Artificial Intelligence

Transcription

1 Artificial Intelligence Dr. Malek Mouhoub Department of Computer Science University of Regina Fall 2005 Malek Mouhoub, CS820 Fall

2 3. State-Space Search 3. State-Space Search Graph Theory Uninformed Search Breath-First Search Depth-First or Backtracking Search Iterative Deepening Heuristic Search And/Or graphs Malek Mouhoub, CS820 Fall

3 Graph Theory Graph Theory Riverbank 1 rb1 River b2 b3 b4 1 Island 1 Island 2 Graph representation i1 b1 i b6 b5 b7 Riverbank 2 rb2 Figure 1: The bridges of Koenigsberg problem Malek Mouhoub, CS820 Fall

4 Graph Theory Graph Theory Graph = (X, U) Direct graph. Rooted graph. tip of leaf node. Path, ancestor, descendant, cycle. Tree. Malek Mouhoub, CS820 Fall

5 Graph Theory Graph Theory Graph representation : Boolean matrix. G[i, j] = 1 if (i, j) U 0 if not List of successors/predecessors. Malek Mouhoub, CS820 Fall

6 Implementing graph search : Uninformed search Implementing graph search : Uninformed search Input : G with lists of successors S, a node s. Output : G with all paths, starting at a, marked. Begin Q = {s} While Q NIL Peek v from Q Mark v For all v S(v) If v is not marked Add v to Q End If End For End While Q is a stack Depth-First Search Q is a Queue Breadth-First Search Q is a Priority Queue Best-First Search Malek Mouhoub, CS820 Fall

7 Implementing graph search : Uninformed search Input : G with lists of successors S, nodes s,t. Output : A path from s to t. Begin Q = {s} m(s) = 0 While Q NIL and m(t) undefined Peek v from Q For all v S(v) If v is undefined m(v ) = v If (v = t) path(v ) Else Add v to Q End If End If End For End While path(v) If m(v) = 0 return (v) Else path(m(v)) v End If Malek Mouhoub, CS820 Fall

8 Breadth-First Search Breadth-First Search Malek Mouhoub, CS820 Fall

9 Breadth-First Search Breadth-First Search Malek Mouhoub, CS820 Fall

10 Depth-First Search Depth-First Search Malek Mouhoub, CS820 Fall

11 Depth-First Search Depth-First Search Malek Mouhoub, CS820 Fall

12 Iterative Deepening Iterative Deepening Successive depth-first searches are conducted, each with depth bounds increasing by 1, until a goal node is found. Enjoys the linear memory requirements of depth-first search while guaranteeing that a goal node of minimal depth will be found. The number if nodes expanded by iterative-deepening search is not many more than would be expanded by breath-first search. Malek Mouhoub, CS820 Fall

13 Iterative Deepening Depth bound = 1 Depth bound = 2 Depth bound = 3 Depth bound = 4 Figure 2: Stages in Iterative-Deepening Search. Malek Mouhoub, CS820 Fall

14 Heuristic search Heuristic search Heuristic evaluation functions f(n) = g(n) + h(n) Example of the 8-puzzle : 1. Depth in the search space + number of tiles out of place. 2. Depth in the search space + minimum number of moves to reach the goal state. Malek Mouhoub, CS820 Fall

15 Heuristic search 1(4) Start node (6) 2 (4) (6) (5) (5) 4 (6) (6) (7) (5) 5 (7) (5) Goal node (5) (7) Figure 3: Depth in the search space + number of tiles out of place Malek Mouhoub, CS820 Fall

16 Heuristic search 1(5) Start node (7) 2 (5) (7) (7) (5) 3 (7) (5) (5) Goal node (5) (7) Figure 4: Depth in the search space + minimum number of moves to reach the goal state. Malek Mouhoub, CS820 Fall

17 Heuristic search Algorithm A Algorithm A : Best-first search using f(n). Algorithm A : Best-first search using f(n) and where : g (n) g(n) h (n) h(n) Malek Mouhoub, CS820 Fall

18 Heuristic search Properties of heuristics Admissibility : Guaranty to find a minimal path to a solution. All A (h(n) h (n)) algorithms are admissible. Breath-fist search algorithm is admissible. f(n) = g(n) + 0. Monotonicity : Any state discovered during the search won t be found later at a cheaper cost. nodes n i n j, h(n i ) h(n j ) cost(n i, n j ). h(goal) = 0. Informedness : if [ node n, h 1 (n) h 2 (n) ] then h 2 is more informed than h 1. The algorithm using heuristic 1 of the 8-puzzle problem is more informed than the breath-first search algorithm. Heuristic 2 of the 8-puzzle problem is more informed than heuristic 1. Malek Mouhoub, CS820 Fall

19 Heuristic search MinMax algorithm Two-person games : Difficulty to develop search algorithms (unpredictable opponent). Searching the space of possible moves and counter moves by the opponent. Guiding the play along a path to a winning state. MinMax algorithm : Max : Min : the player trying to win (maximize his advantage). the opponent trying to minimize Max s score. Malek Mouhoub, CS820 Fall

20 Heuristic search MinMax algorithm MinMax algorithm : 1. Descend to the tip of leaf nodes of the tree. 2. Propagate values back up the tree. (a) If the parent state is Max, give it the maximum value among its children. (b) If the parent is a Min node, give it the minimum value of its children. Malek Mouhoub, CS820 Fall

21 Heuristic search MIN 7 1 MAX MIN MAX MIN MAX Figure 5: Exhaustive minmax for the game of nim. Malek Mouhoub, CS820 Fall

22 Heuristic search MAX MIN MAX MIN Step 1 Figure 6: MinMax to a hypothetical state space. Malek Mouhoub, CS820 Fall

23 Heuristic search MAX MIN MAX MIN Step 2 Figure 7: MinMax to a hypothetical state space. Malek Mouhoub, CS820 Fall

24 Heuristic search MAX MIN MAX MIN Step 3 Figure 8: MinMax to a hypothetical state space. Malek Mouhoub, CS820 Fall

25 Heuristic search MAX MIN MAX MIN Step 4 Figure 9: MinMax to a hypothetical state space. Malek Mouhoub, CS820 Fall

26 Heuristic search Alpha-Beta algorithm MAX 3 MIN MAX MIN Figure 10: Alpha-Beta pruning applied to a hypothetical state space. Malek Mouhoub, CS820 Fall

27 And/Or graphs And/Or graphs (Large,Medium,Small) ==> (Large,Medium,Small) on 1 on 3 (111)==>(333) (111)==>(122) (122)==>(322) Primitive (322)==>(333) (111)==>(113) (113)==>(123) (123)==>(122) (322)==>(321) (321)==>(331) (331)==>(333) Primitive Primitive Primitive Primitive Primitive Primitive Figure 11: A subproblem graph reducing the Tower-of-Hanoi puzzle to primitive one-move puzzles. Malek Mouhoub, CS820 Fall