1. Mahgul Gulzai Moomal Umer Rabail Hafeez
Heuristic Search
Mahgul Gulzai
Moomal Umer
Rabail Hafeez

2. Outline Introduction Definition of heuristic search
Algorithms of heuristic search
Best first search
Hill climbing strategy
Implementing heuristic evaluation functions
Admissibility, Monotonicity and Informedness
Using heuristic in games
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3. Introduction
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”
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4. Definition
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.
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5. AI and Heuristic Search
Heuristic Search is used in AI in two situations:
When a problem doesn’t have an exact solution.
There is an exact solution but the computational cost of finding it exceeds the limit.
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6. Example- Tic Tac Toe 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! =
That is a measure of the amount of work that would have to be done by a brute-force search.
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7. First three levels of the tic-tac-toe state space reduced by symmetry
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8. The “most wins” heuristic applied to the first children in tic-tac-toe
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9. Heuristically reduced state space for tic-tac-toe
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CSC411
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10. Simple Heuristic for Tic Tac Toe: Move to the square in which X has the most winning lines
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.
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11. Implementation of Heuristic Search
Heuristic search is implemented in two parts:
The heuristic measure.
The search algorithm.
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12. Hill Climbing
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.
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13. Potential Problems with Simple Hill Climbing
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
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14. The local maximum problem for hill-climbing with 3-level look ahead
Insert fig 4.4
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15. The Best-First Search
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
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16. The best-first search algorithm
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17. Heuristic search of a hypothetical state space
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18. A trace of the execution of best-first-search
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19. Heuristic search of a hypothetical state space with open and closed states highlighted
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20. Implement Heuristic Evaluation Function
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)
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21. The start state, first moves, and goal state for an example-8 puzzle
Insert fig 4.12
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22. Three heuristics applied to states in the 8-puzzle
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23. Heuristic Design
Use the limited information available in a single state to make intelligent choices
Empirical, judgment, and intuition
Must be its actual performance on problem instances
The 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 n
h(n) – from the current state n to the goal state
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24. The heuristic f applied to states in the 8-puzzle
Insert fig 4.15
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25. State space generated in heuristic search of the 8-puzzle graph
Insert fig 4.16
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26. The successive stages of open and closed that generate the graph are:
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27. Open and closed as they appear after the 3rd iteration of heuristic search
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28. Analysis of the evaluation function
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.
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29. Heuristic search and expert systems
Expert System employ confidence measures to select the conclusions with the highest likelihood of the success through heuristics implementation.
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30. Admissibility, Monotonicity, and Informedness
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.
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31. Admissibility and Algorithm A*
Insert def box page 146
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32. Monotonicity and Informedness
Insert def box, pg 147
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33. 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.
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34. Minimax Procedure Games A variant of game nim
Two players attempting to win
Two opponents are referred to as MAX and MIN
A 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
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35. Exhaustive Search
State space for a variant of nim. Each state partitions the seven matches into one or more piles
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36. Maxmin Search Principles Value propagating rules
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 states
Value 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
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37. Exhaustive minimax for the game of nim
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.
Insert fig 4.20
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38. Minmaxing to Fixed Ply Depth
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
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39. Minimax to a hypothetical state space
Minimax to a hypothetical state space. Leaf states show heuristic values; internal states show backed-up values.
Insert fig 4.21
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40. Heuristic measuring conflict applied to states of tic-tac-toe
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41. Two-ply minimax applied to the opening move of tic-tac-toe
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42. Two ply minimax, and one of two possible MAX second moves
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43. Two-ply minimax applied to X’s move near the end of the game
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44. Alpha-Beta Procedure Alpha-beta pruning to improve search efficiency
Proceeds in a depth-first fashion and creates two values alpha and beta during the search
Alpha associated with MAX nodes, and never decreases
Beta associated with MIN nodes, never increases
To 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 ancestors
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45. Artificial Intelligence www.csc.csudh.edu