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Heuristic Search

Heuristic Search. Mahgul Gulzai Moomal Umer Rabail Hafeez. Outline. Introduction Definition of heuristic search Algorithms of heuristic search Best first search Hill climbing strategy Implementing heuristic evaluation functions Admissibility, Monotonicity and Informedness

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Heuristic Search

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  1. Heuristic Search MahgulGulzai MoomalUmer RabailHafeez

  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 Artificial Intelligence www.csc.csudh.edu

  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” Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  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! = 60480. • That is a measure of the amount of work that would have to be done by a brute-force search. Artificial Intelligence www.csc.csudh.edu

  7. First three levels of the tic-tac-toe state space reduced by symmetry Artificial Intelligence www.csc.csudh.edu

  8. The “most wins” heuristic applied to the first children in tic-tac-toe Artificial Intelligence www.csc.csudh.edu

  9. Heuristically reduced state space for tic-tac-toe Artificial Intelligence Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  11. Implementation of Heuristic Search • Heuristic search is implemented in two parts: • The heuristic measure. • The search algorithm. Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  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 Artificial Intelligence www.csc.csudh.edu

  14. The local maximum problem for hill-climbing with 3-level look ahead Artificial Intelligence www.csc.csudh.edu

  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 Artificial Intelligence www.csc.csudh.edu

  16. The best-first search algorithm Artificial Intelligence www.csc.csudh.edu

  17. Heuristic search of a hypothetical state space Artificial Intelligence www.csc.csudh.edu

  18. A trace of the execution of best-first-search Artificial Intelligence www.csc.csudh.edu

  19. Heuristic search of a hypothetical state space with open and closed states highlighted Artificial Intelligence www.csc.csudh.edu

  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) Artificial Intelligence www.csc.csudh.edu

  21. The start state, first moves, and goal state for an example-8 puzzle Artificial Intelligence www.csc.csudh.edu

  22. Three heuristics applied to states in the 8-puzzle Artificial Intelligence www.csc.csudh.edu

  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 Artificial Intelligence www.csc.csudh.edu

  24. The heuristic f applied to states in the 8-puzzle Artificial Intelligence www.csc.csudh.edu

  25. State space generated in heuristic search of the 8-puzzle graph Artificial Intelligence www.csc.csudh.edu

  26. The successive stages of open and closed that generate the graph are: Artificial Intelligence www.csc.csudh.edu

  27. Open and closed as they appear after the 3rd iteration of heuristic search Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  31. Admissibility and Algorithm A* Artificial Intelligence www.csc.csudh.edu

  32. Monotonicity and Informedness Artificial Intelligence www.csc.csudh.edu

  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. Artificial Intelligence www.csc.csudh.edu

  34. 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 Artificial Intelligence www.csc.csudh.edu

  35. Exhaustive Search State space for a variant of nim. Each state partitions the seven matches into one or more piles Artificial Intelligence www.csc.csudh.edu

  36. 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 Artificial Intelligence www.csc.csudh.edu

  37. 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. Artificial Intelligence www.csc.csudh.edu

  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 ishow to value the nodes at the predefined level – heuristics • Propagating values is similar • Maximum children for MAX nodes • Minimum children for MIN nodes Artificial Intelligence www.csc.csudh.edu

  39. Minimax to a hypothetical state space. Leaf states show heuristic values; internal states show backed-up values. Artificial Intelligence www.csc.csudh.edu

  40. Heuristic measuring conflict applied to states of tic-tac-toe Artificial Intelligence www.csc.csudh.edu

  41. Two-ply minimax applied to the opening move of tic-tac-toe Artificial Intelligence www.csc.csudh.edu

  42. Two ply minimax, and one of two possible MAX second moves Artificial Intelligence www.csc.csudh.edu

  43. Two-ply minimax applied to X’s move near the end of the game Artificial Intelligence www.csc.csudh.edu

  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 Artificial Intelligence www.csc.csudh.edu

  45. Artificial Intelligence www.csc.csudh.edu

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