Mahgul Gulzai Moomal Umer Rabail Hafeez. Introduction Definition of heuristic search Algorithms of heuristic search Best first search Hill climbing strategy.

Mahgul Gulzai

Moomal Umer

Rabail Hafeez

Introduction Definition of heuristic searchAlgorithms of heuristic search

Best first search Hill climbing strategy Implementing heuristic evaluation functions

Admissibility, Monotonicity and Informedness Using heuristic in games

Artificial Intelligence www.csc.csudh.edu

Human generally consider number of alternative strategies on their way to solving a problem .

To obtain the best possible strategy, humans use search.

For example : A chess player consider number of possible moves, A Doctor examine several possible diagnoses .

Human Problem solving seems to be based on judgmental rules that guide our search to those portion of state Space that seems some how promising.

These rules are known as “Heuristics”


George Polya defines Heuristics as, “The study of methods and rules of discovery and invention”.

A heuristic is a method that might not always find the best solution . but is guaranteed to find a good solution in

reasonable time. By sacrificing completeness it increases

efficiency. Useful in solving tough problems which

could not be solved any other way. solutions take an infinite time or very long time

to compute. Artificial Intelligence www.csc.csudh.edu

Heuristic Search is used in AI in two situations:1.When a problem doesn’t have an exact

solution.2.There is an exact solution but the

computational cost of finding it exceeds the limit.


Consider the game of tic-tac-toe. Even if we use symmetry to reduce the

search space of redundant moves, the number of possible paths through the search space is something like 12 x 7! = 60480.

That is a measure of the amount of work that would have to be done by a brute-force search.


First three levels of the tic-tac-toe state space First three levels of the tic-tac-toe state space reduced by symmetryreduced by symmetry


The “most wins” heuristic applied to the first children in tic-tac-toe


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Heuristically reduced state space for tic-tac-toe


Using this rule, we can see that a corner square has heuristic value of 3, a side square has a heuristic value of 2, but the centre square has a heuristic value of 4.

So we can prune the left and right branches of the search tree.

This removes 2/3 of the search space on the first move.

If we apply the heuristic at each level of the search, we will remove most of the states from consideration thereby greatly improving the efficiency of the search.


Heuristic search is implemented in two parts: The heuristic measure. The search algorithm.


Use heuristic to move only to states that are better than the current state.

Always move to better state when possible.The process ends when all operators have

been applied and none of the resulting states are better than the current state.


• Will terminate when at local optimum.• The order of application of operators can

make a big difference.• Can’t see past a single move in the state

space


The local maximum problem for hill-climbing with 3-level look ahead


Also heuristic search – use heuristic (evaluation) function to select the best state to explore

Can be implemented with a priority queue Breadth-first implemented with a queue Depth-first implemented with a stack


The best-The best-first first search search algorithmalgorithm


Heuristic search of a hypothetical state space


A trace of the execution of best-first-search


Heuristic search of a hypothetical state space with open and closed states highlighted


Heuristics can be evaluated in different ways

8-puzzle problem Heuristic 1: count the tiles out of places

compared with the goal state Heuristic 2: sum all the distances by which the

tiles are out of pace, one for each square a tile must be moved to reach its position in the goal state

Heuristic 3: multiply a small number (say, 2) times each direct tile reversal (where two adjacent tiles must be exchanged to be in the order of the goal)


The start state, first moves, and goal state for an example-8 puzzle


Three heuristics applied to states in the 8-puzzle


Use the limited information available in a single state to make intelligent choices

Empirical, judgment, and intuitionMust be its actual performance on problem instancesThe solution path consists of two parts: from the

starting state to the current state, and from the current state to the goal state

The first part can be evaluated using the known information

The second part must be estimated using unknown information

The total evaluation can be f(n) = g(n) + h(n)

g(n) – from the starting state to the current state nh(n) – from the current state n to the goal stateArtificial Intelligence www.csc.csudh.edu

The heuristic f applied to states in the 8-puzzle


State space generated in heuristic search of the 8-puzzle graph


The successive stages of open and closed that generate the graph are:


Open and closed as they appear after the 3rd iteration of heuristic search


evaluation function for the states in a search space, you are interested in two things:

g(n): How far is state n from the start state? h(n): How far is state n from a goal state? Evaluation function. This gives us the

following evaluation function: f(n) = g(n) + h(n) where g(n) measures the actual length of the path from the start state to the state n, and h(n) is a heuristic estimate of the distance from a state n to a goal state.


Expert System employ confidence measures to select the conclusions with the highest likelihood of the success through heuristics implementation.


A best-first search algorithm guarantee to find a best path, if exists, if the algorithm is admissible.

A best-first search algorithm is admissible if its heuristic function h is monotone.


Admissibility and Algorithm A*Admissibility and Algorithm A*


Monotonicity and InformednessMonotonicity and Informedness


Comparison of state space searched using heuristic search with space searched by breadth-first search. The proportion of the graph searched heuristically is shaded. The optimal search selection is in bold. Heuristic used is f(n) = g(n) + h(n) where h(n) is tiles out of place.Artificial Intelligence www.csc.csudh.edu

Games Two players attempting to win Two opponents are referred to as MAX and

MINA variant of game nim

A number of tokens on a table between the 2 opponents

Each player divides a pile of tokens into two nonempty piles of different sizes

The player who cannot make division losses


State space for a variant of nim. Each state partitions the seven matches into one or more piles

Exhaustive SearchExhaustive Search


Principles MAX tries to win by maximizing her score, moves to a

state that is best for MAX MIN, the opponent, tries to minimize the MAX’s score,

moves to a state that is worst for MAX Both share the same information MIN moves first The terminating state that MAX wins is scored 1,

otherwise 0 Other states are valued by propagating the value of

terminating statesValue propagating rules

If the parent state is a MAX node, it is given the maximum value among its children

If the parent state is a MIN state, it is given the minimum value of its children


Exhaustive minimax for the game of nim. Bold lines indicate forced win for MAX. Each node is marked with its derived value (0 or 1) under minimax.


If cannot expand the state space to terminating (leaf) nodes (explosive), can use the fixed ply depth

Search to a predefined number, n, of levels from the starting state, n-ply look-ahead

The problem is how to value the nodes at the predefined level – heuristics

Propagating values is similar Maximum children for MAX nodes Minimum children for MIN nodes


Minimax to a hypothetical state space. Leaf states show heuristic values; internal states show backed-up values.


Heuristic measuring conflict applied to states of tic-tac-toe


Two-ply minimax applied to the opening move of tic-tac-toe


Two ply minimax, and one of two possible MAX second moves


Two-ply minimax applied to X’s move near the end of the game


Alpha-beta pruning to improve search efficiencyProceeds in a depth-first fashion and creates two

values alpha and beta during the searchAlpha associated with MAX nodes, and never

decreasesBeta associated with MIN nodes, never increasesTo begin, descend to full ply depth in a depth-first

search, and apply heuristic evaluation to a state and all its siblings. The value propagation is the same as minimax procedure

Next, descend to other grandchildren and terminate exploration if any of their values is >= this beta value

Terminating criteria Below any MIN node having beta <= alpha of any of its MAX

ancestors Below any MAX node having alpha >= beta of any of its MIN

ancestorsArtificial Intelligence www.csc.csudh.edu


Mahgul Gulzai Moomal Umer Rabail Hafeez. Introduction Definition of heuristic search Algorithms of heuristic search Best first search Hill climbing strategy.

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