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Project 1: Search
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CS 221: Artificial Intelligence

Fall 2011
Lecture 2: Search

(Slides from Dan Klein, with help from Stuart Russell, Andrew Moore, Teg Grenager, Peter Norvig)

Outline
Problem-solving agents (Book: 3.1)
Problem types and problem formulation

Search trees and state space graphs (3.3) Uninformed search (3.4)
Depth-first, Breadth-first, Uniform cost Search graphs

Informed search (3.5)

Greedy search, A* search Heuristics, admissibility
3

Agents
act = AgentFn(percept) sensors agent fn

actuators
4

Reflex Agents
Reflex agents:
Choose action based on current state May have memory or a model of the world s current state; use percept Do not consider the future consequences of their actions Act on how the world IS

Strategy: Dots: WNES No Dots: NESW

Can a reflex agent be rational (optimal)?

Goal Based Agents

Goal-based agents:
Plan ahead Ask what if Decisions based on (hypothesized) consequences of actions Must have a model of how the world evolves in response to actions Act on how the world WOULD BE

WWGBAD?

WWGBAD?

Problem types
Fully observable, deterministic
single-belief-state problem

Non-observable
sensorless (conformant) problem

Partially observable/non-deterministic
contingency problem interleave search and execution

Unknown state space

exploration problem execution first
9

Search Problems
A search problem consists of:
A state space
N, 1

A transition model
E, 1

A start state, goal test, and path cost function

A solution is a sequence of actions (a plan) which transforms the start state to a goal state

Transition Models
Successor function
Successors( ) = {(N, 1, ), (E, 1, )}

Actions and Results

Actions( Result( Cost( ) = {N, E} , N) = ; Result( , N, ) = 1; Cost( , E) = , E, )=1

Example: Romania
State space:
Cities

Successor function:
Go to adj city with cost = dist

Start state:
Arad

Goal test:
Is state == Bucharest?

Solution?

State Space Graphs

State space graph: A mathematical representation of a search problem
For every search problem, there s a corresponding state space graph The successor function is represented by arcs
S

a b d h p q r c e f

This can be large or infinite, so we won t create it in memory

Ridiculously tiny search graph for a tiny search problem

Exponential State Space Sizes

Search Problem: Eat all of the food Pacman positions: 10 x 12 = 120 Food count: 30

Ghost positions: 12 Pacman facing: up, down, left, right

Search Trees
N, 1 E, 1

A search tree:
This is a what if tree of plans and outcomes Start state at the root node Children correspond to successors Nodes contain states, correspond to paths to those states For most problems, we can never actually build the whole tree

Another Search Tree

Search:
Expand out possible plans Maintain a frontier of unexpanded plans Try to expand as few tree nodes as possible

General Tree Search

Important ideas:
Frontier (aka fringe) Expansion Exploration strategy
Detailed pseudocode is in the book!

Main question: which frontier nodes to explore?

State Space vs. Search Tree

a b d c e h p q r f

Each NODE in in the search tree is an entire PATH in the state space.

d We construct both on demand and we construct as little as possible. b a c a p q h q c a e r f p q h q

e r f c a
G

p q

States vs. Nodes

Nodes in state space graphs are problem states
Represent an abstracted state of the world Have successors, can be goal / non-goal, have multiple predecessors

Nodes in search trees are paths

Represent a path (sequence of actions) which results in the node s state Have a problem state and one parent, a path length, (a depth) & a cost The same problem state may be achieved by multiple search tree nodes

State Space Graph

Search Tree
Parent
Depth 5

Action Node
Depth 6

Depth First Search

Strategy: expand deepest node first Implementation: Frontier is a LIFO stack S
p q a b d h r c e f

d b a c a p q h q c a e r f
G

e h p q q c a r f
G

p q

[demo: dfs]

Breadth First Search

Strategy: expand shallowest node first Implementation: Fringe is a FIFO queue
S

a b d

G c e h q r f

p
S

d Search Tiers b a c a p q h q c a e r f
G

e h p q q c a r f
G

p q

[demo: bfs]

Santayana s Warning
Those who cannot remember the past are condemned to repeat it. George Santayana Failure to detect repeated states can cause exponentially more work (why?)

Graph Search
In BFS, for example, we shouldn t bother expanding the circled nodes (why?)
S

d b a c a p q h q c a e r f
G

e h p q q c a r f
G

p q

Graph Search
Very simple fix: never expand a state twice

Can this wreck completeness? Lowest cost?

Graph Search Hints

Graph search is almost always better than tree search (when not?) Implement explored as a dict or set Implement frontier as priority Q and set

Costs on Actions
2 b 1 3
START

GOAL

2 c 8 2 d 9 4 h 4 q r 8 2 e f 1 3 2

1 p

15

Notice that BFS finds the shortest path in terms of number of transitions. It does not find the least-cost path. We will quickly cover an algorithm which does find the least-cost path.

Uniform Cost Search

2 Expand cheapest node first: Frontier is a priority queue S 1
S
p b a

G 8
c

1
d

2
h

2 1
r f

9
q

15 9

0 e p q 1 16

d b 4 Cost contours a 6 c a

3 11 e 5 h 13 r 7 p q f 8
G 10

h 17 r 11 p q q c a f
G

q 11 c a

Uniform Cost Issues

Remember: explores increasing cost contours The good: UCS is complete and optimal! The bad:
Explores options in every direction No information about goal location
c1 c2 c3

Start

Goal

[demos: ucs, ucs2]

Uniform Cost Search

What will UCS do for this graph?
0 1
START

b 0 1
GOAL

What does this mean for completeness?

AI Lesson

To do more, Know more

Search Heuristics
Any estimate of how close a state is to a goal Designed for a particular search problem Examples: Manhattan distance, Euclidean distance

10 5 11.2

Heuristics

Greedy Best First Search

Expand the node that seems closest to goal

What can go wrong?

[demos: gbf1, gbf2]

Greedy goes wrong

Best First / Greedy Search

Strategy: expand the closest node to the goal
2 b h=11 3 S h=12 1 p h=11 15 1 d h=8 4 q h=9 4 a h=8 8 9 h 2 G c h=5 2 h=4 e 1 h=6 3 r
[demos: gbf1, gbf2]

2 f 5 h=6

h=0 5

h=4

Combining UCS and Greedy

Uniform-cost orders by path cost, or backward cost g(n) Best-first orders by distance to goal, or forward cost h(n)
5 1 S h=6 c h=7 1 1 1 b h=6 a h=5 3 d h=2 e h=1 2

G h=0

A* Search orders by the sum: f(n) = g(n) + h(n)

A* Search Progress

source: wikipedia page for A* Algorithm; by Subh83

When should A* terminate?

Should we stop when we enqueue a goal?
2 S
h=3

A
h=2

2 G
h=0

B
h=1

No: only stop when we dequeue a goal

Is A* Optimal?
1 A
h=6

3
h=0

h=7

G 5

What went wrong?

Actual bad path cost (5) < estimate good path cost (1+6)

We need estimates (h=7) to be less than actual (5) costs!

Admissible Heuristics
A heuristic h is admissible (optimistic) if:

where

is the true cost to a nearest goal

Never overestimate!

Creating Admissible Heuristics

Most of the work in solving hard search problems optimally is in coming up with admissible heuristics Often, admissible heuristics are solutions to relaxed problems, where new actions are available

366

15 Inadmissible heuristics are often useful too (why?)

Optimality of A*: Blocking

Notation: g(n) = cost to node n h(n) = estimated cost from n to the nearest goal (heuristic) f(n) = g(n) + h(n) = estimated total cost via n G*: a lowest cost goal node G: another goal node

Optimality of A*: Blocking

Proof: What could go wrong? We d have to have to pop a suboptimal goal G off the frontier before G* This can t happen: Imagine a suboptimal goal G is on the queue Some node n which is a subpath of G* must also be on the frontier (why?) n will be popped before G

Properties of A*
Uniform-Cost
b

A*
b

UCS vs A* Contours
Uniform-cost expanded in all directions
Start Goal

A* expands mainly toward the goal, but does hedge its bets to ensure optimality

Start

Goal

[demos: conu, cona]

Example: 8 Puzzle

What are the states? How many states? What are the actions? What states can I reach from the start state? What should the costs be?

8 Puzzle
Heuristic: Number tiles misplaced Why is it admissible? h(start) = 8 This is a relaxed-problem heuristic:
UCS

Average nodes expanded when optimal path has length 4 steps 8 steps 12 steps

112 TILES 13

6,300 39

3.6 x 106 227

Move A to B if adjacent(A,B) and empty(B)

8 Puzzle
What if we had an easier 8-puzzle where any tile could slide one step at any time, ignoring other tiles? Total Manhattan distance Why admissible? h(start) = 3 + 1 + 2 + = 18 Relaxed problem:
TILES
MANHATTAN

Average nodes expanded when optimal path has length 4 steps 8 steps 12 steps

13 12

39 25

227 73

Move A to B if adjacent(A,B) and empty(B)

Trivial Heuristics, Dominance

Dominance: ha hc if

Heuristics form a semi-lattice:

Max of admissible heuristics is admissible

Trivial heuristics
Bottom of lattice is the zero heuristic (what does this give us?) Top of lattice is the exact heuristic

Other A* Applications
Path finding / routing problems Resource planning problems Robot motion planning Language analysis Machine translation Speech recognition

Summary: A*
A* uses both backward costs, g(n), and (estimates of) forward costs, h(n) A* is optimal with admissible heuristics Heuristic design is key: often use relaxed problems A* is not the final word in search algorithms (but it does get the final word for today)