67 from probGraphicalModels import bn2,Al,Fi,Le,Re,Sm,Ta
68 bn2g = Gibbs_sampling(bn2)
69 ## bn2g.query(Ta,{Re:True},number_samples=100000)

Exercise 8.4 Change the code so that it can have multiple query variables. Make
the list of query variable be an input to the algorithm, so that the default value is
the list of all non-observed variables.
Exercise 8.5 In this algorithm, explain where it computes the probability of a
variable given its Markov blanket. Instead of returning the average of the samples
for the query variable, it is possible to return the average estimate of the probabil-
ity of the query variable given its Markov blanket. Does this converge to the same
answer as the given code? Does it converge faster, slower, or the same?

8.7 Hidden Markov Models

This code for hidden Markov models is independent of the graphical mod-
els code, to keep it simple. Section 8.8 gives code that models hidden Markov
models, and more generally, dynamic belief networks, using the graphical mod-
els code.
This HMM code assumes there are multiple Boolean observation variables
that depend on the current state and are independent of each other given the
state.
probHMM.py — Hidden Markov Model
11 import random
12 from probStochSim import sample_one, sample_multiple
13
14 class HMM(object):
15 def __init__(self, states, obsvars,pobs,trans,indist):
16 """A hidden Markov model.
17 states - set of states
18 obsvars - set of observation variables
19 pobs - probability of observations, pobs[i][s] is P(Obs_i=True | State=s)
20 trans - transition probability - trans[i][j] gives P(State=j | State=i)
21 indist - initial distribution - indist[s] is P(State_0 = s)
22 """
23 self.states = states
24 self.obsvars = obsvars
25 self.pobs = pobs
26 self.trans = trans
27 self.indist = indist
Consider the following example. Suppose you want to unobtrusively keep
track of an animal in a triangular enclosure using sound. Suppose you have
3 microphones that provide unreliable (noisy) binary information at each time
step. The animal is either close to one of the 3 points of the triangle or in the
middle of the triangle.

http://aipython.org Version 0.7.5 June 19, 2018

 158 8. Reasoning Under Uncertainty

probHMM.py — (continued)

29 # state
30 # 0=middle, 1,2,3 are corners
31 states1 = {'middle', 'c1', 'c2', 'c3'} # states
32 obs1 = {'m1','m2','m3'} # microphones

The observation model is as follows. If the animal is in a corner, it will

be detected by the microphone at that corner with probability 0.6, and will be
independently detected by each of the other microphones with a probability of
0.1. If the animal is in the middle, it will be detected by each microphone with
a probability of 0.4.

probHMM.py — (continued)

34 # pobs gives the observation model:

35 #pobs[mi][state] is P(mi=on | state)
36 closeMic=0.6; farMic=0.1; midMic=0.4
37 pobs1 = {'m1':{'middle':midMic, 'c1':closeMic, 'c2':farMic, 'c3':farMic}, # mic 1
38 'm2':{'middle':midMic, 'c1':farMic, 'c2':closeMic, 'c3':farMic}, # mic 2
39 'm3':{'middle':midMic, 'c1':farMic, 'c2':farMic, 'c3':closeMic}} # mic 3

The transition model is as follows: If the animal is in a corner it stays in

the same corner with probability 0.80, goes to the middle with probability 0.1
or goes to one of the other corners with probability 0.05 each. If it is in the
middle, it stays in the middle with probability 0.7, otherwise it moves to one
the corners, each with probability 0.1.

probHMM.py — (continued)

41 # trans specifies the dynamics

42 # trans[i] is the distribution over states resulting from state i
43 # trans[i][j] gives P(S=j | S=i)
44 sm=0.7; mmc=0.1 # transition probabilities when in middle
45 sc=0.8; mcm=0.1; mcc=0.05 # transition probabilities when in a corner
46 trans1 = {'middle':{'middle':sm, 'c1':mmc, 'c2':mmc, 'c3':mmc}, # was in middle
47 'c1':{'middle':mcm, 'c1':sc, 'c2':mcc, 'c3':mcc}, # was in corner 1
48 'c2':{'middle':mcm, 'c1':mcc, 'c2':sc, 'c3':mcc}, # was in corner 2
49 'c3':{'middle':mcm, 'c1':mcc, 'c2':mcc, 'c3':sc}} # was in corner 3

Initially the animal is in one of the four states, with equal probability.

probHMM.py — (continued)

51 # initially we have a uniform distribution over the animal's state

52 indist1 = {st:1.0/len(states1) for st in states1}
53
54 hmm1 = HMM(states1, obs1, pobs1, trans1, indist1)

8.7.1 Exact Filtering for HMMs

A HMM VE filter has a current state distribution which can be updated by ob-
serving or by advancing to the next time.

http://aipython.org Version 0.7.5 June 19, 2018

 8.7. Hidden Markov Models 159

probHMM.py — (continued)

56 from utilities import Displayable

57
58 class HMM_VE_filter(Displayable):
59 def __init__(self,hmm):
60 self.hmm = hmm
61 self.state_dist = hmm.indist
62
63 def filter(self, obsseq):
64 """updates and returns the state distribution following the sequence of
65 observations in obsseq using variable elimination.
66
67 Note that it first advances time.
68 This is what is required if it is called sequentially.
69 If that is not what is wanted initially, do an observe first.
70 """
71 for obs in obsseq:
72 self.advance() # advance time
73 self.observe(obs) # observe
74 return self.state_dist
75
76 def observe(self, obs):
77 """updates state conditioned on observations.
78 obs is a list of values for each observation variable"""
79 for i in self.hmm.obsvars:
80 self.state_dist = {st:self.state_dist[st]*(self.hmm.pobs[i][st]
81 if obs[i] else (1-self.hmm.pobs[i][st]))
82 for st in self.hmm.states}
83 norm = sum(self.state_dist.values()) # normalizing constant
84 self.state_dist = {st:self.state_dist[st]/norm for st in self.hmm.states}
85 self.display(2,"After observing",obs,"state distribution:",self.state_dist)
86
87 def advance(self):
88 """advance to the next time"""
89 nextstate = {st:0.0 for st in self.hmm.states} # distribution over next states
90 for j in self.hmm.states: # j ranges over next states
91 for i in self.hmm.states: # i ranges over previous states
92 nextstate[j] += self.hmm.trans[i][j]*self.state_dist[i]
93 self.state_dist = nextstate

The following are some queries for hmm1.

probHMM.py — (continued)

95 hmm1f1 = HMM_VE_filter(hmm1)
96 # hmm1f1.filter([{'m1':0, 'm2':1, 'm3':1}, {'m1':1, 'm2':0, 'm3':1}])
97 ## HMM_VE_filter.max_display_level = 2 # show more detail in displaying
98 # hmm1f2 = HMM_VE_filter(hmm1)
99 # hmm1f2.filter([{'m1':1, 'm2':0, 'm3':0}, {'m1':0, 'm2':1, 'm3':0}, {'m1':1, 'm2':0, 'm3':0},
100 # {'m1':0, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':0},
101 # {'m1':0, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':1}, {'m1':0, 'm2':0, 'm3':1},
102 # {'m1':0, 'm2':0, 'm3':1}])

http://aipython.org Version 0.7.5 June 19, 2018

 160 8. Reasoning Under Uncertainty

103 # hmm1f3 = HMM_VE_filter(hmm1)

104 # hmm1f3.filter([{'m1':1, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':0}, {'m1':1, 'm2':0, 'm3':0}, {'m1':1
105
106 # How do the following differ in the resulting state distribution?
107 # Note they start the same, but have different initial observations.
108 ## HMM_VE_filter.max_display_level = 1 # show less detail in displaying
109 # for i in range(100): hmm1f1.advance()
110 # hmm1f1.state_dist
111 # for i in range(100): hmm1f3.advance()
112 # hmm1f3.state_dist

Exercise 8.6 The localization example in the book is a controlled HMM, where
there is a given action at each time and the transition depends on the action.
Change the code to allow for controlled HMMs. Hint: the action only influences
the state transition.
Exercise 8.7 The representation assumes that there are a list of Boolean obser-
vations. Extend the representation so that the each observation variable can have
multiple discrete values. You need to choose a representation for the model, and
change the algorithm.

8.7.2 Particle Filtering for HMMs

In this implementation a particle is just a state. If you want to do some form
of smooting, a particle should probably be a history of states. This maintains,
particles, an array of states, weights an array of (non-negative) real numbers,
such that weights[i] is the weight of particles[i].
probHMM.py — (continued)

113 from utilities import Displayable

114 from probStochSim import resample
115
116 class HMM_particle_filter(Displayable):
117 def __init__(self,hmm,number_particles=1000):
118 self.hmm = hmm
119 self.particles = [sample_one(hmm.indist)
120 for i in range(number_particles)]
121 self.weights = [1 for i in range(number_particles)]
122
123 def filter(self, obsseq):
124 """returns the state distribution following the sequence of
125 observations in obsseq using particle filtering.
126
127 Note that it first advances time.
128 This is what is required if it is called after previous filtering.
129 If that is not what is wanted initially, do an observe first.
130 """
131 for obs in obsseq:
132 self.advance() # advance time
133 self.observe(obs) # observe

http://aipython.org Version 0.7.5 June 19, 2018

 8.7. Hidden Markov Models 161

134 self.resample_particles()
135 self.display(2,"After observing", str(obs),
136 "state distribution:", self.histogram(self.particles))
137 self.display(1,"Final state distribution:", self.histogram(self.particles))
138 return self.histogram(self.particles)
139
140 def advance(self):
141 """advance to the next time.
142 This assumes that all of the weights are 1."""
143 self.particles = [sample_one(self.hmm.trans[st])
144 for st in self.particles]
145
146 def observe(self, obs):
147 """reweight the particles to incorporate observations obs"""
148 for i in range(len(self.particles)):
149 for obv in obs:
150 if obs[obv]:
151 self.weights[i] *= self.hmm.pobs[obv][self.particles[i]]
152 else:
153 self.weights[i] *= 1-self.hmm.pobs[obv][self.particles[i]]
154
155 def histogram(self, particles):
156 """returns list of the probability of each state as represented by
157 the particles"""
158 tot=0
159 hist = {st: 0.0 for st in self.hmm.states}
160 for (st,wt) in zip(self.particles,self.weights):
161 hist[st]+=wt
162 tot += wt
163 return {st:hist[st]/tot for st in hist}
164
165 def resample_particles(self):
166 """resamples to give a new set of particles."""
167 self.particles = resample(self.particles, self.weights, len(self.particles))
168 self.weights = [1] * len(self.particles)

The following are some queries for hmm1.

probHMM.py — (continued)

170 hmm1pf1 = HMM_particle_filter(hmm1)

171 # HMM_particle_filter.max_display_level = 2 # show each step
172 # hmm1pf1.filter([{'m1':0, 'm2':1, 'm3':1}, {'m1':1, 'm2':0, 'm3':1}])
173 # hmm1pf2 = HMM_particle_filter(hmm1)
174 # hmm1pf2.filter([{'m1':1, 'm2':0, 'm3':0}, {'m1':0, 'm2':1, 'm3':0}, {'m1':1, 'm2':0, 'm3':0},
175 # {'m1':0, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':0},
176 # {'m1':0, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':1}, {'m1':0, 'm2':0, 'm3':1},
177 # {'m1':0, 'm2':0, 'm3':1}])
178 # hmm1pf3 = HMM_particle_filter(hmm1)
179 # hmm1pf3.filter([{'m1':1, 'm2':0, 'm3':0}, {'m1':0, 'm2':0, 'm3':0}, {'m1':1, 'm2':0, 'm3':0}, {'

Exercise 8.8 A form of importance sampling can be obtained by not resampling.

http://aipython.org Version 0.7.5 June 19, 2018

 162 8. Reasoning Under Uncertainty

Is it better or worse than particle filtering? Hint: you need to think about how
they can be compared. Is the comparison different if there are more states than
particles?

Exercise 8.9 Extend the particle filtering code to continuous variables and ob-
servations. In particular, suppose the state transition is a linear function with
Gaussian noise of the previous state, and the observations are linear functions
with Gaussian noise of the state. You may need to research how to sample from a
Gaussian distribution.

8.7.3 Generating Examples

The following code is useful for generating examples.

probHMM.py — (continued)

181 def simulate(hmm,horizon):

182 """returns a pair of (state sequence, observation sequence) of length horizon.
183 for each time t, the agent is in state_sequence[t] and
184 observes observation_sequence[t]
185 """
186 state = sample_one(hmm.indist)
187 obsseq=[]
188 stateseq=[]
189 for time in range(horizon):
190 stateseq.append(state)
191 newobs = {obs:sample_one({0:1-hmm.pobs[obs][state],1:hmm.pobs[obs][state]})
192 for obs in hmm.obsvars}
193 obsseq.append(newobs)
194 state = sample_one(hmm.trans[state])
195 return stateseq,obsseq
196
197 def simobs(hmm,stateseq):
198 """returns observation sequence for the state sequence"""
199 obsseq=[]
200 for state in stateseq:
201 newobs = {obs:sample_one({0:1-hmm.pobs[obs][state],1:hmm.pobs[obs][state]})
202 for obs in hmm.obsvars}
203 obsseq.append(newobs)
204 return obsseq
205
206 def create_eg(hmm,n):
207 """Create an annotated example for horizon n"""
208 seq,obs = simulate(hmm,n)
209 print("True state sequence:",seq)
210 print("Sequence of observations:\n",obs)
211 hmmfilter = HMM_VE_filter(hmm)
212 dist = hmmfilter.filter(obs)
213 print("Resulting distribution over states:\n",dist)

http://aipython.org Version 0.7.5 June 19, 2018

 8.8. Dynamic Belief Networks 163

8.8 Dynamic Belief Networks

A dynamic belief network consists of:

• A set of features. A variable is a feature-time pair.

• An initial distribution over the features at time 0. This is a belief network

with all variables being time 0 variables.

• A specification of the dynamics. Here we define the how the variables

one time depend on variables at that time and the previous time, in such
a way that the graph is acyclic.

There are a number of ways that reasoning can be carried out in a DBN,
including:

• Rolling out the DBN for some time period, and using standard belief net-
work inference. The latest time that needs to be in the rolled out network
is the time of the latest observation or the time of a query (whichever is
later). This allows us to observe any variables at any time and query any
variables at any time. However, the unrolled Bayesian network may be
very large. We also need to construct multiple copies of each feature.

• Just representing the variables “now”. In this approach we can observe

and query the current variables. We can them move to the next time.
This does not allow for arbitrary historical queries (about the past or the
future), but can be much simpler.

Here we will implement the second of these.

probDBN.py — Dynamic belief networks
11 from probVariables import Variable
12 from probGraphicalModels import Graphical_model
13 from probFactors import Prob, Factor_rename
14 from probVE import VE
15 from utilities import Displayable
16
17 class DBN_variable(Variable):
18 """A random variable that incorporates
19
20 A variable can have both a name and an index. The index defaults to 1.
21 Equality is true if they are both the name and the index are the same."""
22 def __init__(self,name,domain=[False,True],index=1):
23 Variable.__init__(self,name,domain)
24 self.index = index
25 self.previous = None
26
27 def __lt__(self,other):
28 if self.name != other.name:
29 return self.name<other.name

http://aipython.org Version 0.7.5 June 19, 2018

 164 8. Reasoning Under Uncertainty

30 else:
31 return self.index<other.index
32
33 def __gt__(self,other):
34 return other<self
35
36 def __str__(self):
37 # if self.index==1:
38 # return self.name
39 # else:
40 return self.name+"_"+str(self.index)
41
42 __repr__ = __str__
43
44 def variable_pair(name,domain=[False,True]):
45 """returns a variable and its predecessor. This is used to define 2-stage DBNs
46
47 If the name is X, it returns the pair of variables X0,X"""
48 var = DBN_variable(name,domain)
49 var0 = DBN_variable(name,domain,index=0)
50 var.previous = var0
51 return var0, var

probDBN.py — (continued)

53 class DBN(Displayable):
54 """The class of stationary Dynamic Bayesian networks.
55
56 * vars1 is a list of current variables (each must have
57 previous variable).
58 * transition_factors is a list of factors for P(X|parents) where X
59 is a current variable and parents is a list of current or previous variables.
60 * init_factors is a list of factors for P(X|parents) where X is a
61 current variable and parents can only include current variables
62 The graph of transition factors + init factors must be acyclic.
63
64 """
65 def __init__(self,vars1, transition_factors=None, init_factors=None):
66 self.vars1 = vars1
67 self.vars0 = [v.previous for v in vars1]
68 self.transition_factors = transition_factors
69 self.init_factors = init_factors
70 self.var_index = {} # var_index[v] is the index of variable v
71 for i,v in enumerate(vars1):
72 self.var_index[v]=i
Here is a 3 variable DBN:
probDBN.py — (continued)

74 A0,A1 = variable_pair("A")
75 B0,B1 = variable_pair("B")
76 C0,C1 = variable_pair("C")

http://aipython.org Version 0.7.5 June 19, 2018

 8.8. Dynamic Belief Networks 165

77
78 # dynamics
79 pc = Prob(C1,[B1,C0],[0.03,0.97,0.38,0.62,0.23,0.77,0.78,0.22])
80 pb = Prob(B1,[A0,A1],[0.5,0.5,0.77,0.23,0.4,0.6,0.83,0.17])
81 pa = Prob(A1,[A0,B0],[0.1,0.9,0.65,0.35,0.3,0.7,0.8,0.2])
82
83 # initial distribution
84 pa0 = Prob(A1,[],[0.9,0.1])
85 pb0 = Prob(B1,[A1],[0.3,0.7,0.8,0.2])
86 pc0 = Prob(C1,[],[0.2,0.8])
87
88 dbn1 = DBN([A1,B1,C1],[pa,pb,pc],[pa0,pb0,pc0])
Here is the animal example
probDBN.py — (continued)

90 from probHMM import closeMic, farMic, midMic, sm, mmc, sc, mcm, mcc
91
92 Pos_0,Pos_1 = variable_pair("Position",domain=[0,1,2,3])
93 Mic1_0,Mic1_1 = variable_pair("Mic1")
94 Mic2_0,Mic2_1 = variable_pair("Mic2")
95 Mic3_0,Mic3_1 = variable_pair("Mic3")
96
97 # conditional probabilities - see hmm for the values of sm,mmc, etc
98 ppos = Prob(Pos_1, [Pos_0],
99 [sm, mmc, mmc, mmc, #was in middle
100 mcm, sc, mcc, mcc, #was in corner 1
101 mcm, mcc, sc, mcc, #was in corner 2
102 mcm, mcc, mcc, sc]) #was in corner 3
103 pm1 = Prob(Mic1_1, [Pos_1], [1-midMic, midMic, 1-closeMic, closeMic,
104 1-farMic, farMic, 1-farMic, farMic])
105 pm2 = Prob(Mic2_1, [Pos_1], [1-midMic, midMic, 1-farMic, farMic,
106 1-closeMic, closeMic, 1-farMic, farMic])
107 pm3 = Prob(Mic3_1, [Pos_1], [1-midMic, midMic, 1-farMic, farMic,
108 1-farMic, farMic, 1-closeMic, closeMic])
109 ipos = Prob(Pos_1,[], [0.25, 0.25, 0.25, 0.25])
110 dbn_an =DBN([Pos_1,Mic1_1,Mic2_1,Mic3_1],
111 [ppos, pm1, pm2, pm3],
112 [ipos, pm1, pm2, pm3])

probDBN.py — (continued)

114 class DBN_VE_filter(VE):

115 def __init__(self,dbn):
116 self.dbn = dbn
117 self.current_factors = dbn.init_factors
118 self.current_obs = {}
119
120 def observe(self, obs):
121 """updates the current observations with obs.
122 obs is a variable:value dictionary where variable is a current
123 variable.

http://aipython.org Version 0.7.5 June 19, 2018

 166 8. Reasoning Under Uncertainty

124 """
125 assert all(self.current_obs[var]==obs[var] for var in obs
126 if var in self.current_obs),"inconsistent current observations"
127 self.current_obs.update(obs)
128
129 def query(self,var):
130 """returns the posterior probability of current variable var"""
131 return VE(Graphical_model(self.dbn.vars1,self.current_factors)).query(var,self.current_obs)
132
133 def advance(self):
134 """advance to the next time"""
135 prev_factors = [self.make_previous(fac) for fac in self.current_factors]
136 prev_obs = {var.previous:val for var,val in self.current_obs.items()}
137 two_stage_factors = prev_factors + self.dbn.transition_factors
138 self.current_factors = self.elim_vars(two_stage_factors,self.dbn.vars0,prev_obs)
139 self.current_obs = {}
140
141 def make_previous(self,fac):
142 """Creates new factor from fac where the current variables in fac
143 are renamed to previous variables.
144 """
145 return Factor_rename(fac, {var.previous:var for var in fac.variables})
146
147 def elim_vars(self,factors, vars, obs):
148 for var in vars:
149 if var in obs:
150 factors = [self.project_observations(fac,obs) for fac in factors]
151 else:
152 factors = self.eliminate_var(factors, var)
153 return factors
Example queries:
probDBN.py — (continued)

155 df = DBN_VE_filter(dbn1)
156 #df.observe({B1:True}); df.advance(); df.observe({C1:False})
157 #df.query(B1)
158 #df.advance()
159 #df.query(B1)
160 dfa = DBN_VE_filter(dbn_an)
161 # dfa.observe({Mic1_1:0, Mic2_1:1, Mic3_1:1})
162 # dfa.advance()
163 # dfa.observe({Mic1_1:1, Mic2_1:0, Mic3_1:1})
164 # dfa.query(Pos_1)

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 9

Planning with Uncertainty

9.1 Decision Networks

The decision network code builds on the representation for belief networks of
Chapter 8.
We first allow for factors that define the utility. Here the utility is a function
of the variables in vars, and the table is a list that enumerates the values as in
Section 8.2.
decnNetworks.py — Representations for Decision Networks
11 from probGraphicalModels import Graphical_model
12 from probFactors import Factor_stored
13 from probVariables import Variable
14 from probFactors import Prob
15
16 class Utility(Factor_stored):
17 """A factor defined by a utility"""
18 def __init__(self,vars,table):
19 """Creates a factor on vars from the table.
20 The table is ordered according to vars.
21 """
22 Factor_stored.__init__(self,vars,table)
23 assert self.size==len(table),"Table size incorrect "+str(self)
A decision variable is a like a random variable with a string name, and a do-
main, which is a list of possible values. The decision variable also includes the
parents, a list of the variables whose value will be known when the decision is
made.
decnNetworks.py — (continued)

25 class DecisionVariable(Variable):
26 def __init__(self,name,domain,parents):

167
 168 9. Planning with Uncertainty

27 Variable.__init__(self,name,domain)
28 self.parents = parents
29 self.all_vars = set(parents) | {self}
A decision network is a graphical model where the variables can be random
variables or decision variables. In the factors we assume there is one utility
factor.
decnNetworks.py — (continued)

31 class DecisionNetwork(Graphical_model):
32 def __init__(self,vars=None,factors=None):
33 """vars is a list of variables
34 factors is a list of factors (instances of Prob and Utility)
35 """
36 Graphical_model.__init__(self,vars,factors)
VE DN is variable elimination for decision networks. The method optimize is
used to optimize all the decisions. Note that optimize requires a legal emimina-
tion ordering of the random and decision variables, otherwise it will give an
exception. (A decision node can only be maximized if the variables that are not
its parents have already been eliminated.)
decnNetworks.py — (continued)

38 from probFactors import factor_times, Factor_stored

39 from probVE import VE
40
41 class VE_DN(VE):
42 """Variable Elimination for Decision Networks"""
43 def __init__(self,dn=None):
44 """dn is a decision network"""
45 VE.__init__(self,dn)
46 self.dn = dn
47
48 def optimize(self,elim_order=None,obs={}):
49 if elim_order == None:
50 elim_order = self.gm.variables
51 policy = []
52 proj_factors = [self.project_observations(fact,obs)
53 for fact in self.dn.factors]
54 for v in elim_order:
55 if isinstance(v,DecisionVariable):
56 to_max = [fac for fac in proj_factors
57 if v in fac.variables and set(fac.variables) <= v.all_vars]
58 assert len(to_max)==1, "illegal variable order "+str(elim_order)+" at "+str(v)
59 newFac = Factor_max(v, to_max[0])
60 policy.append(newFac.decision_fun)
61 proj_factors = [fac for fac in proj_factors if fac is not to_max[0]]+[newFac]
62 self.display(2,"maximizing",v,"resulting factor",newFac.brief() )
63 self.display(3,newFac)
64 else:
65 proj_factors = self.eliminate_var(proj_factors, v)

http://aipython.org Version 0.7.5 June 19, 2018

 9.1. Decision Networks 169

66 assert len(proj_factors)==1,"Should there be only one element of proj_factors?"

67 value = proj_factors[0].get_value({})
68 return value,policy

decnNetworks.py — (continued)

70 class Factor_max(Factor_stored):
71 """A factor obtained by maximizing a variable in a factor.
72 Also builds a decision_function. This is based on Factor_sum.
73 """
74
75 def __init__(self, dvar, factor):
76 """dvar is a decision variable.
77 factor is a factor that contains dvar and only parents of dvar
78 """
79 self.dvar = dvar
80 self.factor = factor
81 vars = [v for v in factor.variables if v is not dvar]
82 Factor_stored.__init__(self,vars,None)
83 self.values = [None]*self.size
84 self.decision_fun = Factor_DF(dvar,vars,[None]*self.size)
85
86 def get_value(self,assignment):
87 """lazy implementation: if saved, return saved value, else compute it"""
88 index = self.assignment_to_index(assignment)
89 if self.values[index]:
90 return self.values[index]
91 else:
92 max_val = float("-inf") # -infinity
93 new_asst = assignment.copy()
94 for elt in self.dvar.domain:
95 new_asst[self.dvar] = elt
96 fac_val = self.factor.get_value(new_asst)
97 if fac_val>max_val:
98 max_val = fac_val
99 best_elt = elt
100 self.values[index] = max_val
101 self.decision_fun.values[index] = best_elt
102 return max_val

A decision function is a stored factor.

decnNetworks.py — (continued)

104 class Factor_DF(Factor_stored):

105 """A decision function"""
106 def __init__(self,dvar, vars, values):
107 Factor_stored.__init__(self,vars,values)
108 self.dvar = dvar
109 self.name = str(dvar) # Used in printing

The fire decision network of Figure 9.1 is represented as:

http://aipython.org Version 0.7.5 June 19, 2018

 170 9. Planning with Uncertainty

Tampering Fire Utility

Smoke
Alarm

Leaving See_smoke

Check_smoke

Report Call

Figure 9.1: Fire Decision Network

decnNetworks.py — (continued)

111 boolean = [False, True]

112 Al = Variable("Alarm", boolean)
113 Fi = Variable("Fire", boolean)
114 Le = Variable("Leaving", boolean)
115 Re = Variable("Report", boolean)
116 Sm = Variable("Smoke", boolean)
117 Ta = Variable("Tamper", boolean)
118 SS = Variable("See Sm", boolean)
119 CS = DecisionVariable("Ch Sm", boolean,{Re})
120 Call = DecisionVariable("Call", boolean,{SS,CS,Re})
121
122 f_ta = Prob(Ta,[],[0.98,0.02])
123 f_fi = Prob(Fi,[],[0.99,0.01])
124 f_sm = Prob(Sm,[Fi],[0.99,0.01,0.1,0.9])
125 f_al = Prob(Al,[Fi,Ta],[0.9999, 0.0001, 0.15, 0.85, 0.01, 0.99, 0.5, 0.5])
126 f_lv = Prob(Le,[Al],[0.999, 0.001, 0.12, 0.88])
127 f_re = Prob(Re,[Le],[0.99, 0.01, 0.25, 0.75])
128 f_ss = Prob(SS,[CS,Sm],[1,0,1,0,1,0,0,1])
129
130 ut = Utility([CS,Fi,Call],[0,-200,-5000,-200,-20,-220,-5020,-220])
131
132 dnf = DecisionNetwork([Ta,Fi,Al,Le,Sm,Call,SS,CS,Re],[f_ta,f_fi,f_sm,f_al,f_lv,f_re,f_ss,ut])
133 # v,p = VE_DN(dnf).optimize()
134 # for df in p: print(df,"\n")

The following is the representation of the cheating decision of Figure 9.2. Note
that we keep the names of the variables short (less than 8 characters) so that
tables Python prints look good.

http://aipython.org Version 0.7.5 June 19, 2018

 9.1. Decision Networks 171

Watched Punishment

Caught1 Caught2

Cheat1 Cheat2 Utility

Grade1 Grade2

Final Grade

Figure 9.2: Cheating Decision Network

decnNetworks.py — (continued)

136 grades = ["A","B","C","F"]

137 Wa = Variable("Watched", boolean)
138 CC1 = Variable("Caught1", boolean)
139 CC2 = Variable("Caught2", boolean)
140 Pun = Variable("Punish",["None","Suspension","Recorded"])
141 Gr1 = Variable("Grade_1",grades)
142 Gr2 = Variable("Grade_2",grades)
143 GrF = Variable("Fin_Grd",grades)
144 Ch1 = DecisionVariable("Cheat_1", boolean,set()) #no parents
145 Ch2 = DecisionVariable("Cheat_2", boolean,{Ch1,CC1})
146
147 p_wa = Prob(Wa,[],[0.7, 0.3])
148 p_cc1 = Prob(CC1,[Wa,Ch1],[1.0, 0.0, 0.9, 0.1, 1.0, 0.0, 0.5, 0.5])
149 p_cc2 = Prob(CC2,[Wa,Ch2],[1.0, 0.0, 0.9, 0.1, 1.0, 0.0, 0.5, 0.5])
150 p_pun = Prob(Pun,[CC1,CC2],[1.0, 0.0, 0.0, 0.5, 0.4, 0.1, 0.6, 0.2, 0.2, 0.2, 0.5, 0.3])
151 p_gr1 = Prob(Gr1,[Ch1], [0.2, 0.3, 0.3, 0.2, 0.5, 0.3, 0.2, 0.0])
152 p_gr2 = Prob(Gr2,[Ch2], [0.2, 0.3, 0.3, 0.2, 0.5, 0.25, 0.25, 0.0])
153 p_fg = Prob(GrF,[Gr1,Gr2],
154 [1.0, 0.0, 0.0, 0.0, 0.5, 0.5, 0.0, 0.0, 0.25, 0.5, 0.25, 0.0, 0.25,
155 0.25, 0.25, 0.25, 0.5, 0.5, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.5,
156 0.5, 0.0, 0.0, 0.25, 0.5, 0.25, 0.25, 0.5, 0.25, 0.0, 0.0, 0.5, 0.5,
157 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.25, 0.75, 0.25, 0.5, 0.25, 0.0,
158 0.0, 0.25, 0.5, 0.25, 0.0, 0.0, 0.25, 0.75, 0.0, 0.0, 0.0, 1.0])
159 utc = Utility([Pun,GrF],[100,90,70,50,40,20,10,0,70,60,40,20])

http://aipython.org Version 0.7.5 June 19, 2018

 172 9. Planning with Uncertainty

160
161 cheat_dn = DecisionNetwork([Pun,CC2,Wa,GrF,Gr2,Gr1,Ch2,CC1,Ch1],
162 [p_wa, p_cc1, p_cc2, p_pun, p_gr1, p_gr2,p_fg,utc])
163
164 # VE_DN.max_display_level = 3 # if you want to show lots of detail
165 # v,p = VE_DN(cheat_dn).optimize(); print(v)
166 # for df in p: print(df,"\n") # print decision functions

9.2 Markov Decision Processes

We will represent a Markov decision process (MDP) directly, rather than using
the variable elimination code, as we did for decision networks.
States and actions are represented as lists of strings. The data structures for
transitions, rewards, q-values, etc., use the index of the state or the action. The
names of the state with index i is in states[i], and the name of action with index
i is in actions[i].
mdpProblem.py — Representations for Markov Decision Processes
11 from utilities import argmax
12
13 class MDP(object):
14 def __init__(self, states, actions, trans, reward, discount):
15 """states is a list or tuple of states.
16 actions is a list or tuple of actions
17 trans[s][a][s'] represents P(s'|a,s)
18 reward[s][a] gives the expected reward of doing a in state s
19 discount is a real in the range [0,1]
20 """
21 self.states = states
22 self.actions = actions
23 self.trans = trans
24 self.reward = reward
25 self.discount = discount
26 self.v0 = [0 for s in states] # initial value function
2 state partying example:
mdpExamples.py — MDP Examples
11 from mdpProblem import MDP
12 #### Partying Decision Example ####
13
14 # States: Healthy Sick
15 # Actions: Relax Party
16
17 # trans[s][a][s'] gives P(s'|a,s)
18 # Relax Party
19 trans2 = (((0.95,0.05), (0.7, 0.3)), # Healthy
20 ((0.5,0.5), (0.1, 0.9)) # Sick
21 )

http://aipython.org Version 0.7.5 June 19, 2018

 9.3. Value Iteration 173

22
23 # reward[s][a] gives the expected reward of doing a in state s.
24 reward2 = ((7,10),(0,2))
25
26 healthy2 = MDP(['Healthy','Sick'], ['Relax','Party'], trans2, reward2, discount=0.8)
Tiny game from Example 11.7 and Figure 11.8 of Poole and Mackworth,
2010:
mdpExamples.py — (continued)

28 ## Tiny Game from Example 11.7 and Figure 11.8 of Poole and Mackworth, 2010 #
29
30 # actions up right upC left
31 transt = (((0.1,0.1,0.8,0,0,0), (0,1,0,0,0,0), (0,0,1,0,0,0), (1,0,0,0,0,0)), #s0
32 ((0.1,0.1,0,0.8,0,0), (0,1,0,0,0,0), (0,0,0,1,0,0), (1,0,0,0,0,0)),
#s1
33 ((0,0,0.1,0.1,0.8,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,1,0,0,0)),
#s2
34 ((0,0,0.1,0.1,0,0.8), (0,0,0,1,0,0), (0,0,0,0,0,1), (0,0,1,0,0,0)),
#s3
35 ((0.1,0,0,0,0.8,0.1), (0,0,0,0,0,1), (0,0,0,0,1,0), (1,0,0,0,0,0)),
#s4
36 ((0,0,0,0,0.1,0.9), (0,0,0,0,0,1), (0,0,0,0,0,1), (0,0,0,0,1,0)) ) #s5
37
38 # actions up rt upC left
39 rewardt = ((-0.1, 0, -1, -1), #s0
40 (-0.1, -1, -2, 0), #s1
41 (-10, 0, -1, -100), #s2
42 (-0.1, -1, -1, 0), #s3
43 (-1, 0, -2, 10), #s4
44 (-1, -1, -2, 0)) #s5
45
46 mdpt = MDP(['s0','s1','s2','s3','s4','s5'], # states
47 ['up', 'right', 'upC', 'left'], # actions
48 transt, rewardt, discount=0.9)

9.3 Value Iteration

This implements value iteration, storing V.
This uses indexes of the states and actions (not the names). A value function
is list, v, such that v[s] is the value for state with index s. Similarly a policy pi
is represented as a list where pi[s], where s is the index of a state, returns the
index of the action.
mdpProblem.py — (continued)

28 def vi1(self,v):
29 """carry out one iteration of value iteration and
30 returns a value function (a list of a value for each state).
31 v is the previous value function.

http://aipython.org Version 0.7.5 June 19, 2018

 174 9. Planning with Uncertainty

32 """
33 return [max([self.reward[s][a]+self.discount*product(self.trans[s][a],v)
34 for a in range(len(self.actions))])
35 for s in range(len(self.states))]
36
37 def vi(self,v0,n):
38 """carries out n iterations of value iteration starting with value v0.
39
40 Returns a value function
41 """
42 val = self.v0
43 for i in range(n):
44 val= self.vi1(val)
45 return val
46
47 def policy(self,v):
48 """returns an optimal policy assuming the next value function is v
49 v is a list of values for each state
50 returns a list of the indexes of optimal actions for each state
51 """
52 return [argmax(enumerate([self.reward[s][a]+self.discount*product(self.trans[s][a],v)
53 for a in range(len(self.actions))]))
54 for s in range(len(self.states))]
55
56 def q(self,v):
57 """returns the one-step-lookahead q-value assuming the next value function is v
58 v is a list of values for each state
59 returns a list of q values for each state. so that q[s][a] represents Q(s,a)
60 """
61 return [[self.reward[s][a]+self.discount*product(self.trans[s][a],v)
62 for a in range(len(self.actions))]
63 for s in range(len(self.states))]

mdpProblem.py — (continued)

65 def product(l1,l2):
66 """returns the dot product of l1 and l2"""
67 return sum([i1*i2 for (i1,i2) in zip(l1,l2)])
The following gives a trace for the examples:
mdpExamples.py — (continued)

50 def trace(mdp,numiter):
51 print("Q values are shown as",[[st+"_"+ac for ac in mdp.actions] for st in mdp.states])
52 print("One step lookahead Q-values:")
53 print(mdp.q(mdp.v0))
54 print("Values are for the states:", mdp.states)
55 print("One step lookahead values:")
56 print(mdp.vi(mdp.v0,1))
57 print("Two step lookahead Q-values:")
58 print(mdp.q(mdp.vi(mdp.v0,1)))
59 print("Two step lookahead values:")

http://aipython.org Version 0.7.5 June 19, 2018

 9.3. Value Iteration 175

60 print(mdp.vi(mdp.v0,2))
61 vfin = mdp.vi(mdp.v0,numiter)
62 print("After",numiter,"iterations, values:")
63 print(vfin)
64 print("After",numiter,"iterations, Q-values:")
65 print(mdp.q(vfin))
66 print("After",numiter,"iterations, Policy:",
67 [st+"->"+mdp.actions[act] for (st,act) in zip(mdp.states ,mdp.policy(vfin))])
68
69 # Try the following:
70 # trace(healthy2,10)

Exercise 9.1 Implement value iteration that stores the Q-values rather than the
V-values. Does it work better than storing V? (What might better mean?)
Exercise 9.2 Implement asynchronous value iteration. Try a number of differ-
ent ways to choose the states and actions to update (e.g., sweeping through the
state-action pairs, choosing them at random). Note that the best way may be to
determine which states have had their Q-values change the most, and then update
the previous ones, but that is not so straightforward to implement, because you
need to find those previous states.

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 10

Learning with Uncertainty

10.1 K-means
The k-means learner maintains two lists that suffice as sufficient statistics to
classify examples, and to learn the classification:

• class counts is a list such that class counts[c] is the number of examples in
the training set with class = c.

• feature sum is a list such that feature sum[i][c] is sum of the values for the
i’th feature i for members of class c. The average value of the ith feature
in class i is
feature sum[i][c]
class counts[c]

The class is initialized by randomly assigning examples to classes, and updat-

ing the statistics for class counts and feature sum.
learnKMeans.py — k-means learning
11 from learnProblem import Data_set, Learner, Data_from_file
12 import random
13 import matplotlib.pyplot as plt
14
15 class K_means_learner(Learner):
16 def __init__(self,dataset, num_classes):
17 self.dataset = dataset
18 self.num_classes = num_classes
19 self.random_initialize()
20
21 def random_initialize(self):

177
 178 10. Learning with Uncertainty

22 # class_counts[c] is the number of examples with class=c

23 self.class_counts = [0]*self.num_classes
24 # feature_sum[i][c] is the sum of the values of feature i for class c
25 self.feature_sum = [[0]*self.num_classes
26 for feat in self.dataset.input_features]
27 for eg in self.dataset.train:
28 cl = random.randrange(self.num_classes) # assign eg to random class
29 self.class_counts[cl] += 1
30 for (ind,feat) in enumerate(self.dataset.input_features):
31 self.feature_sum[ind][cl] += feat(eg)
32 self.num_iterations = 0
33 self.display(1,"Initial class counts: ",self.class_counts)
The distance from (the mean of) a class to an example is the sum, over all
fratures, of the sum-of-squares differences of the class mean and the example
value.
learnKMeans.py — (continued)

35 def distance(self,cl,eg):
36 """distance of the eg from the mean of the class"""
37 return sum( (self.class_prediction(ind,cl)-feat(eg))**2
38 for (ind,feat) in enumerate(self.dataset.input_features))
39
40 def class_prediction(self,feat_ind,cl):
41 """prediction of the class cl on the feature with index feat_ind"""
42 if self.class_counts[cl] == 0:
43 return 0 # there are no examples so we can choose any value
44 else:
45 return self.feature_sum[feat_ind][cl]/self.class_counts[cl]
46
47 def class_of_eg(self,eg):
48 """class to which eg is assigned"""
49 return (min((self.distance(cl,eg),cl)
50 for cl in range(self.num_classes)))[1]
51 # second element of tuple, which is a class with minimum distance
One step of k-means updates the class counts and feature sum. It uses the old
values to determine the classes, and so the new values for class counts and
feature sum. At the end it determines whether the values of these have changes,
and then replaces the old ones with the new ones. It returns an indicator of
whether the values are stable (have not changed).
learnKMeans.py — (continued)

53 def k_means_step(self):
54 """Updates the model with one step of k-means.
55 Returns whether the assignment is stable.
56 """
57 new_class_counts = [0]*self.num_classes
58 # feature_sum[i][c] is the sum of the values of feature i for class c
59 new_feature_sum = [[0]*self.num_classes
60 for feat in self.dataset.input_features]

http://aipython.org Version 0.7.5 June 19, 2018

 10.1. K-means 179

61 for eg in self.dataset.train:
62 cl = self.class_of_eg(eg)
63 new_class_counts[cl] += 1
64 for (ind,feat) in enumerate(self.dataset.input_features):
65 new_feature_sum[ind][cl] += feat(eg)
66 stable = (new_class_counts == self.class_counts) and (self.feature_sum == new_feature_sum)
67 self.class_counts = new_class_counts
68 self.feature_sum = new_feature_sum
69 self.num_iterations += 1
70 return stable
71
72
73 def learn(self,n=100):
74 """do n steps of k-means, or until convergence"""
75 i=0
76 stable = False
77 while i<n and not stable:
78 stable = self.k_means_step()
79 i += 1
80 self.display(1,"Iteration",self.num_iterations,
81 "class counts: ",self.class_counts," Stable=",stable)
82 return stable
83
84 def show_classes(self):
85 """sorts the data by the class and prints in order.
86 For visualizing small data sets
87 """
88 class_examples = [[] for i in range(self.num_classes)]
89 for eg in self.dataset.train:
90 class_examples[self.class_of_eg(eg)].append(eg)
91 print("Class","Example",sep='\t')
92 for cl in range(self.num_classes):
93 for eg in class_examples[cl]:
94 print(cl,*eg,sep='\t')
95
96 def plot_error(self, maxstep=20):
97 """Plots the sum-of-suares error as a function of the number of steps"""
98 plt.ion()
99 plt.xlabel("step")
100 plt.ylabel("Ave sum-of-squares error")
101 train_errors = []
102 if self.dataset.test:
103 test_errors = []
104 for i in range(maxstep):
105 self.learn(1)
106 train_errors.append( sum(self.distance(self.class_of_eg(eg),eg)
107 for eg in self.dataset.train)
108 /len(self.dataset.train))
109 if self.dataset.test:
110 test_errors.append( sum(self.distance(self.class_of_eg(eg),eg)

http://aipython.org Version 0.7.5 June 19, 2018

 180 10. Learning with Uncertainty

111 for eg in self.dataset.test)

112 /len(self.dataset.test))
113 plt.plot(range(1,maxstep+1),train_errors,
114 label=str(self.num_classes)+" classes. Training set")
115 if self.dataset.test:
116 plt.plot(range(1,maxstep+1),test_errors,
117 label=str(self.num_classes)+" classes. Test set")
118 plt.legend()
119 plt.draw()
120
121 %data = Data_from_file('data/emdata1.csv', num_train=10, target_index=2000) % trivial example
122 data = Data_from_file('data/emdata2.csv', num_train=10, target_index=2000)
123 %data = Data_from_file('data/emdata0.csv', num_train=14, target_index=2000) % example from textbook
124 kml = K_means_learner(data,2)
125 num_iter=4
126 print("Class assignment after",num_iter,"iterations:")
127 kml.learn(num_iter); kml.show_classes()
128
129 # Plot the error
130 # km2=K_means_learner(data,2); km2.plot_error(20) # 2 classes
131 # km3=K_means_learner(data,3); km3.plot_error(20) # 3 classes
132 # km13=K_means_learner(data,13); km13.plot_error(20) # 13 classes
133
134 # data = Data_from_file('data/carbool.csv', target_index=2000,boolean_features=True)
135 # kml = K_means_learner(data,3)
136 # kml.learn(20); kml.show_classes()
137 # km3=K_means_learner(data,3); km3.plot_error(20) # 3 classes
138 # km3=K_means_learner(data,30); km3.plot_error(20) # 30 classes

Exercise 10.1 Change boolean features = True flag to allow for numerical features.
K-means assumes the features are numerical, so we want to make non-numerical
features into numerical features (using characteristic functions) but we probably
don’t want to change numerical features into Boolean.

Exercise 10.2 If there are many classes, some of the classes can become empty
(e.g., try 100 classes with carbool.csv). Implement a way to put some examples
into a class, if possible. Two ideas are:

(a) Initialize the classes with actual examples, so that the classes will not start
empty. (Do the classes become empty?)

(b) In class prediction, we test whether the code is empty, and make a prediction
of 0 for an empty class. It is possible to make a different prediction to “steal”
an example (but you should make sure that a class has a consistent value for
each feature in a loop).

Make your own suggestions, and compare it with the original, and whichever of
these you think may work better.

http://aipython.org Version 0.7.5 June 19, 2018

 10.2. EM 181

10.2 EM
In the following definition, a class, c, is a integer in range [0, num classes). i is
an index of a feature, so feat[i] is the ith feature, and a feature is a function from
tuples to values. val is a value of a feature.
A model consists of 2 lists, which form the sufficient statistics:

• class counts is a list such that class counts[c] is the number of tuples with
class = c, where each tuple is weighted by its probability, i.e.,

class counts[c] = ∑ P(t)

t:class(t)=c

• feature counts is a list such that feature counts[i][val][c] is the weighted count
of the number of tuples t with feat[i](t) = val and class(t) = c, each tuple
is weighted by its probability, i.e.,

feature counts[i][val][c] = ∑ P(t)

t:feat[i](t)=val andclass(t)=c

learnEM.py — EM Learning
11 from learnProblem import Data_set, Learner, Data_from_file
12 import random
13 import math
14 import matplotlib.pyplot as plt
15
16 class EM_learner(Learner):
17 def __init__(self,dataset, num_classes):
18 self.dataset = dataset
19 self.num_classes = num_classes
20 self.class_counts = None
21 self.feature_counts = None

The function em step goes though the training examples, and updates these
counts. The first time it is run, when there is no model, it uses random distri-
butions.
learnEM.py — (continued)

23 def em_step(self, orig_class_counts, orig_feature_counts):

24 """updates the model."""
25 class_counts = [0]*self.num_classes
26 feature_counts = [{val:[0]*self.num_classes
27 for val in feat.frange}
28 for feat in self.dataset.input_features]
29 for tple in self.dataset.train:
30 if orig_class_counts: # a model exists
31 tpl_class_dist = self.prob(tple, orig_class_counts, orig_feature_counts)
32 else: # initially, with no model, return a random distribution

http://aipython.org Version 0.7.5 June 19, 2018

 182 10. Learning with Uncertainty

33 tpl_class_dist = random_dist(self.num_classes)
34 for cl in range(self.num_classes):
35 class_counts[cl] += tpl_class_dist[cl]
36 for (ind,feat) in enumerate(self.dataset.input_features):
37 feature_counts[ind][feat(tple)][cl] += tpl_class_dist[cl]
38 return class_counts, feature_counts
prob computes the probability of a class for a tuple, given the current statistics.

P(c | tple) ∝ P(c) ∗ ∏ P(Xi =tple(i) | c)

i
class counts[c] feature counts[i][feati (tple)][c]
len(self .dataset) ∏
= ∗
i
class counts[c]

len(self .dataset) is a constant (independent of c). class counts[c] can be taken out
of the product, but needs to be raised to the power of the number of features,
and one of them cancels.
learnEM.py — (continued)

40 def prob(self,tple,class_counts,feature_counts):
41 """returns a distribution over the classes for the original tuple in the current model
42 """
43 feats = self.dataset.input_features
44 unnorm = [prod(feature_counts[i][feat(tple)][c]
45 for (i,feat) in enumerate(feats))/(class_counts[c]**(len(feats)-1))
46 for c in range(self.num_classes)]
47 thesum = sum(unnorm)
48 return [un/thesum for un in unnorm]
learn does n steps of EM:
learnEM.py — (continued)

50 def learn(self,n):
51 """do n steps of em"""
52 for i in range(n):
53 self.class_counts,self.feature_counts = self.em_step(self.class_counts,
54 self.feature_counts)
The following is for visualizing the classes. It prints the dataset ordered by the
probability of class c.
learnEM.py — (continued)

56 def show_class(self,c):
57 """sorts the data by the class and prints in order.
58 For visualizing small data sets
59 """
60 sorted_data = sorted((self.prob(tpl,self.class_counts,self.feature_counts)[c],
61 ind, # preserve ordering for equal probabilities
62 tpl)
63 for (ind,tpl) in enumerate(self.dataset.train))
64 for cc,r,tpl in sorted_data:
65 print(cc,*tpl,sep='\t')

http://aipython.org Version 0.7.5 June 19, 2018

 10.2. EM 183

The following are for evaluating the classes.

The probability of a tuple can be evaluated by marginalizing over the classes:

P(tple) = ∑ P(c) ∗ ∏ P(Xi =tple(i) | c)

c i
cc[c] fc[i][feati (tple)][c]
=∑
len(self .dataset) ∏
∗
c i
cc[c]

where cc is the class count and fc is feature count. len(self .dataset) can be dis-
tributed out of the sum, and cc[c] can be taken out of the product:

1 1
=
len(self .dataset) ∑ cc[c]#feats−1 ∗ ∏ fc[i][feati (tple)][c]
c i

Given the probability of each tuple, we can evaluate the logloss, as the negative
of the log probability:
learnEM.py — (continued)

67 def logloss(self,tple):
68 """returns the logloss of the prediction on tple, which is -log(P(tple))
69 based on the current class counts and feature counts
70 """
71 feats = self.dataset.input_features
72 res = 0
73 cc = self.class_counts
74 fc = self.feature_counts
75 for c in range(self.num_classes):
76 res += prod(fc[i][feat(tple)][c]
77 for (i,feat) in enumerate(feats))/(cc[c]**(len(feats)-1))
78 if res>0:
79 return -math.log2(res/len(self.dataset.train))
80 else:
81 return float("inf") #infinity
82
83 def plot_error(self, maxstep=20):
84 """Plots the logloss error as a function of the number of steps"""
85 plt.ion()
86 plt.xlabel("step")
87 plt.ylabel("Ave Logloss (bits)")
88 train_errors = []
89 if self.dataset.test:
90 test_errors = []
91 for i in range(maxstep):
92 self.learn(1)
93 train_errors.append( sum(self.logloss(tple) for tple in self.dataset.train)
94 /len(self.dataset.train))
95 if self.dataset.test:
96 test_errors.append( sum(self.logloss(tple) for tple in self.dataset.test)
97 /len(self.dataset.test))
98 plt.plot(range(1,maxstep+1),train_errors,

http://aipython.org Version 0.7.5 June 19, 2018

 184 10. Learning with Uncertainty

99 label=str(self.num_classes)+" classes. Training set")

100 if self.dataset.test:
101 plt.plot(range(1,maxstep+1),test_errors,
102 label=str(self.num_classes)+" classes. Test set")
103 plt.legend()
104 plt.draw()
105
106 def prod(L):
107 """returns the product of the elements of L"""
108 res = 1
109 for e in L:
110 res *= e
111 return res
112
113 def random_dist(k):
114 """generate k random numbers that sum to 1"""
115 res = [random.random() for i in range(k)]
116 s = sum(res)
117 return [v/s for v in res]
118
119 data = Data_from_file('data/emdata2.csv', num_train=10, target_index=2000)
120 eml = EM_learner(data,2)
121 num_iter=2
122 print("Class assignment after",num_iter,"iterations:")
123 eml.learn(num_iter); eml.show_class(0)
124
125 # Plot the error
126 # em2=EM_learner(data,2); em2.plot_error(40) # 2 classes
127 # em3=EM_learner(data,3); em3.plot_error(40) # 3 classes
128 # em13=EM_learner(data,13); em13.plot_error(40) # 13 classes
129
130 # data = Data_from_file('data/carbool.csv', target_index=2000,boolean_features=False)
131 # [f.frange for f in data.input_features]
132 # eml = EM_learner(data,3)
133 # eml.learn(20); eml.show_class(0)
134 # em3=EM_learner(data,3); em3.plot_error(60) # 3 classes
135 # em3=EM_learner(data,30); em3.plot_error(60) # 30 classes

Exercise 10.3 For the EM data, where there are naturally 2 classes, 3 classes does
better on the training set after a while than 2 classes, but worse on the test set.
Explain why. Hint: look what the 3 classes are. Use ”em3.show class(i)” for each
of the classes i ∈ [0, 3).
Exercise 10.4 Write code to plot the logloss as a function of the number of classes
(from 1 to say 15) for a fixed number of iterations. (From the experience with the
existing code, think about how many iterations is appropriate.)

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 11

Multiagent Systems

11.1 Minimax
Here we consider two-player zero-sum games. Here a player only wins when
another player loses. This can be modeled as where there is a single utility
which one agent (the maximizing agent) is trying minimize and the other agent
(the minimizing agent) is trying to minimize.

11.1.1 Creating a two-player game

masProblem.py — A Multiagent Problem

11 from utilities import Displayable
12
13 class Node(Displayable):
14 """A node in a search tree. It has a
15 name a string
16 isMax is True if it is a maximizing node, otherwise it is minimizing node
17 children is the list of children
18 value is what it evaluates to if it is a leaf.
19 """
20 def __init__(self, name, isMax, value, children):
21 self.name = name
22 self.isMax = isMax
23 self.value = value
24 self.allchildren = children
25
26 def isLeaf(self):
27 """returns true of this is a leaf node"""
28 return self.allchildren is None
29

185
 186 11. Multiagent Systems

30 def children(self):
31 """returns the list of all children."""
32 return self.allchildren
33
34 def evaluate(self):
35 """returns the evaluation for this node if it is a leaf"""
36 return self.value
The following gives the tree from Figure 10.5 of the book. Note how 888 is used
as a value here, but never appears in the trace.
masProblem.py — (continued)

38 fig10_5 = Node("a",True,None, [
39 Node("b",False,None, [
40 Node("d",True,None, [
41 Node("h",False,None, [
42 Node("h1",True,7,None),
43 Node("h2",True,9,None)]),
44 Node("i",False,None, [
45 Node("i1",True,6,None),
46 Node("i2",True,888,None)])]),
47 Node("e",True,None, [
48 Node("j",False,None, [
49 Node("j1",True,11,None),
50 Node("j2",True,12,None)]),
51 Node("k",False,None, [
52 Node("k1",True,888,None),
53 Node("k2",True,888,None)])])]),
54 Node("c",False,None, [
55 Node("f",True,None, [
56 Node("l",False,None, [
57 Node("l1",True,5,None),
58 Node("l2",True,888,None)]),
59 Node("m",False,None, [
60 Node("m1",True,4,None),
61 Node("m2",True,888,None)])]),
62 Node("g",True,None, [
63 Node("n",False,None, [
64 Node("n1",True,888,None),
65 Node("n2",True,888,None)]),
66 Node("o",False,None, [
67 Node("o1",True,888,None),
68 Node("o2",True,888,None)])])])])
The following is a representation of a magic-sum game, where players take
turns picking a number in the range [1, 9], and the first player to have 3 num-
bers that sum to 15 wins. Note that this is a syntactic variant of tic-tac-toe or
naughts and crosses. To see this, consider the numbers on a magic square (Fig-
ure 11.1); 3 numbers that add to 15 correspond exactly to the winning positions
of tic-tac-toe played on the magic square.
Note that we do not remove symmetries. (What are the symmetries? How

http://aipython.org Version 0.7.5 June 19, 2018

 11.1. Minimax 187

6 1 8
7 5 3
2 9 4

Figure 11.1: Magic Square

do the symmetries of tic-tac-toe translate here?)

masProblem.py — (continued)

70
71 class Magic_sum(Node):
72 def __init__(self, xmove=True, last_move=None,
73 available=[1,2,3,4,5,6,7,8,9], x=[], o=[]):
74 """This is a node in the search for the magic-sum game.
75 xmove is True if the next move belongs to X.
76 last_move is the number selected in the last move
77 available is the list of numbers that are available to be chosen
78 x is the list of numbers already chosen by x
79 o is the list of numbers already chosen by o
80 """
81 self.isMax = self.xmove = xmove
82 self.last_move = last_move
83 self.available = available
84 self.x = x
85 self.o = o
86 self.allchildren = None #computed on demand
87 lm = str(last_move)
88 self.name = "start" if not last_move else "o="+lm if xmove else "x="+lm
89
90 def children(self):
91 if self.allchildren is None:
92 if self.xmove:
93 self.allchildren = [
94 Magic_sum(xmove = not self.xmove,
95 last_move = sel,
96 available = [e for e in self.available if e is not sel],
97 x = self.x+[sel],
98 o = self.o)
99 for sel in self.available]
100 else:
101 self.allchildren = [
102 Magic_sum(xmove = not self.xmove,
103 last_move = sel,
104 available = [e for e in self.available if e is not sel],
105 x = self.x,
106 o = self.o+[sel])
107 for sel in self.available]
108 return self.allchildren
109

http://aipython.org Version 0.7.5 June 19, 2018

 188 11. Multiagent Systems

110 def isLeaf(self):

111 """A leaf has no numbers available or is a win for one of the players.
112 We only need to check for a win for o if it is currently x's turn,
113 and only check for a win for x if it is o's turn (otherwise it would
114 have been a win earlier).
115 """
116 return (self.available == [] or
117 (sum_to_15(self.last_move,self.o)
118 if self.xmove
119 else sum_to_15(self.last_move,self.x)))
120
121 def evaluate(self):
122 if self.xmove and sum_to_15(self.last_move,self.o):
123 return -1
124 elif not self.xmove and sum_to_15(self.last_move,self.x):
125 return 1
126 else:
127 return 0
128
129 def sum_to_15(last,selected):
130 """is true if last, toegether with two other elements of selected sum to 15.
131 """
132 return any(last+a+b == 15
133 for a in selected if a != last
134 for b in selected if b != last and b != a)

11.1.2 Minimax and α-β Pruning

This is a naive depth-first minimax algorithm:

masMiniMax.py — Minimax search with alpha-beta pruning

11 def minimax(node):
12 """returns the value of node, and a best path for the agents
13 """
14 if node.isLeaf():
15 return node.evaluate(),None
16 elif node.isMax:
17 max_score = -999
18 max_path = None
19 for C in node.children():
20 score,path = minimax(C,depth+1)
21 if score > max_score:
22 max_score = score
23 max_path = C.name,path
24 return max_score,max_path
25 else:
26 min_score = 999
27 min_path = None
28 for C in node.children():
29 score,path = minimax(C,depth+1)
30 if score < min_score:

http://aipython.org Version 0.7.5 June 19, 2018

 11.1. Minimax 189

31 min_score = score
32 min_path = C.name,path
33 return min_score,min_path
The following is a depth-first minimax with α-β pruning. It returns the
value for a node as well as a best path for the agents.
masMiniMax.py — (continued)

35 def minimax_alpha_beta(node,alpha,beta,depth=0):
36 """node is a Node, alpha and beta are cutoffs, depth is the depth
37 returns value, path
38 where path is a sequence of nodes that results in the value"""
39 node.display(2," "*depth,"minimax_alpha_beta(",node.name,", ",alpha, ", ", beta,")")
40 best=None # only used if it will be pruned
41 if node.isLeaf():
42 node.display(2," "*depth,"returning leaf value",node.evaluate())
43 return node.evaluate(),None
44 elif node.isMax:
45 for C in node.children():
46 score,path = minimax_alpha_beta(C,alpha,beta,depth+1)
47 if score >= beta: # beta pruning
48 node.display(2," "*depth,"pruned due to beta=",beta,"C=",C.name)
49 return score, None
50 if score > alpha:
51 alpha = score
52 best = C.name, path
53 node.display(2," "*depth,"returning max alpha",alpha,"best",best)
54 return alpha,best
55 else:
56 for C in node.children():
57 score,path = minimax_alpha_beta(C,alpha,beta,depth+1)
58 if score <= alpha: # alpha pruning
59 node.display(2," "*depth,"pruned due to alpha=",alpha,"C=",C.name)
60 return score, None
61 if score < beta:
62 beta=score
63 best = C.name,path
64 node.display(2," "*depth,"returning min beta",beta,"best=",best)
65 return beta,best
Testing:
masMiniMax.py — (continued)

67 from masProblem import fig10_5, Magic_sum, Node

68
69 # Node.max_display_level=2 # print detailed trace
70 # minimax_alpha_beta(fig10_5, -9999, 9999,0)
71 # minimax_alpha_beta(Magic_sum(), -9999, 9999,0)
72
73 #To see how much time alpha-beta pruning can save over minimax, uncomment the following:
74 ## import timeit
75 ## timeit.Timer("minimax(Magic_sum())",setup="from __main__ import minimax, Magic_sum"

http://aipython.org Version 0.7.5 June 19, 2018

 190 11. Multiagent Systems

76 ## ).timeit(number=1)
77 ## trace=False
78 ## timeit.Timer("minimax_alpha_beta(Magic_sum(), -9999, 9999,0)",
79 ## setup="from __main__ import minimax_alpha_beta, Magic_sum"
80 ## ).timeit(number=1)

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 12

Reinforcement Learning

12.1 Representing Agents and Environments

When the learning agent does an action in the environment, it observes a (state, reward)
pair from the environment. The state is the world state; this is the fully observ-
able assumption.
An RL environment implements a do(action) method that returns a (state, reward)
pair.
rlProblem.py — Representations for Reinforcement Learning
11 import random
12 from utilities import Displayable, flip
13
14 class RL_env(Displayable):
15 def __init__(self,actions,state):
16 self.actions = actions # set of actions
17 self.state = state # initial state
18
19 def do(self, action):
20 """do action
21 returns state,reward
22 """
23 raise NotImplementedError("RL_env.do") # abstract method
Here is the definition of the simple 2-state, 2-action party/relax decision.
rlProblem.py — (continued)

25 class Healthy_env(RL_env):
26 def __init__(self):
27 RL_env.__init__(self,["party","relax"], "healthy")
28
29 def do(self, action):

191
 192 12. Reinforcement Learning

30 """updates the state based on the agent doing action.

31 returns state,reward
32 """
33 if self.state=="healthy":
34 if action=="party":
35 self.state = "healthy" if flip(0.7) else "sick"
36 reward = 10
37 else: # action=="relax"
38 self.state = "healthy" if flip(0.95) else "sick"
39 reward = 7
40 else: # self.state=="sick"
41 if action=="party":
42 self.state = "healthy" if flip(0.1) else "sick"
43 reward = 2
44 else:
45 self.state = "healthy" if flip(0.5) else "sick"
46 reward = 0
47 return self.state,reward

12.1.1 Simulating an environment from an MDP

Given the definition for an MDP (page 172), Env from MDP takes in an MDP
and simulates the environment with those dynamics.
Note that the MDP does not contain enough information to simulate a sys-
tem, because it loses any dependency between the rewards and the resulting
state; here we assume the agent always received the average reward for the
state and action.
rlProblem.py — (continued)

49 class Env_from_MDP(RL_env):
50 def __init__(self, mdp):
51 initial_state = mdp.states[0]
52 RL_env.__init__(self,mdp.actions, initial_state)
53 self.mdp = mdp
54 self.action_index = {action:index for (index,action) in enumerate(mdp.actions)}
55 self.state_index = {state:index for (index,state) in enumerate(mdp.states)}
56
57 def do(self, action):
58 """updates the state based on the agent doing action.
59 returns state,reward
60 """
61 action_ind = self.action_index[action]
62 state_ind = self.state_index[self.state]
63 self.state = pick_from_dist(self.mdp.trans[state_ind][action_ind], self.mdp.states)
64 reward = self.mdp.reward[state_ind][action_ind]
65 return self.state, reward
66
67 def pick_from_dist(dist,values):
68 """

http://aipython.org Version 0.7.5 June 19, 2018

 12.1. Representing Agents and Environments 193

4 P1 R P2

3 M

2 M

1 M M M

0 P3 P4

0 1 2 3 4

Figure 12.1: Monster game

69 e.g. pick_from_dist([0.3,0.5,0.2],['a','b','c']) should pick 'a' with probability 0.3, etc.

70 """
71 ran = random.random()
72 i=0
73 while ran>dist[i]:
74 ran -= dist[i]
75 i += 1
76 return values[i]

12.1.2 Simple Game

This is for the game depicted in Figure 12.1.
rlSimpleEnv.py — Simple game
11 import random
12 from utilities import flip
13 from rlProblem import RL_env
14
15 class Simple_game_env(RL_env):
16 xdim = 5
17 ydim = 5
18
19 vwalls = [(0,3), (0,4), (1,4)] # vertical walls right of these locations
20 hwalls = [] # not implemented
21 crashed_reward = -1
22
23 prize_locs = [(0,0), (0,4), (4,0), (4,4)]
24 prize_apears_prob = 0.3
25 prize_reward = 10
26

http://aipython.org Version 0.7.5 June 19, 2018

 194 12. Reinforcement Learning

27 monster_locs = [(0,1), (1,1), (2,3), (3,1), (4,2)]

28 monster_appears_prob = 0.4
29 monster_reward_when_damaged = -10
30 repair_stations = [(1,4)]
31
32 actions = ["up","down","left","right"]
33
34 def __init__(self):
35 # State:
36 self.x = 2
37 self.y = 2
38 self.damaged = False
39 self.prize = None
40 # Statistics
41 self.number_steps = 0
42 self.total_reward = 0
43 self.min_reward = 0
44 self.min_step = 0
45 self.zero_crossing = 0
46 RL_env.__init__(self, Simple_game_env.actions,
47 (self.x, self.y, self.damaged, self.prize))
48 self.display(2,"","Step","Tot Rew","Ave Rew",sep="\t")
49
50 def do(self,action):
51 """updates the state based on the agent doing action.
52 returns state,reward
53 """
54 reward = 0.0
55 # A prize can appear:
56 if self.prize is None and flip(self.prize_apears_prob):
57 self.prize = random.choice(self.prize_locs)
58 # Actions can be noisy
59 if flip(0.4):
60 actual_direction = random.choice(self.actions)
61 else:
62 actual_direction = action
63 # Modeling the actions given the actual direction
64 if actual_direction == "right":
65 if self.x==self.xdim-1 or (self.x,self.y) in self.vwalls:
66 reward += self.crashed_reward
67 else:
68 self.x += 1
69 elif actual_direction == "left":
70 if self.x==0 or (self.x-1,self.y) in self.vwalls:
71 reward += self.crashed_reward
72 else:
73 self.x += -1
74 elif actual_direction == "up":
75 if self.y==self.ydim-1:
76 reward += self.crashed_reward

http://aipython.org Version 0.7.5 June 19, 2018

 12.1. Representing Agents and Environments 195

77 else:
78 self.y += 1
79 elif actual_direction == "down":
80 if self.y==0:
81 reward += self.crashed_reward
82 else:
83 self.y += -1
84 else:
85 raise RuntimeError("unknown_direction "+str(direction))
86
87 # Monsters
88 if (self.x,self.y) in self.monster_locs and flip(self.monster_appears_prob):
89 if self.damaged:
90 reward += self.monster_reward_when_damaged
91 else:
92 self.damaged = True
93 if (self.x,self.y) in self.repair_stations:
94 self.damaged = False
95
96 # Prizes
97 if (self.x,self.y) == self.prize:
98 reward += self.prize_reward
99 self.prize = None
100
101 # Statistics
102 self.number_steps += 1
103 self.total_reward += reward
104 if self.total_reward < self.min_reward:
105 self.min_reward = self.total_reward
106 self.min_step = self.number_steps
107 if self.total_reward>0 and reward>self.total_reward:
108 self.zero_crossing = self.number_steps
109 self.display(2,"",self.number_steps,self.total_reward,
110 self.total_reward/self.number_steps,sep="\t")
111
112 return (self.x, self.y, self.damaged, self.prize), reward

12.1.3 Evaluation and Plotting

rlPlot.py — RL Plotter
11 import matplotlib.pyplot as plt
12
13 def plot_rl(ag, label=None, yplot='Total', step_size=None,
14 steps_explore=1000, steps_exploit=1000, xscale='linear'):
15 """
16 plots the agent ag
17 label is the label for the plot
18 yplot is 'Average' or 'Total'
19 step_size is the number of steps between each point plotted

http://aipython.org Version 0.7.5 June 19, 2018

 196 12. Reinforcement Learning

20 steps_explore is the number of steps the agent spends exploring

21 steps_exploit is the number of steps the agent spends exploiting
22 xscale is 'log' or 'linear'
23
24 returns total reward when exploring, total reward when exploiting
25 """
26 assert yplot in ['Average','Total']
27 if step_size is None:
28 step_size = max(1,(steps_explore+steps_exploit)//500)
29 if label is None:
30 label = ag.label
31 ag.max_display_level,old_mdl = 1,ag.max_display_level
32 plt.ion()
33 plt.xscale(xscale)
34 plt.xlabel("step")
35 plt.ylabel(yplot+" reward")
36 steps = [] # steps
37 rewards = [] # return
38 ag.restart()
39 step = 0
40 while step < steps_explore:
41 ag.do(step_size)
42 step += step_size
43 steps.append(step)
44 if yplot == "Average":
45 rewards.append(ag.acc_rewards/step)
46 else:
47 rewards.append(ag.acc_rewards)
48 acc_rewards_exploring = ag.acc_rewards
49 ag.explore,explore_save = 0,ag.explore
50 while step < steps_explore+steps_exploit:
51 ag.do(step_size)
52 step += step_size
53 steps.append(step)
54 if yplot == "Average":
55 rewards.append(ag.acc_rewards/step)
56 else:
57 rewards.append(ag.acc_rewards)
58 plt.plot(steps,rewards,label=label)
59 plt.legend(loc="upper left")
60 plt.draw()
61 ag.max_display_level = old_mdl
62 ag.explore=explore_save
63 return acc_rewards_exploring, ag.acc_rewards-acc_rewards_exploring

http://aipython.org Version 0.7.5 June 19, 2018

 12.2. Q Learning 197

12.2 Q Learning
To run the Q-learning demo, in folder “aipython”, load “rlQTest.py”,
and copy and paste the example queries at the bottom of that file. This
assumes Python 3.

rlQLearner.py — Q Learning
11 import random
12 from utilities import Displayable, argmax, flip
13
14 class RL_agent(Displayable):
15 """An RL_Agent
16 has percepts (s, r) for some state s and real reward r
17 """

rlQLearner.py — (continued)

19 class Q_learner(RL_agent):
20 """A Q-learning agent has
21 belief-state consisting of
22 state is the previous state
23 q is a {(state,action):value} dict
24 visits is a {(state,action):n} dict. n is how many times action was done in state
25 acc_rewards is the accumulated reward
26
27 it observes (s, r) for some world-state s and real reward r
28 """

rlQLearner.py — (continued)

30 def __init__(self, env, discount, explore=0.1, fixed_alpha=True, alpha=0.2,

31 alpha_fun=lambda k:1/k,
32 qinit=0, label="Q_learner"):
33 """env is the environment to interact with.
34 discount is the discount factor
35 explore is the proportion of time the agent will explore
36 fixed_alpha specifies whether alpha is fixed or varies with the number of visits
37 alpha is the weight of new experiences compared to old experiences
38 alpha_fun is a function that computes alpha from the number of visits
39 qinit is the initial value of the Q's
40 label is the label for plotting
41 """
42 RL_agent.__init__(self)
43 self.env = env
44 self.actions = env.actions
45 self.discount = discount
46 self.explore = explore
47 self.fixed_alpha = fixed_alpha
48 self.alpha = alpha
49 self.alpha_fun = alpha_fun

http://aipython.org Version 0.7.5 June 19, 2018

 198 12. Reinforcement Learning

50 self.qinit = qinit
51 self.label = label
52 self.restart()
restart is used to make the learner relearn everything. This is used by the plot-
ter to create new plots.
rlQLearner.py — (continued)

54 def restart(self):
55 """make the agent relearn, and reset the accumulated rewards
56 """
57 self.acc_rewards = 0
58 self.state = self.env.state
59 self.q = {}
60 self.visits = {}
do takes in the number of steps.
rlQLearner.py — (continued)

62 def do(self,num_steps=100):
63 """do num_steps of interaction with the environment"""
64 self.display(2,"s\ta\tr\ts'\tQ")
65 alpha = self.alpha
66 for i in range(num_steps):
67 action = self.select_action(self.state)
68 next_state,reward = self.env.do(action)
69 if not self.fixed_alpha:
70 k = self.visits[(self.state, action)] = self.visits.get((self.state, action),0)+1
71 alpha = self.alpha_fun(k)
72 self.q[(self.state, action)] = (
73 (1-alpha) * self.q.get((self.state, action),self.qinit)
74 + alpha * (reward + self.discount
75 * max(self.q.get((next_state, next_act),self.qinit)
76 for next_act in self.actions)))
77 self.display(2,self.state, action, reward, next_state,
78 self.q[(self.state, action)], sep='\t')
79 self.state = next_state
80 self.acc_rewards += reward
select action us used to select the next action to perform. This can be reimple-
mented to give a different exploration strategy.
rlQLearner.py — (continued)

82 def select_action(self, state):

83 """returns an action to carry out for the current agent
84 given the state, and the q-function
85 """
86 if flip(self.explore):
87 return random.choice(self.actions)
88 else:
89 return argmax((next_act, self.q.get((state, next_act),self.qinit))
90 for next_act in self.actions)

http://aipython.org Version 0.7.5 June 19, 2018

 12.2. Q Learning 199

Exercise 12.1 Implement a soft-max action selection. Choose a temperature that

works well for the domain. Explain how you picked this temperature. Compare
the epsilon-greedy, soft-max and optimism in the face of uncertainty.
Exercise 12.2 Implement SARSA. Hint: it does not do a max in do. Instead it
needs to choose next act before it does the update.

12.2.1 Testing Q-learning

The first tests are for the 2-action 2-state
rlQTest.py — RL Q Tester
11 from rlProblem import Healthy_env
12 from rlQLearner import Q_learner
13 from rlPlot import plot_rl
14
15 env = Healthy_env()
16 ag = Q_learner(env, 0.7)
17 ag_opt = Q_learner(env, 0.7, qinit=100, label="optimistic" ) # optimistic agent
18 ag_exp_l = Q_learner(env, 0.7, explore=0.01, label="less explore")
19 ag_exp_m = Q_learner(env, 0.7, explore=0.5, label="more explore")
20 ag_disc = Q_learner(env, 0.9, qinit=100, label="disc 0.9")
21 ag_va = Q_learner(env, 0.7, qinit=100,fixed_alpha=False,alpha_fun=lambda k:10/(9+k),label="alpha=1
22
23 # ag.max_display_level = 2
24 # ag.do(20)
25 # ag.q # get the learned q-values
26 # ag.max_display_level = 1
27 # ag.do(1000)
28 # ag.q # get the learned q-values
29 # plot_rl(ag,yplot="Average")
30 # plot_rl(ag_opt,yplot="Average")
31 # plot_rl(ag_exp_l,yplot="Average")
32 # plot_rl(ag_exp_m,yplot="Average")
33 # plot_rl(ag_disc,yplot="Average")
34 # plot_rl(ag_va,yplot="Average")
35
36 from mdpExamples import mdpt
37 from rlProblem import Env_from_MDP
38 envt = Env_from_MDP(mdpt)
39 agt = Q_learner(envt, 0.8)
40 # agt.do(20)
41
42 from rlSimpleEnv import Simple_game_env
43 senv = Simple_game_env()
44 sag1 = Q_learner(senv,0.9,explore=0.2,fixed_alpha=True,alpha=0.1)
45 # plot_rl(sag1,steps_explore=100000,steps_exploit=100000,label="alpha="+str(sag1.alpha))
46 sag2 = Q_learner(senv,0.9,explore=0.2,fixed_alpha=False)
47 # plot_rl(sag2,steps_explore=100000,steps_exploit=100000,label="alpha=1/k")
48 sag3 = Q_learner(senv,0.9,explore=0.2,fixed_alpha=False,alpha_fun=lambda k:10/(9+k))
49 # plot_rl(sag3,steps_explore=100000,steps_exploit=100000,label="alpha=10/(9+k)")

http://aipython.org Version 0.7.5 June 19, 2018

 200 12. Reinforcement Learning

12.3 Model-based Reinforcement Learner

To run the demo, in folder “aipython”, load “rlModelLearner.py”, and
copy and paste the example queries at the bottom of that file. This
assumes Python 3.

A model-based reinforcement learner builds a Markov decision process model

of the domain, simultaneously learns the model and plans with that model.
The model-based reinforcement learner used the following data structures:

• q[s, a] is dictionary that, given a (s, a) pair returns the Q-value, the esti-
mate of the future (discounted) value of being in state s and doing action
a.

• r[s, a] is dictionary that, given a (s, a) pair returns the average reward
from doing a in state s.

• t[s, a, s0 ] is dictionary that, given a (s, a, s0 ) tuple returns the number of

times a was done in state s, with the result being state s0 .

• visits[s, a] is dictionary that, given a (s, a) pair returns the number of times
action a was carried out in state s.

• res states[s, a] is dictionary that, given a (s, a) pair returns the list of re-
sulting states that have occurred when action a was carried out in state s.
This is used in the asynchronous value iteration to determine the s0 states
to sum over.

• visits list is a list of (s, a) pair that have been carried out. This is used
to ensure there is no divide-by zero in the asynchronous value iteration.
Note that this could be constructed from r, visits or res states by enumer-
ating the keys, but needs to be a list for random.choice, and we don’t want
to keep recreating it.

rlModelLearner.py — Model-based Reinforcement Learner

11 import random
12 from rlQLearner import RL_agent
13 from utilities import Displayable, argmax, flip
14
15 class Model_based_reinforcement_learner(RL_agent):
16 """A Model-based reinforcement learner
17 """
18
19 def __init__(self, env, discount, explore=0.1, qinit=0,
20 updates_per_step=10, label="MBR_learner"):
21 """env is the environment to interact with.
22 discount is the discount factor
23 explore is the proportion of time the agent will explore

http://aipython.org Version 0.7.5 June 19, 2018

 12.3. Model-based Reinforcement Learner 201

24 qinit is the initial value of the Q's

25 updates_per_step is the number of AVI updates per action
26 label is the label for plotting
27 """
28 RL_agent.__init__(self)
29 self.env = env
30 self.actions = env.actions
31 self.discount = discount
32 self.explore = explore
33 self.qinit = qinit
34 self.updates_per_step = updates_per_step
35 self.label = label
36 self.restart()

rlModelLearner.py — (continued)

38 def restart(self):
39 """make the agent relearn, and reset the accumulated rewards
40 """
41 self.acc_rewards = 0
42 self.state = self.env.state
43 self.q = {} # {(st,action):q_value} map
44 self.r = {} # {(st,action):reward} map
45 self.t = {} # {(st,action,st_next):count} map
46 self.visits = {} # {(st,action):count} map
47 self.res_states = {} # {(st,action):set_of_states} map
48 self.visits_list = [] # list of (st,action)
49 self.previous_action = None

rlModelLearner.py — (continued)

51 def do(self,num_steps=100):
52 """do num_steps of interaction with the environment
53 for each action, do updates_per_step iterations of asynchronous value iteration
54 """
55 for step in range(num_steps):
56 pst = self.state # previous state
57 action = self.select_action(pst)
58 self.state,reward = self.env.do(action)
59 self.acc_rewards += reward
60 self.t[(pst,action,self.state)] = self.t.get((pst, action,self.state),0)+1
61 if (pst,action) in self.visits:
62 self.visits[(pst,action)] += 1
63 self.r[(pst,action)] += (reward-self.r[(pst,action)])/self.visits[(pst,action)]
64 self.res_states[(pst,action)].add(self.state)
65 else:
66 self.visits[(pst,action)] = 1
67 self.r[(pst,action)] = reward
68 self.res_states[(pst,action)] = {self.state}
69 self.visits_list.append((pst,action))
70 st,act = pst,action #initial state-action pair for AVI
71 for update in range(self.updates_per_step):

http://aipython.org Version 0.7.5 June 19, 2018

 202 12. Reinforcement Learning

72 self.q[(st,act)] = self.r[(st,act)]+self.discount*(
73 sum(self.t[st,act,rst]/self.visits[st,act]*
74 max(self.q.get((rst,nact),self.qinit) for nact in self.actions)
75 for rst in self.res_states[(st,act)]))
76 st,act = random.choice(self.visits_list)

rlModelLearner.py — (continued)

78 def select_action(self, state):

79 """returns an action to carry out for the current agent
80 given the state, and the q-function
81 """
82 if flip(self.explore):
83 return random.choice(self.actions)
84 else:
85 return argmax((next_act, self.q.get((state, next_act),self.qinit))
86 for next_act in self.actions)

rlModelLearner.py — (continued)

88 from rlQTest import senv # simple game environment

89 mbl1 = Model_based_reinforcement_learner(senv,0.9,updates_per_step=10)
90 # plot_rl(mbl1,steps_explore=100000,steps_exploit=100000,label="model-based(10)")
91 mbl2 = Model_based_reinforcement_learner(senv,0.9,updates_per_step=1)
92 # plot_rl(mbl2,steps_explore=100000,steps_exploit=100000,label="model-based(1)")

Exercise 12.3 If there was only one update per step, the algorithm can be made
simpler and use less space. Explain how. Does it make it more efficient? Is it
worthwhile having more than one update per step for the games implemented
here?
Exercise 12.4 It is possible to implement the model-based reinforcement learner
by replacing q, r, visits, res states with a single dictionary that returns a tuple
(q, r, v, tm) where q, r and v are numbers, and tm is a map from resulting states
into counts. Does this make the algorithm easier to understand? Does this make
the algorithm more efficient?
Exercise 12.5 If the states and the actions were mapped into integers, the dictio-
naries could be implemented more efficiently as arrays. This entails an extra step
in specifying problems. Implement this for the simple game. Is it more efficient?

12.4 Reinforcement Learning with Features

To run the demo, in folder “aipython”, load “rlFeatures.py”, and copy
and paste the example queries at the bottom of that file. This assumes
Python 3.

12.4.1 Representing Features

A feature is a function from state and action. To construct the features for a
domain, we construct a function that takes a state and an action and returns the

http://aipython.org Version 0.7.5 June 19, 2018

 12.4. Reinforcement Learning with Features 203

list of all feature values for that state and action. This feature set is redesigned
for each problem.
get features(state, action) returns the feature values appropriate for the sim-
ple game.

rlSimpleGameFeatures.py — Feature-based Reinforcement Learner

11 from rlSimpleEnv import Simple_game_env
12 from rlProblem import RL_env
13
14 def get_features(state,action):
15 """returns the list of feature values for the state-action pair
16 """
17 assert action in Simple_game_env.actions
18 (x,y,d,p) = state
19 # f1: would go to a monster
20 f1 = monster_ahead(x,y,action)
21 # f2: would crash into wall
22 f2 = wall_ahead(x,y,action)
23 # f3: action is towards a prize
24 f3 = towards_prize(x,y,action,p)
25 # f4: damaged and action is toward repair station
26 f4 = towards_repair(x,y,action) if d else 0
27 # f5: damaged and towards monster
28 f5 = 1 if d and f1 else 0
29 # f6: damaged
30 f6 = 1 if d else 0
31 # f7: not damaged
32 f7 = 1-f6
33 # f8: damaged and prize ahead
34 f8 = 1 if d and f3 else 0
35 # f9: not damaged and prize ahead
36 f9 = 1 if not d and f3 else 0
37 features = [1,f1,f2,f3,f4,f5,f6,f7,f8,f9]
38 for pr in Simple_game_env.prize_locs+[None]:
39 if p==pr:
40 features += [x, 4-x, y, 4-y]
41 else:
42 features += [0, 0, 0, 0]
43 # fp04 feature for y when prize is at 0,4
44 # this knows about the wall to the right of the prize
45 if p==(0,4):
46 if x==0:
47 fp04 = y
48 elif y<3:
49 fp04 = y
50 else:
51 fp04 = 4-y
52 else:
53 fp04 = 0
54 features.append(fp04)

http://aipython.org Version 0.7.5 June 19, 2018

 204 12. Reinforcement Learning

55 return features
56
57 def monster_ahead(x,y,action):
58 """returns 1 if the location expected to get to by doing
59 action from (x,y) can contain a monster.
60 """
61 if action == "right" and (x+1,y) in Simple_game_env.monster_locs:
62 return 1
63 elif action == "left" and (x-1,y) in Simple_game_env.monster_locs:
64 return 1
65 elif action == "up" and (x,y+1) in Simple_game_env.monster_locs:
66 return 1
67 elif action == "down" and (x,y-1) in Simple_game_env.monster_locs:
68 return 1
69 else:
70 return 0
71
72 def wall_ahead(x,y,action):
73 """returns 1 if there is a wall in the direction of action from (x,y).
74 This is complicated by the internal walls.
75 """
76 if action == "right" and (x==Simple_game_env.xdim-1 or (x,y) in Simple_game_env.vwalls):
77 return 1
78 elif action == "left" and (x==0 or (x-1,y) in Simple_game_env.vwalls):
79 return 1
80 elif action == "up" and y==Simple_game_env.ydim-1:
81 return 1
82 elif action == "down" and y==0:
83 return 1
84 else:
85 return 0
86
87 def towards_prize(x,y,action,p):
88 """action goes in the direction of the prize from (x,y)"""
89 if p is None:
90 return 0
91 elif p==(0,4): # take into account the wall near the top-left prize
92 if action == "left" and (x>1 or x==1 and y<3):
93 return 1
94 elif action == "down" and (x>0 and y>2):
95 return 1
96 elif action == "up" and (x==0 or y<2):
97 return 1
98 else:
99 return 0
100 else:
101 px,py = p
102 if p==(4,4) and x==0:
103 if (action=="right" and y<3) or (action=="down" and y>2) or (action=="up" and y<2):
104 return 1

http://aipython.org Version 0.7.5 June 19, 2018

 12.4. Reinforcement Learning with Features 205

105 else:
106 return 0
107 if (action == "up" and y<py) or (action == "down" and py<y):
108 return 1
109 elif (action == "left" and px<x) or (action == "right" and x<px):
110 return 1
111 else:
112 return 0
113
114 def towards_repair(x,y,action):
115 """returns 1 if action is towards the repair station.
116 """
117 if action == "up" and (x>0 and y<4 or x==0 and y<2):
118 return 1
119 elif action == "left" and x>1:
120 return 1
121 elif action == "right" and x==0 and y<3:
122 return 1
123 elif action == "down" and x==0 and y>2:
124 return 1
125 else:
126 return 0
127
128 def simp_features(state,action):
129 """returns a list of feature values for the state-action pair
130 """
131 assert action in Simple_game_env.actions
132 (x,y,d,p) = state
133 # f1: would go to a monster
134 f1 = monster_ahead(x,y,action)
135 # f2: would crash into wall
136 f2 = wall_ahead(x,y,action)
137 # f3: action is towards a prize
138 f3 = towards_prize(x,y,action,p)
139 return [1,f1,f2,f3]

12.4.2 Feature-based RL learner

This learns a linear function approximation of the Q-values. It requires the
function get features that given a state and an action returns a list of values for
all of the features. Each environment requires this function to be provided.
rlFeatures.py — Feature-based Reinforcement Learner
11 import random
12 from rlQLearner import RL_agent
13 from utilities import Displayable, argmax, flip
14
15 class SARSA_LFA_learner(RL_agent):
16 """A SARSA_LFA learning agent has
17 belief-state consisting of

http://aipython.org Version 0.7.5 June 19, 2018

 206 12. Reinforcement Learning

18 state is the previous state

19 q is a {(state,action):value} dict
20 visits is a {(state,action):n} dict. n is how many times action was done in state
21 acc_rewards is the accumulated reward
22
23 it observes (s, r) for some world-state s and real reward r
24 """
25 def __init__(self, env, get_features, discount, explore=0.2, step_size=0.01,
26 winit=0, label="SARSA_LFA"):
27 """env is the feature environment to interact with
28 get_features is a function get_features(state,action) that returns the list of feature values
29 discount is the discount factor
30 explore is the proportion of time the agent will explore
31 step_size is gradient descent step size
32 winit is the initial value of the weights
33 label is the label for plotting
34 """
35 RL_agent.__init__(self)
36 self.env = env
37 self.get_features = get_features
38 self.actions = env.actions
39 self.discount = discount
40 self.explore = explore
41 self.step_size = step_size
42 self.winit = winit
43 self.label = label
44 self.restart()
restart() is used to make the learner relearn everything. This is used by the
plotter to create new plots.
rlFeatures.py — (continued)

46 def restart(self):
47 """make the agent relearn, and reset the accumulated rewards
48 """
49 self.acc_rewards = 0
50 self.state = self.env.state
51 self.features = self.get_features(self.state, list(self.env.actions)[0])
52 self.weights = [self.winit for f in self.features]
53 self.action = self.select_action(self.state)
do takes in the number of steps.
rlFeatures.py — (continued)

55 def do(self,num_steps=100):
56 """do num_steps of interaction with the environment"""
57 self.display(2,"s\ta\tr\ts'\tQ\tdelta")
58 for i in range(num_steps):
59 next_state,reward = self.env.do(self.action)
60 self.acc_rewards += reward
61 next_action = self.select_action(next_state)
62 feature_values = self.get_features(self.state,self.action)

http://aipython.org Version 0.7.5 June 19, 2018

 12.4. Reinforcement Learning with Features 207

63 oldQ = dot_product(self.weights, feature_values)

64 nextQ = dot_product(self.weights, self.get_features(next_state,next_action))
65 delta = reward + self.discount * nextQ - oldQ
66 for i in range(len(self.weights)):
67 self.weights[i] += self.step_size * delta * feature_values[i]
68 self.display(2,self.state, self.action, reward, next_state,
69 dot_product(self.weights, feature_values), delta, sep='\t')
70 self.state = next_state
71 self.action = next_action
72
73 def select_action(self, state):
74 """returns an action to carry out for the current agent
75 given the state, and the q-function.
76 This implements an epsilon-greedy approach
77 where self.explore is the probability of exploring.
78 """
79 if flip(self.explore):
80 return random.choice(self.actions)
81 else:
82 return argmax((next_act, dot_product(self.weights,
83 self.get_features(state,next_act)))
84 for next_act in self.actions)
85
86 def show_actions(self,state=None):
87 """prints the value for each action in a state.
88 This may be useful for debugging.
89 """
90 if state is None:
91 state = self.state
92 for next_act in self.actions:
93 print(next_act,dot_product(self.weights, self.get_features(state,next_act)))
94
95 def dot_product(l1,l2):
96 return sum(e1*e2 for (e1,e2) in zip(l1,l2))

Test code:
rlFeatures.py — (continued)

99 from rlQTest import senv # simple game environment

100 from rlSimpleGameFeatures import get_features, simp_features
101 from rlPlot import plot_rl
102
103 fa1 = SARSA_LFA_learner(senv, get_features, 0.9, step_size=0.01)
104 #fa1.max_display_level = 2
105 #fa1.do(20)
106 #plot_rl(fa1,steps_explore=10000,steps_exploit=10000,label="SARSA_LFA(0.01)")
107 fas1 = SARSA_LFA_learner(senv, simp_features, 0.9, step_size=0.01)
108 #plot_rl(fas1,steps_explore=10000,steps_exploit=10000,label="SARSA_LFA(simp)")

Exercise 12.6 How does the step-size affect performance? Try different step sizes
(e.g., 0.1, 0.001, other sizes in between). Explain the behaviour you observe. Which

http://aipython.org Version 0.7.5 June 19, 2018

 208 12. Reinforcement Learning

step size works best for this example. Explain what evidence you are basing your
prediction on.
Exercise 12.7 Does having extra features always help? Does it sometime help?
Does whether it helps depend on the step size? Give evidence for your claims.
Exercise 12.8 For each of the following first predict, then plot, then explain the
behavour you observed:
(a) SARSA LFA, Model-based learning (with 1 update per step) and Q-learning
for 10,000 steps 20% exploring followed by 10,000 steps 100% exploiting
(b) SARSA LFA, model-based learning and Q-learning for
i) 100,000 steps 20% exploring followed by 100,000 steps 100% exploit
ii) 10,000 steps 20% exploring followed by 190,000 steps 100% exploit
(c) Suppose your goal was to have the best accumulated reward after 200,000
steps. You are allowed to change the exploration rate at a fixed number of
steps. For each of the methods, which is the best position to start exploiting
more? Which method is better? What if you wanted to have the best reward
after 10,000 or 1,000 steps?
Based on this evidence, explain when it is preferable to use SARSA LFA, Model-
based learner, or Q-learning.
Important: you need to run each algorithm more than once. Your explanation
should include the variability as well as the typical behavior.

12.5 Learning to coordinate - UNFINISHED!!!!

Coordinating agents should implement the agent architecture. However, in
that architecture, an agent calls the environment. That architecture was cho-
sen because it was simple. However, it does not really work when there are
multiple agents. In such cases, a coroutining architecture is more appropriate.
We assume there is an x-player, and a y-player. game[xa][ya][ag] gives value
to the agent ag (ag=for the x-player) of the strategy of the x-agent doing xa and
the y-agent doing ya.
learnCoordinate.py — Learning to Coordinate
11 from learnProblem import Learner
12
13 soccer = [[(-0.6,0.6),(-0.3,0.3)],[(-0.2,0.2),(-0.9,0.9)]]]
14 football = [[(2,1),(0,0)],[(0,0),(1,2)]]
15 prisoners_game = [[(100,100),(0,1100)],[(1100,0),(1000,1000)]]]
16
17 class Policy_hill_climbing(Learner):
18 def __init__(self,game)

http://aipython.org Version 0.7.5 June 19, 2018

 Chapter 13

Relational Learning

13.1 Collaborative Filtering

Based on gradient descent algorithm of Koren, Y., Bell, R. and Volinsky, C.,
Matrix Factorization Techniques for Recommender Systems, IEEE Computer
2009.
This assumes the form of the dataset from movielens (http://grouplens.
org/datasets/movielens/). The rating are a set of (user, item, rating, timestamp)
tuples.

relnCollFilt.py — Latent Property-based Collaborative Filtering

11 import random
12 import matplotlib.pyplot as plt
13 import urllib.request
14 from learnProblem import Learner
15 from utilities import Displayable
16
17 class CF_learner(Learner):
18 def __init__(self,
19 rating_set, # a Rating_set object
20 rating_subset = None, # subset of ratings to be used as training ratings
21 test_subset = None, # subset of ratings to be used as test ratings
22 step_size = 0.01, # gradient descent step size
23 reglz = 1.0, # the weight for the regularization terms
24 num_properties = 10, # number of hidden properties
25 property_range = 0.02 # properties are initialized to be between
26 # -property_range and property_range
27 ):
28 self.rating_set = rating_set
29 self.ratings = rating_subset or rating_set.training_ratings # whichever is not empty
30 if test_subset is None:

209
 210 13. Relational Learning

31 self.test_ratings = self.rating_set.test_ratings
32 else:
33 self.test_ratings = test_subset
34 self.step_size = step_size
35 self.reglz = reglz
36 self.num_properties = num_properties
37 self.num_ratings = len(self.ratings)
38 self.ave_rating = (sum(r for (u,i,r,t) in self.ratings)
39 /self.num_ratings)
40 self.users = {u for (u,i,r,t) in self.ratings}
41 self.items = {i for (u,i,r,t) in self.ratings}
42 self.user_bias = {u:0 for u in self.users}
43 self.item_bias = {i:0 for i in self.items}
44 self.user_prop = {u:[random.uniform(-property_range,property_range)
45 for p in range(num_properties)]
46 for u in self.users}
47 self.item_prop = {i:[random.uniform(-property_range,property_range)
48 for p in range(num_properties)]
49 for i in self.items}
50 self.zeros = [0 for p in range(num_properties)]
51 self.iter=0
52
53 def stats(self):
54 self.display(1,"ave sumsq error of mean for training=",
55 sum((self.ave_rating-rating)**2 for (user,item,rating,timestamp)
56 in self.ratings)/len(self.ratings))
57 self.display(1,"ave sumsq error of mean for test=",
58 sum((self.ave_rating-rating)**2 for (user,item,rating,timestamp)
59 in self.test_ratings)/len(self.test_ratings))
60 self.display(1,"error on training set",
61 self.evaluate(self.ratings))
62 self.display(1,"error on test set",
63 self.evaluate(self.test_ratings))

learn carries out num iter steps of gradient descent.

relnCollFilt.py — (continued)

65 def prediction(self,user,item):
66 """Returns prediction for this user on this item.
67 The use of .get() is to handle users or items not in the training set.
68 """
69 return (self.ave_rating
70 + self.user_bias.get(user,0) #self.user_bias[user]
71 + self.item_bias.get(item,0) #self.item_bias[item]
72 + sum([self.user_prop.get(user,self.zeros)[p]*self.item_prop.get(item,self.zeros)[p]
73 for p in range(self.num_properties)]))
74
75 def learn(self, num_iter = 50):
76 """ do num_iter iterations of gradient descent."""
77 for i in range(num_iter):
78 self.iter += 1

http://aipython.org Version 0.7.5 June 19, 2018

 13.1. Collaborative Filtering 211

79 abs_error=0
80 sumsq_error=0
81 for (user,item,rating,timestamp) in random.sample(self.ratings,len(self.ratings)):
82 error = self.prediction(user,item) - rating
83 abs_error += abs(error)
84 sumsq_error += error * error
85 self.user_bias[user] -= self.step_size*error
86 self.item_bias[item] -= self.step_size*error
87 for p in range(self.num_properties):
88 self.user_prop[user][p] -= self.step_size*error*self.item_prop[item][p]
89 self.item_prop[item][p] -= self.step_size*error*self.user_prop[user][p]
90 for user in self.users:
91 self.user_bias[user] -= self.step_size*self.reglz* self.user_bias[user]
92 for p in range(self.num_properties):
93 self.user_prop[user][p] -= self.step_size*self.reglz*self.user_prop[user][p]
94 for item in self.items:
95 self.item_bias[item] -= self.step_size*self.reglz*self.item_bias[item]
96 for p in range(self.num_properties):
97 self.item_prop[item][p] -= self.step_size*self.reglz*self.item_prop[item][p]
98 self.display(1,"Iteration",self.iter,
99 "(Ave Abs,AveSumSq) training =",self.evaluate(self.ratings),
100 "test =",self.evaluate(self.test_ratings))
evaluate evaluates current predictions on the rating set:
relnCollFilt.py — (continued)

102 def evaluate(self,ratings):

103 """returns (avergage_absolute_error, average_sum_squares_error) for ratings
104 """
105 abs_error = 0
106 sumsq_error = 0
107 if not ratings: return (0,0)
108 for (user,item,rating,timestamp) in ratings:
109 error = self.prediction(user,item) - rating
110 abs_error += abs(error)
111 sumsq_error += error * error
112 return abs_error/len(ratings), sumsq_error/len(ratings)

13.1.1 Alternative Formulation

An alternative formulation is to regularize after each update.

13.1.2 Plotting
relnCollFilt.py — (continued)

114 def plot_predictions(self, examples="test"):

115 """
116 examples is either "test" or "training" or the actual examples
117 """

http://aipython.org Version 0.7.5 June 19, 2018

 212 13. Relational Learning

118 if examples == "test":

119 examples = self.test_ratings
120 elif examples == "training":
121 examples = self.ratings
122 plt.ion()
123 plt.xlabel("prediction")
124 plt.ylabel("cumulative proportion")
125 self.actuals = [[] for r in range(0,6)]
126 for (user,item,rating,timestamp) in examples:
127 self.actuals[rating].append(self.prediction(user,item))
128 for rating in range(1,6):
129 self.actuals[rating].sort()
130 numrat=len(self.actuals[rating])
131 yvals = [i/numrat for i in range(numrat)]
132 plt.plot(self.actuals[rating], yvals, label="rating="+str(rating))
133 plt.legend()
134 plt.draw()
This plots a single property. Each (user, item, rating) is plotted where the
x-value is the value of the property for the user, the y-value is the value of the
property for the item, and the rating is plotted at this (x, y) position. That is,
rating is plotted at the (x, y) position (p(user), p(item)).
relnCollFilt.py — (continued)

136 def plot_property(self,

137 p, # property
138 plot_all=False, # true if all points should be plotted
139 num_points=200 # number of random points plotted if not all
140 ):
141 """plot some of the user-movie ratings,
142 if plot_all is true
143 num_points is the number of points selected at random plotted.
144
145 the plot has the users on the x-axis sorted by their value on property p and
146 with the items on the y-axis sorted by their value on property p and
147 the ratings plotted at the corresponding x-y position.
148 """
149 plt.ion()
150 plt.xlabel("users")
151 plt.ylabel("items")
152 user_vals = [self.user_prop[u][p]
153 for u in self.users]
154 item_vals = [self.item_prop[i][p]
155 for i in self.items]
156 plt.axis([min(user_vals)-0.02,
157 max(user_vals)+0.05,
158 min(item_vals)-0.02,
159 max(item_vals)+0.05])
160 if plot_all:
161 for (u,i,r,t) in self.ratings:
162 plt.text(self.user_prop[u][p],

http://aipython.org Version 0.7.5 June 19, 2018

 13.1. Collaborative Filtering 213

163 self.item_prop[i][p],
164 str(r))
165 else:
166 for i in range(num_points):
167 (u,i,r,t) = random.choice(self.ratings)
168 plt.text(self.user_prop[u][p],
169 self.item_prop[i][p],
170 str(r))
171 plt.show()

13.1.3 Creating Rating Sets

A rating set can be read from the Internet or read from a local file. The default
is to read the Movielens 100K dataset from the Internet. It would be more
efficient to save the dataset as a local file, and then set local file = True, as then
it will not need to download the dataset every time the program is run.
relnCollFilt.py — (continued)

173 class Rating_set(Displayable):

174 def __init__(self,
175 date_split=892000000,
176 local_file=False,
177 url="http://files.grouplens.org/datasets/movielens/ml-100k/u.data",
178 file_name="u.data"):
179 self.display(1,"reading...")
180 if local_file:
181 lines = open(file_name,'r')
182 else:
183 lines = (line.decode('utf-8') for line in urllib.request.urlopen(url))
184 all_ratings = (tuple(int(e) for e in line.strip().split('\t'))
185 for line in lines)
186 self.training_ratings = []
187 self.training_stats = {1:0, 2:0, 3:0, 4:0 ,5:0}
188 self.test_ratings = []
189 self.test_stats = {1:0, 2:0, 3:0, 4:0 ,5:0}
190 for rate in all_ratings:
191 if rate[3] < date_split: # rate[3] is timestamp
192 self.training_ratings.append(rate)
193 self.training_stats[rate[2]] += 1
194 else:
195 self.test_ratings.append(rate)
196 self.test_stats[rate[2]] += 1
197 self.display(1,"...read:", len(self.training_ratings),"training ratings and",
198 len(self.test_ratings),"test ratings")
199 tr_users = {user for (user,item,rating,timestamp) in self.training_ratings}
200 test_users = {user for (user,item,rating,timestamp) in self.test_ratings}
201 self.display(1,"users:",len(tr_users),"training,",len(test_users),"test,",
202 len(tr_users & test_users),"in common")
203 tr_items = {item for (user,item,rating,timestamp) in self.training_ratings}
204 test_items = {item for (user,item,rating,timestamp) in self.test_ratings}
205 self.display(1,"items:",len(tr_items),"training,",len(test_items),"test,",

http://aipython.org Version 0.7.5 June 19, 2018

 214 13. Relational Learning

206 len(tr_items & test_items),"in common")

207 self.display(1,"Rating statistics for training set: ",self.training_stats)
208 self.display(1,"Rating statistics for test set: ",self.test_stats)
Sometimes it is useful to plot a property for all (user, item, rating) triples.
There are too many such triples in the data set. The method create top subset
creates a much smaller dataset where this makes sense. It picks the most rated
items, then picks the users who have the most ratings on these items. It is
designed for depicting the meaning of properties, and may not be useful for
other purposes.
relnCollFilt.py — (continued)

210 def create_top_subset(self, num_items = 30, num_users = 30):

211 """Returns a subset of the ratings by picking the most rated items,
212 and then the users that have most ratings on these, and then all of the
213 ratings that involve these users and items.
214 """
215 items = {item for (user,item,rating,timestamp) in self.training_ratings}
216
217 item_counts = {i:0 for i in items}
218 for (user,item,rating,timestamp) in self.training_ratings:
219 item_counts[item] += 1
220
221 items_sorted = sorted((item_counts[i],i) for i in items)
222 top_items = items_sorted[-num_items:]
223 set_top_items = set(item for (count, item) in top_items)
224
225 users = {user for (user,item,rating,timestamp) in self.training_ratings}
226 user_counts = {u:0 for u in users}
227 for (user,item,rating,timestamp) in self.training_ratings:
228 if item in set_top_items:
229 user_counts[user] += 1
230
231 users_sorted = sorted((user_counts[u],u)
232 for u in users)
233 top_users = users_sorted[-num_users:]
234 set_top_users = set(user for (count, user) in top_users)
235 used_ratings = [ (user,item,rating,timestamp)
236 for (user,item,rating,timestamp) in self.training_ratings
237 if user in set_top_users and item in set_top_items]
238 return used_ratings
239
240 movielens = Rating_set()
241 learner0 = CF_learner(movielens, num_properties = 1)
242 #learner0.learn(50)
243 # learner0.plot_predictions(examples = "training")
244 # learner0.plot_predictions(examples = "test")
245 #learner0.plot_property(0)
246 #movielens_subset = movielens.create_top_subset(num_items = 20, num_users = 20)
247 #learner1 = CF_learner(movielens, rating_subset=movielens_subset, test_subset=[], num_properties=1)
248 #learner1.learn(1000)

http://aipython.org Version 0.7.5 June 19, 2018

 13.1. Collaborative Filtering 215

249 #learner1.plot_property(0,plot_all=True)

http://aipython.org Version 0.7.5 June 19, 2018

Index

α-β pruning, 189 Branch and bound, 44

CF learner, 209
A∗ search, 39 CSP, 50
A∗ Search, 42 CSP from STRIPS, 92
action, 81 Clause, 73
agent, 19, 191 Con solver, 58
argmax, 16 Constraint, 49
assignment, 50, 139 DBN, 164
assumable, 78 DBN VE filter, 165
augmented feature, 110 DBN variable, 163
DT learner, 116
batched stochastic gradient descent,
Data from file, 107
127
Data set, 103
blocks world, 83
Data set augmented, 111
Boolean feature, 103
botton-up proof, 75 Data set random, 115
branch-and-bound search, 44 DecisionNetwork, 168
DecisionVariable, 167
class Displayable, 15
Action instance, 95 EM learner, 181
Agent, 19 Env from MDP, 192
Arc, 32 Environment, 20
Askable, 73 Factor, 138
Assumable, 78 Factor DF, 169
Belief network, 143 Factor max, 169
Boosted dataset, 133 Factor observed, 140
Boosting learner, 133 Factor rename, 143

217
218 Index

Factor stored, 140 Rob top layer, 27

Factor sum, 141 Runtime distribution, 70
Forward STRIPS, 86 SARSA LFA learner, 205
FrontierPQ, 40 SLSearcher, 64
Gibbs sampling, 155 STRIPS domain, 82
Graphical model, 143 Sampling inference method, 149
HMM, 157 Search from CSP, 56
HMM VE filter, 158 Search problem, 31
HMM particle filter, 160 Search problem from explicit graph,
Healthye nv, 191 33
Inference method, 144 Search with AC from CSP, 62
KB, 74 Searcher, 39
KBA, 78 SearcherMPP, 43
K fold dataset, 120 Sigmoid layer, 129
K means learner, 177 Simple game env, 193
Layer, 128 State, 85
Learner, 113 Strips, 81
Likelihood weighting, 151 Subgoal, 89
Linear complete layer, 129 TP agent, 22
Linear learner, 122 TP env, 20
Linear learner bsgd, 127 Updatable priority queue, 69
MDP, 172 Utility, 167
Magic sum, 187 VE, 145
Model based reinforcement learner, VE DN, 168
200 Variable, 137
NN, 130 clause, 73
Node, 185 collaborative filtering, 209
POP node, 96 condition, 49
POP search from STRIPS, 97 consistency algorithms, 58
Particle filtering, 152 constraint, 49
Path, 34 constraint satisfaction problem, 49
Planning problem, 82 copy with assign, 61
Plot env, 28 cross validation, 120
Plot prices, 22 CSP, 49
Prob, 142 consistency, 58
Q learner, 197 domain splitting, 60, 62
RL agent, 197 search, 56
RL env, 191 stochastic local search, 63
Rating set, 213 currying, 51
ReLU layer, 130
Regression STRIPS, 90 data set, 103
Rejection sampling, 150 DBN (dynamic belief network), 163
Rob body, 24 decision network, 167
Rob env, 23 decision tree learning, 116
Rob middle layer, 26 deep learning, 128

http://aipython.org Version 0.7.5 June 19, 2018

Index 219

display, 15 probGraphicalModels.py, 143

Displayable, 15 probHMM.py, 157
domain splitting, 60, 62 probMCMC.py, 155
dynamic belief network, 163 probStochSim.py, 147
probVE.py, 145
EM, 181 probVariables.py, 137
environment, 19, 20, 191 pythonDemo.py, 11
example, 103 relnCollFilt.py, 209
explicit graph, 32 rlFeatures.py, 205
rlModelLearner.py, 200
factor, 138 rlPlot.py, 195
factor times, 141 rlProblem.py, 191
feature, 103 rlQLearner.py, 197
file rlQTest.py, 199
agentEnv.py, 23 rlSimpleEnv.py, 193
agentMiddle.py, 26 rlSimpleGameFeatures.py, 203
agentTop.py, 27 searchBranchAndBound.py, 44
agents.py, 19 searchGeneric.py, 39
cspConsistency.py, 58 searchMPP.py, 43
cspExamples.py, 51 searchProblem.py, 31
cspProblem.py, 49 searchTest.py, 46
cspSLS.py, 64 stripsCSPPlanner.py, 92
cspSearch.py, 56 stripsForwardPlanner.py, 85
decnNetworks.py, 167 stripsHeuristic.py, 87
learnBoosting.py, 133 stripsPOP.py, 95
learnCoordinate.py, 208 stripsProblem.py, 81
learnCrossValidation.py, 120 stripsRegressionPlanner.py, 89
learnDT.py, 116 utilities.py, 15
learnEM.py, 181 filtering, 158, 160
learnKMeans.py, 177 forward planning, 85
learnLinear.py, 122
learnLinearBSGD.py, 127 game, 185
learnNN.py, 128 Gibbs sampling, 155
learnNoInputs.py, 113 graphical model, 143
learnProblem.py, 103
logicAssumables.py, 78 heuristic planning, 87, 92
logicBottomUp.py, 75 hidden Markov model, 157
logicProblem.py, 73 hierarchical controller, 23
logicTopDown.py, 77 HMM
masMiniMax.py, 188 exact filtering, 158
masProblem.py, 185 particle filtering, 160
mdpExamples.py, 172 HMM (hidden Markov models), 157
mdpProblem.py, 172
probDBN.py, 163 importance sampling, 152
probFactors.py, 138 ipython, 8

http://aipython.org Version 0.7.5 June 19, 2018

220 Index

k-means, 177 NotImplementedError, 19

knowledge base, 74
partial-order planner, 95
learner, 113 particle filtering, 152
learning, 103–135, 177–184, 191–215 HMMs, 160
batched stochastic gradient de- planning, 81–101, 167–175
scent, 127 CSP, 92
cross validation, 120 decision network, 167
decision tree, 116 forward, 85
deep learning, 128 MDP, 172
EM, 181 partial order, 95
k-means, 177 regression, 89
linear regression, 122 with certainty, 81–101
linear classification, 122 with learning, 200
neural network, 128 with uncertainty, 167–175
no inputs, 113 plotting
reinforcement, 191–208 agents in time, 22
relational, 209 reinforcement learning, 195
supervised, 103–135 robot environment, 28
with uncertainty, 177–184 runtime distribution, 70
likelihood weighting, 151 stochastic simulation, 154
linear regression, 122 predictor, 105
linear classification, 122 proability, 137
proof
magic square, 186 bottom-up, 75
magic-sum game, 186 top-down, 77
Markov Chain Monte Carlo, 155 proposition, 73
Markov decision process, 172 Python, 7
max display level, 15
MCMC, 155 Q learning, 197
MDP, 172, 192
method regression planning, 89
consistent, 51 reinforcement learning, 191–208
holds, 50 environment, 191
maxh, 88 feature-based, 202
zero, 86 model-based, 200
minimax, 185 Q-learning, 197
minimax algorithm, 188 rejection sampling, 150
minsets, 79 relational learning, 209
model-based reinforcement learner, resampling, 153
200 robot
multiagent system, 185 body, 24
multiple path pruning, 43 environment, 23
middle layer, 26
naughts and crosses, 186 plotting, 28
neural network, 128 top layer, 27

http://aipython.org Version 0.7.5 June 19, 2018

Index 221

robot delivery domain, 82

runtime, 13
runtime distribution, 70

sampling, 147
importance sampling, 152
belief networks, 149
likelihood weighting, 151
particle filtering, 152
rejection, 150
scope, 49
search, 31
A∗ , 39
branch-and-bound, 44
multiple path pruning, 43
search with any conflict, 65
search with var pq, 67
sigmoid, 123
stochastic local search, 63
any-conflict, 65
two-stage choice, 66
stochastic simulation, 147

test
SLS, 71
tic-tac-toe, 186
top-down proof, 77

uncertainty, 137
unit tests, 17
updatable priority queue, 68

value iteration, 173

variable, 49, 137
variable elimination (VE), 145
VE, 145
visualize, 15

yield, 12

http://aipython.org Version 0.7.5 June 19, 2018