# Introduction to Artificial Intelligence. Problem Solving and Search

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1 Introduction to Artificial Intelligence Problem Solving and Search Bernhard Beckert UNIVESITÄT KOBLENZ-LANDAU Wintersemester 2003/2004 B. Beckert: Einführung in die KI / KI für IM p.1

2 Outline Problem solving Problem types Problem formulation Example problems Basic search algorithms B. Beckert: Einführung in die KI / KI für IM p.2

3 Problem solving Offline problem solving Acting only with complete knowledge of problem and solution Online problem solving Acting without complete knowledge Here Here we are concerned with offline problem solving only B. Beckert: Einführung in die KI / KI für IM p.3

4 Example: Travelling in omania Scenario On holiday in omania; currently in Arad Flight leaves tomorrow from Bucharest B. Beckert: Einführung in die KI / KI für IM p.4

5 Example: Travelling in omania Scenario On holiday in omania; currently in Arad Flight leaves tomorrow from Bucharest Goal Be in Bucharest B. Beckert: Einführung in die KI / KI für IM p.4

6 Example: Travelling in omania Scenario On holiday in omania; currently in Arad Flight leaves tomorrow from Bucharest Goal Be in Bucharest Formulate problem States: various cities Actions: drive between cities B. Beckert: Einführung in die KI / KI für IM p.4

7 Example: Travelling in omania Scenario On holiday in omania; currently in Arad Flight leaves tomorrow from Bucharest Goal Be in Bucharest Formulate problem States: various cities Actions: drive between cities Solution Appropriate sequence of cities e.g.: Arad, Sibiu, Fagaras, Bucharest B. Beckert: Einführung in die KI / KI für IM p.4

8 Example: Travelling in omania Oradea Neamt Zerind Arad Iasi Sibiu Fagaras Vaslui Timisoara imnicu Vilcea Lugoj Pitesti Mehadia Urziceni Hirsova Dobreta Craiova Bucharest Giurgiu Eforie B. Beckert: Einführung in die KI / KI für IM p.5

9 Problem types Single-state problem observable (at least the initial state) deterministic static discrete Multiple-state problem partially observable (initial state not observable) deterministic static discrete Contingency problem partially observable (initial state not observable) non-deterministic B. Beckert: Einführung in die KI / KI für IM p.6

10 Example: vacuum-cleaner world Single-state Start in: 5 Solution: B. Beckert: Einführung in die KI / KI für IM p.7

11 Example: vacuum-cleaner world Single-state Start in: 5 Solution: [right, suck] B. Beckert: Einführung in die KI / KI für IM p.7

12 Example: vacuum-cleaner world Single-state Start in: 5 Solution: [right, suck] Multiple-state Start in: {1,2,3,4,5,6,7,8} Solution: B. Beckert: Einführung in die KI / KI für IM p.7

13 Example: vacuum-cleaner world Single-state Start in: 5 Solution: [right, suck] Multiple-state Start in: {1,2,3,4,5,6,7,8} Solution: [right, suck, left, suck] right {2,4,6,8} suck {4,8} left {3,7} suck {7} B. Beckert: Einführung in die KI / KI für IM p.7

14 Example: vacuum-cleaner world 1 2 Contingency Murphy s Law: suck can dirty a clean carpet Local sensing: dirty/not dirty at location only Start in: {1,3} Solution: B. Beckert: Einführung in die KI / KI für IM p.8

15 Example: vacuum-cleaner world 1 2 Contingency Murphy s Law: suck can dirty a clean carpet Local sensing: dirty/not dirty at location only Start in: {1,3} Solution: [suck, right, suck] suck {5,7} right {6,8} suck {6,8} Improvement: [suck, right, if dirt then suck] (decide whether in 6 or 8 using local sensing) B. Beckert: Einführung in die KI / KI für IM p.8

16 Single-state problem formulation Defined by the following four items 1. Initial state Example: Arad 2. Successor function S Example: S(Arad) = { gozerind, Zerind, gosibiu, Sibiu,... } 3. Goal test Example: x = Bucharest (explicit test) nodirt(x) (implicit test) 4. Path cost (optional) Example: sum of distances, number of operators executed, etc. B. Beckert: Einführung in die KI / KI für IM p.9

17 Single-state problem formulation Solution A sequence of operators leading from the initial state to a goal state B. Beckert: Einführung in die KI / KI für IM p.10

18 Selecting a state space Abstraction eal world is absurdly complex State space must be abstracted for problem solving (Abstract) state Set of real states (Abstract) operator Complex combination of real actions Example: Arad Zerind represents complex set of possible routes (Abstract) solution Set of real paths that are solutions in the real world B. Beckert: Einführung in die KI / KI für IM p.11

19 Example: The 8-puzzle Start State Goal State States Actions Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.12

20 Example: The 8-puzzle Start State Goal State States integer locations of tiles Actions Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.12

21 Example: The 8-puzzle Start State Goal State States Actions integer locations of tiles left, right, up, down Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.12

22 Example: The 8-puzzle Start State Goal State States Actions Goal test integer locations of tiles left, right, up, down = goal state? Path cost B. Beckert: Einführung in die KI / KI für IM p.12

23 Example: The 8-puzzle Start State Goal State States Actions Goal test Path cost integer locations of tiles left, right, up, down = goal state? 1 per move B. Beckert: Einführung in die KI / KI für IM p.12

24 Example: Vacuum-cleaner L L S S L L L L S S S S L L S S States Actions Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.13

25 Example: Vacuum-cleaner L L S S L L L L S S S S L L S S States integer dirt and robot locations Actions Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.13

26 Example: Vacuum-cleaner L L S S L L L L S S S S L L S S States Actions integer dirt and robot locations left, right, suck, noop Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.13

27 Example: Vacuum-cleaner L L S S L L L L S S S S L L S S States Actions Goal test integer dirt and robot locations left, right, suck, noop not dirty? Path cost B. Beckert: Einführung in die KI / KI für IM p.13

28 Example: Vacuum-cleaner L L S S L L L L S S S S L L S S States Actions Goal test integer dirt and robot locations left, right, suck, noop not dirty? Path cost 1 per operation (0 for noop) B. Beckert: Einführung in die KI / KI für IM p.13

29 Example: obotic assembly P States Actions Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.14

30 Example: obotic assembly P States real-valued coordinates of robot joint angles and parts of the object to be assembled Actions Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.14

31 Example: obotic assembly P States real-valued coordinates of robot joint angles and parts of the object to be assembled Actions continuous motions of robot joints Goal test Path cost B. Beckert: Einführung in die KI / KI für IM p.14

32 Example: obotic assembly P States real-valued coordinates of robot joint angles and parts of the object to be assembled Actions Goal test continuous motions of robot joints assembly complete? Path cost B. Beckert: Einführung in die KI / KI für IM p.14

33 Example: obotic assembly P States real-valued coordinates of robot joint angles and parts of the object to be assembled Actions Goal test Path cost continuous motions of robot joints assembly complete? time to execute B. Beckert: Einführung in die KI / KI für IM p.14

34 Tree search algorithms Offline Simulated exploration of state space in a search tree by generating successors of already-explored states function TEE-SEACH( problem, strategy) returns a solution or failure initialize the search tree using the initial state of problem loop do if there are no candidates for expansion then return failure choose a leaf node for expansion according to strategy if the node contains a goal state then return the corresponding solution else expand the node and add the resulting nodes to the search tree end B. Beckert: Einführung in die KI / KI für IM p.15

38 Implementation: States vs. nodes State A (representation of) a physical configuration Node A data structure constituting part of a search tree (includes parent, children, depth, path cost, etc.) parent State 5 4 Node depth = 6 g = state children B. Beckert: Einführung in die KI / KI für IM p.17

39 Implementation of search algorithms function TEE-SEACH( problem, fringe) returns a solution or failure fringe INSET(MAKE-NODE(INITIAL-STATE[problem]),fringe) loop do if fringe is empty then return failure node EMOVE-FIST(fringe) if GOAL-TEST[problem] applied to STATE(node) succeeds then return node else fringe INSET-ALL(EXPAND(node, problem), fringe) end fringe State Expand queue of nodes not yet considered gives the state that is represented by node creates new nodes by applying possible actions to node B. Beckert: Einführung in die KI / KI für IM p.18

40 Search strategies Strategy Defines the order of node expansion Important properties of strategies completeness time complexity space complexity optimality does it always find a solution if one exists? number of nodes generated/expanded maximum number of nodes in memory does it always find a least-cost solution? Time and space complexity measured in terms of b d maximum branching factor of the search tree depth of a solution with minimal distance to root m maximum depth of the state space (may be ) B. Beckert: Einführung in die KI / KI für IM p.19

41 Uninformed search strategies Uninformed search Use only the information available in the problem definition Frequently used strategies Breadth-first search Uniform-cost search Depth-first search Depth-limited search Iterative deepening search B. Beckert: Einführung in die KI / KI für IM p.20

42 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue A B C D E F G B. Beckert: Einführung in die KI / KI für IM p.21

43 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue A B C D E F G B. Beckert: Einführung in die KI / KI für IM p.21

44 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue A B C D E F G B. Beckert: Einführung in die KI / KI für IM p.21

45 Breadth-first search Idea Expand shallowest unexpanded node Implementation fringe is a FIFO queue, i.e. successors go in at the end of the queue A B C D E F G B. Beckert: Einführung in die KI / KI für IM p.21

46 Breadth-first search: Example omania Arad B. Beckert: Einführung in die KI / KI für IM p.22

47 Breadth-first search: Example omania Arad Zerind Sibiu Timisoara B. Beckert: Einführung in die KI / KI für IM p.22

50 Breadth-first search: Properties Complete Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.23

51 Breadth-first search: Properties Complete Yes (if b is finite) Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.23

52 Breadth-first search: Properties Complete Yes (if b is finite) Time 1 + b + b 2 + b b d + b(b d 1) O(b d+1 ) i.e. exponential in d Space Optimal B. Beckert: Einführung in die KI / KI für IM p.23

53 Breadth-first search: Properties Complete Yes (if b is finite) Time 1 + b + b 2 + b b d + b(b d 1) O(b d+1 ) Space O(b d+1 ) i.e. exponential in d keeps every node in memory Optimal B. Beckert: Einführung in die KI / KI für IM p.23

54 Breadth-first search: Properties Complete Yes (if b is finite) Time 1 + b + b 2 + b b d + b(b d 1) O(b d+1 ) Space O(b d+1 ) i.e. exponential in d keeps every node in memory Optimal Yes (if cost = 1 per step), not optimal in general B. Beckert: Einführung in die KI / KI für IM p.23

55 Breadth-first search: Properties Complete Yes (if b is finite) Time 1 + b + b 2 + b b d + b(b d 1) O(b d+1 ) Space O(b d+1 ) i.e. exponential in d keeps every node in memory Optimal Yes (if cost = 1 per step), not optimal in general Disadvantage Space is the big problem (can easily generate nodes at 5MB/sec so 24hrs = 430GB) B. Beckert: Einführung in die KI / KI für IM p.23

56 omania with step costs in km Oradea 71 Neamt Zerind Arad Timisoara 111 Lugoj 70 Mehadia Dobreta Sibiu 99 Fagaras 80 imnicu 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 imnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374 B. Beckert: Einführung in die KI / KI für IM p.24

57 Uniform-cost search Idea Expand least-cost unexpanded node (costs added up over paths from root to leafs) Implementation fringe is queue ordered by increasing path cost Note Equivalent to depth-first search if all step costs are equal B. Beckert: Einführung in die KI / KI für IM p.25

58 Uniform-cost search Arad B. Beckert: Einführung in die KI / KI für IM p.26

59 Uniform-cost search Arad Zerind Sibiu Timisoara B. Beckert: Einführung in die KI / KI für IM p.26

60 Uniform-cost search Arad Zerind Sibiu Timisoara Arad Oradea B. Beckert: Einführung in die KI / KI für IM p.26

62 Uniform-cost search: Properties Complete Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.27

63 Uniform-cost search: Properties Complete Yes (if step costs positive) Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.27

64 Uniform-cost search: Properties Complete Yes (if step costs positive) Time # of nodes with past-cost less than that of optimal solution Space Optimal B. Beckert: Einführung in die KI / KI für IM p.27

65 Uniform-cost search: Properties Complete Yes (if step costs positive) Time Space # of nodes with past-cost less than that of optimal solution # of nodes with past-cost less than that of optimal solution Optimal B. Beckert: Einführung in die KI / KI für IM p.27

66 Uniform-cost search: Properties Complete Yes (if step costs positive) Time Space Optimal # of nodes with past-cost less than that of optimal solution # of nodes with past-cost less than that of optimal solution Yes B. Beckert: Einführung in die KI / KI für IM p.27

67 Depth-first search Idea Expand deepest unexpanded node Implementation fringe is a LIFO queue (a stack), i.e. successors go in at front of queue Note Depth-first search can perform infinite cyclic excursions Need a finite, non-cyclic search space (or repeated-state checking) B. Beckert: Einführung in die KI / KI für IM p.28

68 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

69 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

70 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

71 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

72 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

73 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

74 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

75 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

76 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

77 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

78 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

79 Depth-first search A B C D E F G H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.29

80 Depth-first search: Example omania Arad B. Beckert: Einführung in die KI / KI für IM p.30

81 Depth-first search: Example omania Arad Zerind Sibiu Timisoara B. Beckert: Einführung in die KI / KI für IM p.30

82 Depth-first search: Example omania Arad Zerind Sibiu Timisoara Arad Oradea B. Beckert: Einführung in die KI / KI für IM p.30

83 Depth-first search: Example omania Arad Zerind Sibiu Timisoara Arad Oradea Zerind Sibiu Timisoara B. Beckert: Einführung in die KI / KI für IM p.30

84 Depth-first search: Properties Complete Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.31

85 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.31

86 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space Optimal B. Beckert: Einführung in die KI / KI für IM p.31

87 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space O(bm) (i.e. linear space) Optimal B. Beckert: Einführung in die KI / KI für IM p.31

88 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space O(bm) (i.e. linear space) Optimal No B. Beckert: Einführung in die KI / KI für IM p.31

89 Depth-first search: Properties Complete Yes: if state space finite No: if state contains infinite paths or loops Time O(b m ) Space O(bm) (i.e. linear space) Optimal No Disadvantage Time terrible if m much larger than d Advantage Time may be much less than breadth-first search if solutions are dense B. Beckert: Einführung in die KI / KI für IM p.31

90 Iterative deepening search Depth-limited search Depth-first search with depth limit B. Beckert: Einführung in die KI / KI für IM p.32

91 Iterative deepening search Depth-limited search Depth-first search with depth limit Iterative deepening search Depth-limit search with ever increasing limits function ITEATIVE-DEEPENING-SEACH( problem) returns a solution or failure inputs: problem /* a problem */ for depth 0 to do result DEPTH-LIMITED-SEACH( problem, depth) if result cutoff then return result end B. Beckert: Einführung in die KI / KI für IM p.32

92 Iterative deepening search with depth limit 0 Limit = 0 A A B. Beckert: Einführung in die KI / KI für IM p.33

93 Iterative deepening search with depth limit 1 Limit = 1 A A A A B C B C B C B C B. Beckert: Einführung in die KI / KI für IM p.34

94 Iterative deepening search with depth limit 2 Limit = 2 A A A A B C B C B C B C D E F G D E F G D E F G D E F G A A A A B C B C B C B C D E F G D E F G D E F G D E F G B. Beckert: Einführung in die KI / KI für IM p.35

95 Iterative deepening search with depth limit 3 Limit = 3 A A A A B C B C B C B C D E F G D E F G D E F G D E F G H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O A A A A B C B C B C B C D E F G D E F G D E F G D E F G H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O A A A A B C B C B C B C D E F G D E F G D E F G D E F G H I J K L M N O H I J K L M N O H I J K L M N O H I J K L M N O B. Beckert: Einführung in die KI / KI für IM p.36

96 Iterative deepening search: Example omania with l = 0 Arad B. Beckert: Einführung in die KI / KI für IM p.37

97 Iterative deepening search: Example omania with l = 1 Arad B. Beckert: Einführung in die KI / KI für IM p.38

98 Iterative deepening search: Example omania with l = 1 Arad Zerind Sibiu Timisoara B. Beckert: Einführung in die KI / KI für IM p.38

99 Iterative deepening search: Example omania with l = 2 Arad B. Beckert: Einführung in die KI / KI für IM p.39

100 Iterative deepening search: Example omania with l = 2 Arad Zerind Sibiu Timisoara B. Beckert: Einführung in die KI / KI für IM p.39

101 Iterative deepening search: Example omania with l = 2 Arad Zerind Sibiu Timisoara Arad Oradea B. Beckert: Einführung in die KI / KI für IM p.39

102 Iterative deepening search: Example omania with l = 2 Arad Zerind Sibiu Timisoara Arad Oradea Arad Oradea Fagaras imnicu Vilcea B. Beckert: Einführung in die KI / KI für IM p.39

103 Iterative deepening search: Example omania with l = 2 Arad Zerind Sibiu Timisoara Arad Oradea Arad Oradea Fagaras imnicu Vilcea Arad Lugoj B. Beckert: Einführung in die KI / KI für IM p.39

104 Iterative deepening search: Properties Complete Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.40

105 Iterative deepening search: Properties Complete Yes Time Space Optimal B. Beckert: Einführung in die KI / KI für IM p.40

106 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 + (d 1)b b d O(b d+1 ) Space Optimal B. Beckert: Einführung in die KI / KI für IM p.40

107 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 + (d 1)b b d O(b d+1 ) Space O(bd) Optimal B. Beckert: Einführung in die KI / KI für IM p.40

108 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 + (d 1)b b d O(b d+1 ) Space O(bd) Optimal Yes (if step cost = 1) B. Beckert: Einführung in die KI / KI für IM p.40

109 Iterative deepening search: Properties Complete Yes Time (d + 1)b 0 + db 1 + (d 1)b b d O(b d+1 ) Space O(bd) Optimal Yes (if step cost = 1) (Depth-First) Iterative-Deepening Search often used in practice for search spaces of large, infinite, or unknown depth. B. Beckert: Einführung in die KI / KI für IM p.40

110 Comparison Criterion Breadthfirst Uniformcost Depthfirst Iterative deepening Complete? Yes Yes No Yes Time b d+1 b d b m b d Space b d+1 b d bm bd Optimal? Yes Yes No Yes B. Beckert: Einführung in die KI / KI für IM p.41

111 Comparison Breadth-first search Iterative deepening search B. Beckert: Einführung in die KI / KI für IM p.42

112 Summary Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored Variety of uninformed search strategies Iterative deepening search uses only linear space and not much more time than other uninformed algorithms B. Beckert: Einführung in die KI / KI für IM p.43

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