1. Artificial Intelligence
Chapter 4
HEURISTIC SEARCH
Contents
Hill-climbing
Dynamic programming
Heuristic search algorithm
Admissibility, Monotonicity, and Informedness
Using Heuristics In Games
Complexity Issues
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2. Artificial Intelligence
Heuristics
Rules for choosing paths in a state space that most likely lead to an acceptable problem solution
Purpose
Reduce the search space
Reasons
May not have exact solutions, need approximations
Computational cost is too high
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3. Artificial Intelligence
First three levels of the tic-tac-toe state space reduced by symmetry
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4. Artificial Intelligence
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
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6. Artificial Intelligence
Hill-Climbing
Analog
Go uphill along the steepest possible path until no farther up
Principle
Expand the current state of the search and evaluate its children
Select the best child, ignore its siblings and parent
No history for backtracking
Problem
Local maxima – not the best solution
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The local maximum problem for hill-climbing with 3-level look ahead
Insert fig 4.4
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8. Artificial Intelligence
Dynamic Programming
Forward-backward searching
Divide-and-conquer: Divided problems into multiple interacting and related subproblems
Address issues of reusing subproblems solutions
An example: Fibonacci series
F(0) = 1; F(1)=1; F(n)=F(n-1)+F(n-2)
Keep track of the computation F(n-1) and F(n-2), and reuse their results to compute F(n)
Compare with recursion, much more efficient
Applications:
String matching
Spell checking
Nature language processing and understanding
planning
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9. Global Alignment of Strings
Find an optimal global alignment of two character strings
Data structure: (n+1)(m+1) array, each element reflects the global alignment success to that point
Three possible costs for the current state
If a character is shifted along in the shorter string for better possible alignment, the cost is 1 and recorded in the column score; and If a new character is inserted, cost is 1 and reflected in the row score
If the characters to be aligned are different, shift and insert, the cost is 2
If identical, the cost is 0
The initialization stage and first step in completing the array for character alignment using dynamic programming.
Insert fig 4.5 WITH BAADDCABDDA
BBADC B A
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10. Artificial Intelligence
The initialization stage and first step in completing the array for character alignment using dynamic programming.
Insert fig 4.5 WITH BAADDCABDDA
BBADC B A
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The completed array reflecting the maximum alignment information for the strings.
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A completed backward component of the dynamic programming example giving one (of several possible) string alignments.
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13. Minimum Edit Difference
Determine the best approximate words of s misspelling word in spelling checker
Specified as the number of character insertion, deletion, and replacements necessary to turn the first string into the second
Cost: 1 for insertion and deletion, and 2 for replacement
Determine the minimum cost difference
Data structure: array
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Initialization of minimum edit difference matrix between intention and execution
Insert fig 4.8
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Array elements are the costs of the minimum editing to that point plus the minimum cost of either an insertion, deletion or replacement
Cost(x, y) = min{ Cost(x-1, y) + 1 (insertion cost),
Cost(x-1, y-1) + 2 (replacement cost),
Cost(x, y-1) + 1 (deletion cost) }
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Complete array of minimum edit difference between intention and execution (of several possible) string alignments.
Intention
ntention delete I, cost 1
etention replace n with e, cost 2
exention replace t with x, cost 2
exenution insert u, cost 1
execution replace n with c, cost 2
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17. Artificial Intelligence
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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18. Artificial Intelligence
The best-first search algorithm
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19. Artificial Intelligence
Heuristic search of a hypothetical state space
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A trace of the execution of best-first-search
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Heuristic search of a hypothetical state space with open and closed states highlighted
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22. 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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The start state, first moves, and goal state for an example-8 puzzle
Insert fig 4.12
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Three heuristics applied to states in the 8-puzzle
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25. Artificial Intelligence
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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The heuristic f applied to states in the 8-puzzle
Insert fig 4.15
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State space generated in heuristic search of the 8-puzzle graph
Insert fig 4.16
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28. Artificial Intelligence
The successive stages of open and closed that generate the graph are:
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Open and closed as they appear after the 3rd iteration of heuristic search
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30. Heuristic Design Summary
f(n) is computed as the sum of g(n) and h(n)
g(n) is the depth of n in the search space and has the search more of a breadth-first flavor.
h(n) is the heuristic estimate of the distance from n to a goal
The h value guides search toward heuristically promising states
The g value grows to determine h and force search back to a shorter path, and thus prevents search from persisting indefinitely on a fruitless path
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31. 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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32. Admissibility and Algorithm A*
Insert def box page 146
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33. Monotonicity and Informedness
Insert def box, pg 147
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34. Artificial Intelligence
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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35. Artificial Intelligence
Minimax Procedure
Games
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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Exhaustive Search
State space for a variant of nim. Each state partitions the seven matches into one or more piles
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37. Artificial Intelligence
Maxmin 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 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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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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39. 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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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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Heuristic measuring conflict applied to states of tic-tac-toe
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Two-ply minimax applied to the opening move of tic-tac-toe
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Two ply minimax, and one of two possible MAX second moves
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Two-ply minimax applied to X’s move near the end of the game
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45. Artificial Intelligence
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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46. Artificial Intelligence
Alpha-beta pruning applied. States without numbers are not evaluated.
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