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Related Terminology
• Performance Measure of Agent − It is the criteria, which
determines how successful an agent is.
• Behavior of Agent − It is the action that agent performs after any
given sequence of percepts.
• Percept − It is agent’s perceptual inputs at a given instance.
• Percept Sequence − It is the history of all that an agent has
perceived till date.
• Agent Function − It is a map from the precept sequence to an
action.

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Characteristics of an Intelligence Agent

• Must sense
• Must act
• Must autonomous (to some extend)
• Must rational
Examples of Agents
• Human agent: eyes, ears, and other organs for sensors; – hands,
legs, mouth, and other body parts for actuators
• Robotic agent: cameras and infrared range finders for sensors; –
various motors, wheels, and speakers for actuators
• A software agent: function (Input) as sensors — function as
actuators (output-Screening)
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MODULE 2
PROBLEM SOLVING

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PROBLEM STATE SPACE

• A Problem is the issue which comes across any system. A solution is
needed to solve that particular problem.
• Technically, a problem is defined by its ‘elements’ and their
‘relations’.
• To provide a formal description of a problem, we need to
understand the following terms:
Define a state space that contains all the possible configurations of
the relevant objects, including some impossible ones.

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Specify one or more states that describe possible situations, from

which the problem solving process may start. These states are
called initial states.
Specify one or more states that would be acceptable solution to
the problem. These states are called goal states.
• A state is a representation of problem elements at a given
moment.
• A State space is the set of all states reachable from the initial
state.

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 A state space consists of the following elements

 A (possibly infinite) set of states
 Out of the possible states, one state represents the start state that is
the initial state of the problem.
 Each state represents some configuration reachable from the start state
 Out of the possible states, some states may be goal states (solutions)
A set of rules
 Applying a rule to the current state, transforms it to another or a new
state in the state space
 All operators may not be applicable to all states in the state space
State spaces are used extensively in Artificial Intelligence (AI) to
represent and solve problems.
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Examples of Problems in Artificial Intelligence

The most prevalent problems that artificial intelligence has
resolved are the following:
1. Chess
2. N-Queen problem
3. Tower of Hanoi Problem
4. Travelling Salesman Problem
5. Water-Jug Problem

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Some Real World Problems

The real world problems that artificial intelligence has resolved are
the following:
1. Route-finding problems
2. Touring problems
3. VLSI layout problem
4. Robot navigation
5. Automatic assembly sequencing
6. Internet searching
7. Speech Recognition System etc.

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Problem characteristics
In order to choose the most appropriate method for a particular
problem, it is necessary to analyze the problem along several
dimensions.
• Is the problem decomposable into a set of independent smaller or
easier sub problems?
• Can solution steps be ignored or at least undone if they prove
unwise?
• Is the problem’s universe predictable?
• Is a good solution to the problem obvious without comparison to
all other possible solutions?
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• Is the desired solution a state of the world or a path to a state?

• Is a large amount of knowledge absolutely required to solve the
problem, or is knowledge important only to constrain the search?
• Can a computer that is simply given the problem return the
solution, or will the solution of the problem require interaction
between the computer and a person?

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An Example:
Scenario:
You want to go on holiday to city ‘Chandigarh’ but currently you live in
city ‘Kottayam’ And the flight for ‘Chandigarh’ leaves from another city
‘Kochi’.
Goal:
• To reach city ‘Kochi’ from ‘Kottayam’ to catch the flight for ‘Chandigarh’.
Formulate Problem:
 States: various cities
 Action: movement between the cities
Solution:
• Appropriate sequence of the cities.
• Say, Kottayam->X->Y->Z->Kochi
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Problem Solving
• Problem-solving is commonly known as the method to reach the
desired goal or finding a solution to a given situation.
• In computer science, problem-solving refers to artificial intelligence
techniques, including various techniques such as:
forming efficient algorithms,
heuristics, and
performing root cause analysis to find desirable solutions.

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• Steps in Problem Solving in AI

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Steps in Problem Solving in AI

• Goal Formulation: It is the first and simplest step in problem-
solving. It organizes the steps/sequence required to formulate one
goal out of multiple goals as well as actions to achieve that goal.
Goal formulation is based on the current situation and the agent’s
performance measure.
• Problem Formulation: It is the most important step of problem-
solving which decides what actions should be taken to achieve the
formulated goal.
• There are following five components involved in problem
formulation:
Initial State: It is the starting state or initial step of the agent
towards its goal.
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Actions: It is the description of the possible actions available to

the agent.
Transition Model: It describes what each action does.
Goal Test: It determines if the given state is a goal state.
Path cost: It assigns a numeric cost to each path that follows the
goal. The problem-solving agent selects a cost function, which
reflects its performance measure.
 Remember, an optimal solution has the lowest path cost among
all the solutions.

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Problem space
• A ‘problem space’ is an abstract space.
• A problem space encompasses all valid states that can be
generated by the application of any combination of operators on
any combination of objects.
• The problem space may contain one or more solutions. A solution
is a combination of operations/actions and objects that achieve the
goals.

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STATE SPACE SEARCH

A state space consists of
• A representation of the states the system can be in.
• A set of operators that can change one state into another state. In
a board game, the operators are the legal moves from any given
state. Often the operators are represented as programs that
change a state representation to represent the new state.
• An initial state.
• A set of final states; some of these may be desirable, others
undesirable. This set is often represented implicitly by a program
that detects terminal states.
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A state space represents a problem in terms of states and

operators that change states.
S : {S ,A, Action(S),Result(S,A),Cost(S,A)}
Where
• S - start, goal state.
• A - set of possible actions.
• Action(S) - the action performed from the set of states.
• Result(S,A) - Final sate
• Cost(S,A) – A constant value either in terms of time or distance.

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Measuring problem-solving performance

• Completeness: Is a solution found if one exists?
• Optimality: Does the strategy find the optimal solution?
• Time Complexity: How long does it take to find a solution?
• Space Complexity: How much memory is needed to perform the
search?
Time and space complexity are measured in terms of problem
difficulty defined by:
• b - branching factor of the search tree.
• d - depth of the shallowest goal node.
• m - maximum length of any path in the state space.
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WATER JUG PROBLEM

In this problem, we use two jugs called four and three; four holds
a maximum of four gallons of water and three a maximum of three
gallons of water. How can we get two gallons of water in the four
jug?

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• The start state is (0, 0) and the goal state is (2, n) where n may be
any but it is limited to three holding from 0 to 3 gallons of water or
empty.
• Three and four shows the name and numerical number shows the
amount of water in jugs for solving the water jug problem.
• The major production rules for solving this problem are shown
below:

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Production Rules:
1. (x,y)=(4,y)
2. (x,y)=(x,3)
3. (x,y)=(x-d,y) if x>0
4. (x,y)=(x,y-d) if y>0
5. (x,y)=(0,y) if x>0
6. (x,y)=(x,0) if y>0
7. (x,y)= (4,y-(4-x)) if y>0
8. (x,y)=(x-(3-y),3) if x>0
9. (x,y)=(x+y,0) , if x+y<=4,y>0

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8 PUZZLE PROBLEM
• The 8 puzzle consists of eight numbered, movable tiles set in a
3x3 frame.
• One cell of the frame is always empty thus making it possible to
move an adjacent numbered tile into the empty cell.
The production system consists of
• Production set
• Working Memory content
• Start State
• Goal State
• Control engine
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SEARCH ALGORITHM TERMINOLOGIES

• Search: Searching is a step by step procedure to solve a search-

problem in a given search space. A search problem can have three
main factors:
 Search Space: Search space represents a set of possible solutions,
which a system may have.
 Start State: It is a state from where agent begins the search.
 Goal test: It is a function which observe the current state and
returns whether the goal state is achieved or not.

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• Search tree: A tree representation of search problem is called

Search tree. The root of the search tree is the root node which is
corresponding to the initial state.
• Actions: It gives the description of all the available actions to the
agent.
• Transition model: A description of what each action do, can be
represented as a transition model.
• Path Cost: It is a function which assigns a numeric cost to each
path.
• Solution: It is an action sequence which leads from the start node
to the goal node.
• Optimal Solution: If a solution has the lowest cost solution among
all solutions.
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Properties of Search Algorithms

1. Completeness: A search algorithm is said to be complete if it
guarantees to return a solution if at least any solution exists for
any random input.
2. Optimality: If a solution found for an algorithm is guaranteed to
be the best solution (lowest path cost) among all other solutions,
then such a solution for is said to be an optimal solution.
3. Time Complexity: Time complexity is a measure of time for an
algorithm to complete its task.
4. Space Complexity: It is the maximum storage space required at
any point during the search, as the complexity of the problem.

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Type of Search Strategies

Uninformed search (Also known as blind search) –Uninformed
search algorithms have no additional information on the goal node
other than the one provided in the problem definition
Informed search ( Also known as heuristic search) – Search
strategies know whether one state is more promising than another.

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XX

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DEPTH FIRST SEARCH (DFS)

• The DFS algorithm is a recursive algorithm that uses the idea of
backtracking. It involves exhaustive searches of all the nodes by
going ahead, if possible, else by backtracking.
• Here, the word backtrack means that when you are moving
forward and there are no more nodes along the current path, you
move backwards on the same path to find nodes to traverse. All
the nodes will be visited on the current path till all the unvisited
nodes have been traversed after which the next path will be
selected.
• This recursive nature of DFS can be implemented using stacks.

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The basic idea is as follows:

Pick a starting node and push all its adjacent nodes into a stack.
Pop a node from stack to select the next node to visit and push all
its adjacent nodes into a stack.
Repeat this process until the stack is empty.
• However, ensure that the nodes that are visited are marked. This
will prevent you from visiting the same node more than once.
• If you do not mark the nodes that are visited and you visit the
same node more than once, you may end up in an infinite loop.

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EXAMPLE

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• DFS is not optimal, meaning the number of steps in reaching the

solution, or the cost spent in reaching it is high.
Time Complexity: Time complexity of DFS will be equivalent to the
node traversed by the algorithm. It is given by:
T(n)= 1+ n2+ n3 +.........+ nm= O(nm)
 Where, m= maximum depth of any node and this can be much
larger than d (Shallowest solution depth)

• Not guaranteed that DFS will give you solution.

• Completeness: DFS will provide completeness feature when state

space is finite.

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BREADTH FIRST SEARCH (BFS)

• In Breadth First Search(BFS), the root node of the graph is
expanded first, then all the successor nodes are expanded and then
their successor and so on i.e. the nodes are expanded level wise
starting at root level.
• Breadth-first search (BFS) is an algorithm for traversing or searching
tree or graph data structures.
• BFS algorithm starts searching from the root node of the tree and
expands all successor nodes at the current level before moving to
nodes of next level.
• Breadth-first search implemented using FIFO queue data structure.
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EXAMPLE
S= source node
K= destination node
Path will be according to BFS concept.
S---> A--->B---->C--->D---->G--->H--->
E---->F---->I ---->K

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FF

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• Time Complexity: Time Complexity of BFS algorithm can be

obtained by the number of nodes traversed in BFS until the
shallowest Node. Where the d= depth of shallowest solution and b
is a node at every state.
T (b) = 1+b2+b3+.......+ bd= O (bd)
• In this procedure at any way it will find the goal.
• There is nothing like useless path in BFS, since it searches level by
level.
• BFS consumes large memory space. Its time complexity is more.

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ITERATIVE DEEPENING DFS

• The iterative deepening algorithm is a combination of DFS and BFS
algorithms.
• This search algorithm finds out the best depth limit and does it by
gradually increasing the limit until a goal is found.
• This algorithm performs depth-first search up to a certain "depth
limit", and it keeps increasing the depth limit after each iteration
until the goal node is found.
• This Search algorithm combines the benefits of Breadth-first
search's fast search and depth-first search's memory efficiency.
• The iterative search algorithm is useful uninformed search when
search space is large, and depth of goal node is unknown.
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Advantages:
• It combines the benefits of BFS and DFS search algorithm in terms
of fast search and memory efficiency.
Disadvantages:
• The main drawback of IDDFS is that it repeats all the work of the
previous phase.

If b is the branching factor, and d is the depth of the goal node or
the depth at which the iteration of IDDFS function terminates, the
time complexity is O( 𝒅 )

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Example 1
• Here in the given tree, the starting node is A and the depth
initialized to 0.
• The goal node is R where we have to find the depth and the path to
reach it. The depth from the figure is 4.
• In this example, we consider the tree as a finite tree, while we can
consider the same procedure for the infinite tree as well.
• We knew that in the algorithm of IDDFS we first do DFS till a
specified depth and then increase the depth at each loop.
• This special step forms the part of DLS or Depth Limited Search.
Thus the following traversal shows the IDDFS search.

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EXAMPLE 2

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INFORMED SEARCH ALGORITHMS

• So far we have talked about the uninformed search algorithms
which looked through search space for all possible solutions of the
problem without having any additional knowledge about search
space.
• But informed search algorithm contains an array of knowledge such
as how far we are from the goal, path cost, how to reach to goal
node, etc. This knowledge help agents to explore less to the search
space and find more efficiently the goal node.
• The informed search algorithm is more useful for large search
space. Informed search algorithm uses the idea of heuristic, so it is
also called Heuristic search.
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Heuristics function
• Heuristic is a function which is used in Informed Search, and it finds
the most promising path.
• It takes the current state of the agent as its input and produces the
estimation of how close agent is from the goal.
• The heuristic method, however, might not always give the best
solution, but it guaranteed to find a good solution in reasonable
time.
• Heuristic function estimates how close a state is to the goal. It is
represented by h(n), and it calculates the cost of an optimal path
between the pair of states.
• The value of the heuristic function is always positive.
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h(n) <= h*(n)

• Here h(n) is heuristic cost, and h*(n) is the estimated cost. Hence
heuristic cost should be less than or equal to the estimated cost.
Pure Heuristic Search
• Pure heuristic search is the simplest form of heuristic search
algorithms. It expands nodes based on their heuristic value h(n). It
maintains two lists, OPEN and CLOSED list. In the CLOSED list, it
places those nodes which have already expanded and in the OPEN
list, it places nodes which have yet not been expanded.
• On each iteration, each node n with the lowest heuristic value is
expanded and generates all its successors and n is placed to the
closed list. The algorithm continues unit a goal state is found.
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Best-first Search Algorithm (Greedy Search)

• Greedy best-first search algorithm always selects the path which
appears best at that moment. It is the combination of depth-first
search and breadth-first search algorithms.
• It uses the heuristic function and search. Best-first search allows us
to take the advantages of both algorithms. With the help of best-
first search, at each step, we can choose the most promising node.
In the best first search algorithm, we expand the node which is
closest to the goal node and the closest cost is estimated by
heuristic function, i.e.
h(n)= g(n)
• Were, h(n)= estimated cost from node n to the goal.
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EXAMPLE

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A* ALGORITHM
• It uses heuristic function h(n), and cost to reach the node n from
the start state g(n).
• A* search algorithm finds the shortest path through the search
space using the heuristic function. This search algorithm expands
less search tree and provides optimal result faster.
• A* algorithm is similar to UCS except that it uses g(n)+h(n) instead
of g(n).
• In A* search algorithm, we use search heuristic as well as the cost
to reach the node. Hence we can combine both costs as following,
and this sum is called as a fitness number.
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Advantages
• A* search algorithm is the best algorithm than other search
algorithms.
• A* search algorithm is optimal and complete.
• This algorithm can solve very complex problems.
Disadvantages
• It does not always produce the shortest path as it mostly based on
heuristics and approximation.
• A* search algorithm has some complexity issues.
• The main drawback of A* is memory requirement as it keeps all
generated nodes in the memory, so it is not practical for various
large-scale problems.

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Algorithm of A* search:
• Step1: Place the starting node in the OPEN list.
• Step 2: Check if the OPEN list is empty or not, if the list is empty then
return failure and stops.
• Step 3: Select the node from the OPEN list which has the smallest value
of evaluation function (g+h), if node n is goal node then return success
and stop, otherwise
• Step 4: Expand node n and generate all of its successors, and put n into
the closed list. For each successor n', check whether n' is already in the
OPEN or CLOSED list, if not then compute evaluation function for n' and
place into Open list.
• Step 5: Else if node n' is already in OPEN and CLOSED, then it should be
attached to the back pointer which reflects the lowest g(n') value.
• Step 6: Return to Step 2.
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Example

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• The heuristic value of all states is given in the below table so we

will calculate the f(n) of each state using the formula :
f(n)= g(n) + h(n)
where g(n) is the cost to reach any node from start state.

Prepared By Mr. EBIN PM, Chandigarh University, Punjab EDULINE 60

Prepared By Mr. EBIN PM, AP, CU PUNJAB 30