Chapter 4 HEURISTIC SEARCH Contents CSC 411 Hillclimbing

Chapter 4 HEURISTIC SEARCH Contents • • • CSC 411 Hill-climbing Dynamic programming Heuristic search algorithm Admissibility, Monotonicity, and Informedness Using Heuristics In Games Complexity Issues Artificial Intelligence 1

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 CSC 411 Artificial Intelligence 2

First three levels of the tic-tac-toe state space reduced by symmetry CSC 411 Artificial Intelligence 3

The “most wins” heuristic applied to the first children in tic-tac-toe CSC 411 Artificial Intelligence 4

Heuristically reduced state space for tic-tac-toe CSC 411 Artificial Intelligence 5

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 CSC 411 Artificial Intelligence 6

The local maximum problem for hill-climbing with 3 -level look ahead CSC 411 Artificial Intelligence 7

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: – – CSC 411 String matching Spell checking Nature language processing and understanding planning Artificial Intelligence 8

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. CSC 411 Artificial Intelligence 9

• The initialization stage and first step in completing the array for character alignment using dynamic programming. CSC 411 Artificial Intelligence 10

The completed array reflecting the maximum alignment information for the strings. CSC 411 Artificial Intelligence 11

A completed backward component of the dynamic programming example giving one (of several possible) string alignments. CSC 411 Artificial Intelligence 12

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 CSC 411 Artificial Intelligence 13

Initialization of minimum edit difference matrix between intention and execution CSC 411 Artificial Intelligence 14

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) } CSC 411 Artificial Intelligence 15

Complete array of minimum edit difference between intention and execution (of several possible) string alignments. Intention delete I, cost 1 etention replace n with e, cost 2 exentionreplace t with x, cost 2 exenution insert u, cost 1 execution replace n with c, cost 2 CSC 411 Artificial Intelligence 16

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 CSC 411 Artificial Intelligence 17

The bestfirst search algorithm CSC 411 Artificial Intelligence 18

Heuristic search of a hypothetical state space CSC 411 Artificial Intelligence 19 13

A trace of the execution of best-first-search CSC 411 Artificial Intelligence 20

Heuristic search of a hypothetical state space with open and closed states highlighted CSC 411 Artificial Intelligence 21

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) CSC 411 Artificial Intelligence 22

The start state, first moves, and goal state for an example-8 puzzle CSC 411 Artificial Intelligence 23

Three heuristics applied to states in the 8 -puzzle CSC 411 Artificial Intelligence 24

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 CSC 411 Artificial Intelligence 25

The heuristic f applied to states in the 8 -puzzle CSC 411 Artificial Intelligence 26

State space generated in heuristic search of the 8 -puzzle graph CSC 411 Artificial Intelligence 27

The successive stages of open and closed that generate the graph are: CSC 411 Artificial Intelligence 28

Open and closed as they appear after the 3 rd iteration of heuristic search CSC 411 Artificial Intelligence 29

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 CSC 411 Artificial Intelligence 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 CSC 411 Artificial Intelligence 31

Admissibility and Algorithm A* CSC 411 Artificial Intelligence 32

Monotonicity and Informedness CSC 411 Artificial Intelligence 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. CSC 411 Artificial Intelligence 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 CSC 411 Artificial Intelligence 35

Exhaustive Search State space for a variant of nim. Each state partitions the seven matches into one or more piles CSC 411 Artificial Intelligence 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 CSC 411 Artificial Intelligence 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. CSC 411 Artificial Intelligence 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 lookahead 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 CSC 411 Artificial Intelligence 39

Minimax to a hypothetical state space. Leaf states show heuristic values; internal states show backed-up values. CSC 411 Artificial Intelligence 40

Heuristic measuring conflict applied to states of tic-tac-toe CSC 411 Artificial Intelligence 41

Two-ply minimax applied to the opening move of tic-tac-toe CSC 411 Artificial Intelligence 42

Two ply minimax, and one of two possible MAX second moves CSC 411 Artificial Intelligence 43

Two-ply minimax applied to X’s move near the end of the game CSC 411 Artificial Intelligence 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 CSC 411 Artificial Intelligence 45

Alpha-beta pruning applied. States without numbers are not evaluated. CSC 411 Artificial Intelligence 46