Problem Solving and Search

Transcription

1 Artificial Intelligence Problem Solving and Search Dae-Won Kim School of Computer Science & Engineering Chung-Ang University

2 Outline Problem-solving agents Problem types Problem formulation Example problems Basic search algorithms

3 Problem-Solving Agents

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5 On holiday In Romania; currently in Arad. Flight leaves tomorrow for Bucharest.

6 Goal: be in Bucharest

7 Solution: sequence of cities

8 Solution:??? Performance measure:???

9 Problem formulation: states various cities actions drive between cities

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11 Problem Formulation: How To vs. Problem Modeling

12 A problem is defined by four items: Initial state Successor function: set of action-state pairs Goal test Path cost

13 A solution is a sequence of actions leading from the initial state to a goal state. Consider a solution in algorithm

14 Problem Formulation: Romania

15 Initial state: Successor function: Goal test: Path cost:

16 Initial state: x = at Arad Successor function: S = {<Arad Zerind,Zerind>, } Goal test: x = at Bucharest Path cost: sum of distances

17 Problem Formulation: Vacuum Cleaner

18 States: Actions: Goal test: Path cost:

19 States: integer dirt and robot locations Actions: left, right, suck, stay Goal test: no dirt Path cost: 1 per action (0 for stay)

20 Problem Formulation: Robot Assembly

21 States: real-valued coordinates of joint angles Actions: continuous motions of robot joints Goal test: complete assembly Path cost: time to execute

22 Problem Formulation: The 8-Puzzle

23 States? Actions? Goal test? Path cost?

24 How to achieve the goal state through the complex state space from the initial state?

25 Answer: Tree Search Algorithms

26 Idea: exploration of state space by generating successors of alreadyexplored states (expanding states)

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28 Implementation: States vs. Nodes

29 A state is (a representation of) a physical configuration

30 A node is data structure constituting part of a search tree includes parents, children, depth, path cost.

31 A search strategy is defined by picking the order of node expansion.

32 Strategies are evaluated along the following dimensions: Completeness Time complexity Space complexity Optimality

33 Uninformed Search Strategies

34 Uninformed search strategies use only the information available in the problem definition. Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search

35 Breath-First Search

36 Expand shallowest unexpanded node. Implementation: FIFO queue

37 Complete? Time complexity? Space complexity? Optimal?

38 Complete? Yes (if b is finite) Time complexity? O(b d+1 ) Space complexity? O(b d+1 ) Optimal? Yes (if cost = 1 per step)

39 Uniform-Cost Search Expand least-cost unexpanded node using queue ordered by path cost Equivalent to BFS if step costs equal.

40 Depth-First Search

41 Expand deepest unexpanded node. Implementation: LIFO queue

42 Complete? Time complexity? Space complexity? Optimal?

43 Complete? No (infinite-depth, loops) Time complexity? O(b m ) Space complexity? O(bm) Optimal? No

44 Depth-Limited Search = DFS with depth limit (L). i.e., nodes at depth (L) have no successors

45 Iterative Deepening Search

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48 Complete? Time complexity? Space complexity? Optimal?

49 Complete? Yes Time complexity? O(b d ) Space complexity? O(bd) Optimal? Yes (if step cost = 1)

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51 Artificial Intelligence Informed Search Methods : The Basics Dae-Won Kim School of Computer Science & Engineering Chung-Ang University

52 Outline Best-first search Greedy search A* search Brach and Bound

53 A strategy is defined by picking the order of node expansion.

54 Informed search strategy can find solutions more efficiently than an uninformed search.

55 It uses problem-specific knowledge beyond the definition of the problem itself.

56 Best-First Search

57 Idea: use an evaluation function for each node.

58 Estimate the desirability of each node Expand most desirable unexpanded node

59 Special cases: Greedy search A* search Branch and Bound

60 Romania Example with Step Costs

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62 Greedy Search

63 We need an evaluation function : heuristic function h(n)

64 h(n) = estimate of cost from n to the closest goal

65 h(n) = straight-line distance from n to Bucharest

66 Greedy search expands the node that appears to be closest to goal.

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71 Properties of greedy search

72 Complete? Time complexity? Space complexity? Optimal?

73 Complete? No (can get stuck in loops) Yes (in finite space with repeated-state checking) Time complexity? O(b m ), A good heuristic is needed. Space complexity? O(b m ), Keeps all nodes in memory. Optimal? No

74 What is A* Search?

75 Idea: avoid expanding paths that are already expensive.

76 Evaluation function: f(n) = g(n) + h(n)

77 g(n) = cost so far to reach n h(n) = estimated cost to goal from n f(n) = estimated total cost through n to goal

78 A* search uses an admissible heuristic. Thus, it is optimal.

79 h(n) h*(n) where h*(n) is the true cost from n. h(n) 0, so h(goal) = 0.

80 e.g., h straight (n) never overestimates the actual distance.

81 Romania Example with A* Search

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88 Properties of A* Search

89 Complete? Time complexity? Space complexity? Optimal?

90 Complete? Yes Time complexity? Exponential in [relative error in h x length of sol.] Space complexity? Keeps all nodes in memory. Optimal? Yes

91 Q: Explain why A* is optimal?

92 Admissible Heuristics for the 8-puzzle

93 Guess a h-function : f = g + h

94 h1(n) = number of misplaced tiles h1(n) = 6

95 h2(n) = total Manhattan distance h2(n) = = 14

96 Admissible Heuristics & Dominance

97 If h2(n) > h1(n) for all n, then h2 dominates h1 and is better for search.

98 IDS = 50,000,000,000 nodes A*(h1) = 39,135 nodes A*(h2) = 1,641 nodes

99 Q: How to find good heuristics?

100 Admissible heuristics can be derived from the exact solution cost of a relaxed version of the problem.

101 The optimal solution cost of a relaxed problem is no greater than the optimal solution cost of the real problem.