Introduction to Artificial Intelligence. Informed Search
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1 Introduction to Artificial Intelligence Informed Search Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU Winter Term 2004/2005 B. Beckert: KI für IM p.1
2 Outline Best-first search A search Heuristics B. Beckert: KI für IM p.2
3 Review: Tree search function TREE-SEARCH( problem, fringe) returns a solution or failure fringe INSERT(MAKE-NODE(INITIAL-STATE[problem]),fringe) loop do if fringe is empty then return failure node REMOVE-FIRST(fringe) if GOAL-TEST[problem] applied to STATE(node) succeeds then return node else fringe INSERT-ALL(EXPAND(node, problem), fringe) end Strategy Defines the order of node expansion B. Beckert: KI für IM p.3
4 Best-first search Idea Use an evaluation function for each node (estimate of desirability ) Expand most desirable unexpanded node Implementation fringe is a queue sorted in decreasing order of desirability Special cases Greedy search A search B. Beckert: KI für IM p.4
5 Romania with step costs in km Oradea 71 Neamt Zerind Arad Timisoara 111 Lugoj 70 Mehadia Dobreta Sibiu 99 Fagaras 80 Rimnicu Vilcea 97 Pitesti Bucharest 90 Craiova Giurgiu 87 Iasi Urziceni Vaslui Hirsova 86 Eforie Straight line distance to Bucharest Arad 366 Bucharest 0 Craiova 160 Dobreta 242 Eforie 161 Fagaras 178 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 98 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374 B. Beckert: KI für IM p.5
6 Greedy search Heuristic Evaluation function h(n) = estimate of cost from n to goal Greedy search expands the node that appears to be closest to goal Example h SLD (n) = straight-line distance from n to Bucharest Note Unlike uniform-cost search the node evaluation function has nothing to do with the nodes explored so far B. Beckert: KI für IM p.6
7 Greedy search: Example Romania Arad 366 B. Beckert: KI für IM p.7
8 Greedy search: Example Romania Arad Sibiu Timisoara Zerind B. Beckert: KI für IM p.7
9 Greedy search: Example Romania Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea B. Beckert: KI für IM p.7
10 Greedy search: Example Romania Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea Sibiu Bucharest B. Beckert: KI für IM p.7
11 Greedy search: Properties Complete Time Space Optimal B. Beckert: KI für IM p.8
12 Greedy search: Properties Complete No Can get stuck in loops Example: Iasi to Oradea Iasi Neamt Iasi Neamt Complete in finite space with repeated-state checking Time Space Optimal B. Beckert: KI für IM p.8
13 Greedy search: Properties Complete No Time O(b m ) Space Optimal Can get stuck in loops Example: Iasi to Oradea Iasi Neamt Iasi Neamt Complete in finite space with repeated-state checking B. Beckert: KI für IM p.8
14 Greedy search: Properties Complete No Time O(b m ) Space O(b m ) Optimal Can get stuck in loops Example: Iasi to Oradea Iasi Neamt Iasi Neamt Complete in finite space with repeated-state checking B. Beckert: KI für IM p.8
15 Greedy search: Properties Complete No Time O(b m ) Space O(b m ) Can get stuck in loops Example: Iasi to Oradea Iasi Neamt Iasi Neamt Complete in finite space with repeated-state checking Optimal No B. Beckert: KI für IM p.8
16 Greedy search: Properties Complete No Time O(b m ) Space O(b m ) Can get stuck in loops Example: Iasi to Oradea Iasi Neamt Iasi Neamt Complete in finite space with repeated-state checking Optimal No Note Worst-case time same as depth-first search, Worst-case space same as breadth-first But a good heuristic can give dramatic improvement B. Beckert: KI für IM p.8
17 A search Idea Avoid expanding paths that are already expensive Evaluation function where f (n) = g(n) + h(n) g(n) = cost so far to reach n h(n) = estimated cost to goal from n f (n) = estimated total cost of path through n to goal B. Beckert: KI für IM p.9
18 A search: Admissibility Admissibility of heuristic h(n) is admissible if h(n) h (n) for all n where h (n) is the true cost from n to goal B. Beckert: KI für IM p.10
19 A search: Admissibility Admissibility of heuristic h(n) is admissible if h(n) h (n) for all n where h (n) is the true cost from n to goal Also required h(n) 0 for all n In particular: h(g) = 0 for goal G B. Beckert: KI für IM p.10
20 A search: Admissibility Admissibility of heuristic h(n) is admissible if h(n) h (n) for all n where h (n) is the true cost from n to goal Also required h(n) 0 for all n In particular: h(g) = 0 for goal G Example Straight-line distance never overestimates the actual road distance B. Beckert: KI für IM p.10
21 A search: Admissibility Theorem A search with admissible heuristic is optimal B. Beckert: KI für IM p.11
22 A search example Arad 366=0+366 B. Beckert: KI für IM p.12
23 A search example Arad Sibiu 393= Timisoara Zerind 447= = B. Beckert: KI für IM p.12
24 A search example Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = = = B. Beckert: KI für IM p.12
25 A search example Arad Sibiu Timisoara Zerind 447= = Arad Fagaras 646= = Oradea 671= Rimnicu Vilcea Craiova Pitesti Sibiu 526= = = B. Beckert: KI für IM p.12
26 A search example Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = Sibiu Bucharest Craiova Pitesti Sibiu 591= = = = = B. Beckert: KI für IM p.12
27 A search example Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = Sibiu Bucharest Craiova Pitesti Sibiu 591= = = = Bucharest Craiova Rimnicu Vilcea 418= = = B. Beckert: KI für IM p.12
28 A search: f -contours A gradually adds f -contours of nodes O Z N A I T S R F V L P D M C 420 G B U H E B. Beckert: KI für IM p.13
29 Optimality of A search: Proof Start n G G 2 Suppose a suboptimal goal G 2 has been generated Let n be an unexpanded node on a shortest path to an optimal goal G f (G 2 ) = g(g 2 ) since h(g 2 ) = 0 > g(g) since G 2 suboptimal = g(n) + h (n) g(n) + h(n) since h is admissible = f (n) Thus, A never selects G 2 for expansion B. Beckert: KI für IM p.14
30 A search: Properties Complete Time Space Optimal B. Beckert: KI für IM p.15
31 A search: Properties Complete Yes (unless there are infinitely many nodes n with f (n) f (G)) Time Space Optimal B. Beckert: KI für IM p.15
32 A search: Properties Complete Time Yes (unless there are infinitely many nodes n with f (n) f (G)) Exponential in [relative error in h length of solution] Space Optimal B. Beckert: KI für IM p.15
33 A search: Properties Complete Time Space Yes (unless there are infinitely many nodes n with f (n) f (G)) Exponential in [relative error in h length of solution] Same as time Optimal B. Beckert: KI für IM p.15
34 A search: Properties Complete Time Space Optimal Yes (unless there are infinitely many nodes n with f (n) f (G)) Exponential in [relative error in h length of solution] Same as time Yes B. Beckert: KI für IM p.15
35 A search: Properties Complete Time Space Optimal Yes (unless there are infinitely many nodes n with f (n) f (G)) Exponential in [relative error in h length of solution] Same as time Yes Note A expands all nodes with A expands some nodes with A expands no nodes with f (n) < C f (n) = C f (n) > C B. Beckert: KI für IM p.15
36 Admissible heuristics: Example 8-puzzle Start State Goal State Addmissible heuristics h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) B. Beckert: KI für IM p.16
37 Admissible heuristics: Example 8-puzzle Start State Goal State Addmissible heuristics h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) In the example h 1 (S) = h 2 (S) = B. Beckert: KI für IM p.16
38 Admissible heuristics: Example 8-puzzle Start State Goal State Addmissible heuristics h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) In the example h 1 (S) = 6 h 2 (S) = B. Beckert: KI für IM p.16
39 Admissible heuristics: Example 8-puzzle Start State Goal State Addmissible heuristics h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance (i.e., no. of squares from desired location of each tile) In the example h 1 (S) = 6 h 2 (S) = = 14 B. Beckert: KI für IM p.16
40 Dominance Definition h 1,h 2 two admissible heuristics h 2 dominates h 1 if h 2 (n) h 1 (n) for all n B. Beckert: KI für IM p.17
41 Dominance Definition h 1,h 2 two admissible heuristics h 2 dominates h 1 if h 2 (n) h 1 (n) for all n Theorem If h 2 dominates h 1, then h 2 is better for search than h 1. B. Beckert: KI für IM p.17
42 Dominance: Example 8-puzzle Typical search costs d = 14 IDS 3,473,941 nodes A (h 1 ) A (h 2 ) 539 nodes 113 nodes d = 24 IDS too many nodes A (h 1 ) A (h 2 ) 39,135 nodes 1,641 nodes d: depth of first solution IDS: iterative deepening search B. Beckert: KI für IM p.18
43 Relaxed problems Finding good admissible heuristics is an art! Deriving admissible heuristics Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem B. Beckert: KI für IM p.19
44 Relaxed problems Finding good admissible heuristics is an art! Deriving admissible heuristics Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem Example If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then we get heuristic h 1 If the rules are relaxed so that a tile can move to any adjacent square, then we get heuristic h 2 B. Beckert: KI für IM p.19
45 Relaxed problems Finding good admissible heuristics is an art! Deriving admissible heuristics Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem Example If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then we get heuristic h 1 If the rules are relaxed so that a tile can move to any adjacent square, then we get heuristic h 2 Key point The optimal solution cost of a relaxed problem is not greater than the optimal solution cost of the real problem B. Beckert: KI für IM p.19
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