Artificial Intelligence Lecture : Heuristic Search-I Ghulam Irtaza Sheikh– Dept CS-BZU Adapted from original course material & others

Today Hill-climbing Heuristic search algorithm Heuristic Evaluation Function

Heuristic “A commonsense rule (or set of rules) intended to increase the probability of solving some problem”

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 Fallible

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

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

Heuristically reduced state space for tic-tac-toe

Today Hill-climbing Heuristic search algorithm Heuristic Evaluation Function

Hill-Climbing Analog Principle Problem 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

The local maximum problem for hill-climbing with 3-level look ahead Insert fig 4.4

Today Hill-climbing Heuristic search algorithm Heuristic Evaluation Function

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

Best-First Search

Best-First / Greedy Search Expand the node that seems closest… What can go wrong?

Best-First / Greedy Search A common case: Best-first takes you straight to the goal on a wrong path Worst-case: like a badly-guided DFS in the worst case Can explore everything Can get stuck in loops if no cycle checking Like DFS in completeness (finite states w/ cycle checking) b … b …

Best First Search What do we need to do to make it complete? Algorithm Complete Optimal Time Space Greedy Best-First Search Y* N O(bm) O(bm) b … m What do we need to do to make it complete? Can we make it optimal?

The best-first search algorithm

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

Today Hill-climbing Heuristic search algorithm Heuristic Evaluation Function

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)

The start state, first moves, and goal state for an example-8 puzzle Insert fig 4.12

Three heuristics applied to states in the 8-puzzle

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

The heuristic f applied to states in the 8-puzzle Insert fig 4.15

State space generated in heuristic search of the 8-puzzle graph Insert fig 4.16

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

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