CS 5100: Founda.ons of Ar.ficial Intelligence

Transcription

1 CS 5100: Founda.ons of Ar.ficial Intelligence Search Problems and Solutions Prof. Amy Sliva October 6, 2011

2 Outline Review inference in FOL Most general unidiers Conversion to CNF UniDication algorithm Backward chaining Search Problems Uninformed search Informed search (maybe )

3 Review: Find a MGU for the following pairs (if one exists) 1. Isa(Fido,Dog) Isa(x, Dog) 2. Isa(x,Dog) Isa(y,z) 3. Likes(x,Owner(x)) Likes(John, Owner(Fido)) 4. Likes(x,Owner(y)) Likes(John, z) 5. Likes(x,Owner(y)) Likes(Fido, Father(John))

4 Review: Find a MGU for the following pairs (if one exists) 1. Isa(Fido,Dog) Isa(x, Dog) θ = {x/fido} 2. Isa(x,Dog) Isa(y,z) θ = {x/y, z/dog} 3. Likes(x,Owner(x)) Likes(John, Owner(Fido)) None 4. Likes(x,Owner(y)) Likes(John, z) θ = {x/john, z/owner(y)} 5. Likes(x,Owner(y)) Likes(Fido, Father(John)) None

5 Review: Steps in conver.ng FOL to CNF Remove implications Move negation inward Standardize variable names Skolemize (drop existential quantidiers) x y z [ Knows(x, z) Knows(y, z) ] becomes x y [Knows(x, F(x,y)) Knows(y, F(x,y)) ] x z y [ Knows(x, z) Knows(y, z) ] becomes x y [Knows(x, F(x)) Knows(y, F(x)) ] Drop universal quantidiers Distribute over

6 Conversion to CNF Examples 1. y z [ City(z) Team- from(y, z) Winner(y, SuperBowl) ] 2. x y [ Student(x) TakingClass(x, y) z [ Grade(z) GradeFor(x, y, z)] ] 3. w [ z [ House(z) ( Owns(w, z) Rents(w, z) ) ] CreditWorthy(w) ]

7 Conversion to CNF Examples 1. y z [ City(z) Team- from(y, z) Winner(y, SuperBowl) ] City(g2) Team- from(g1, g2) Winner(g1, SuperBowl) 2. x y [ Student(x) TakingClass(x, y) z [ Grade(z) GradeFor(x, y, z)] ] ( Student(x) TakingClass(x, y) Grade(z)) ( Student(x) TakingClass(x, y) GradeFor(x, y, z)) 3. w [ z [ House(z) ( Owns(w, z) Rents(w, z) ) ] CreditWorthy(w) ] ( House(z) Owns(w, z) CreditWorthy(w)) ( House(z) Rents(w, z) CreditWorthy(w))

8 Unifica.on Algorithm and Op(x) = Op(y) then

9 Unifica.on Algorithm (cont.) Using compose

10 Unifica.on algorithm in ac.on Unify Likes(y,x) with Likes(x,John)

11 Unifica.on algorithm in ac.on and Op(x) = Op(y) then call UNIFY(Likes(y,x), Likes(x,John), [ ]) return UNIFY([y,x], [x,john], [ ])

12 Unifica.on algorithm in ac.on and Op(x) = Op(y) then call UNIFY([y,x], [x,john], [ ]) return UNIFY([x], [John], UNIFY([y],[x],[ ]))

13 Unifica.on algorithm in ac.on and Op(x) = Op(y) then call UNIFY(y,x,[ ]) return UNIFY- VAR(y,x,[ ])

14 Unifica.on algorithm in ac.on # Using compose call UNIFY- VAR(y,x,[ ]) return θ = {y/x}

15 Unifica.on algorithm in ac.on Unify Likes(y,x) with Likes(x,John) First step: θ = {y/x}

16 Unifica.on algorithm in ac.on and Op(x) = Op(y) then call UNIFY([y,x], [x,john], [ ]) return UNIFY(x, John, θ = {y/x})

17 Unifica.on algorithm in ac.on and Op(x) = Op(y) then call UNIFY(x, John, θ = {y/x}) return UNIFY- VAR(x,John, θ = {y/x})

18 Unifica.on algorithm in ac.on # Using compose call UNIFY- VAR(x,John, θ = {y/x})

19 Unifica.on algorithm in ac.on Unify Likes(y,x) with Likes(x,John) First step: θ = {y/x} Second step: unify- var(x, John, θ) add{x/john} to θ yields what? Push(newpair, θ) à {x/john, y/x} Compose(newpair, θ): s.t. when the lhs (L) of the new pair appears in the rhs (R) of an existing element of theta, replace R with subst(newpair, R)à {x/john, y/john}

20 Unifica.on algorithm in ac.on # Using compose call UNIFY- VAR(x,John, θ = {y/x}) return θ = {x/john, y/john}

21 Backward chaining algorithm

22 Some slight changes/correc.ons NOTE: Should be: for each r, θ combination in KB where An empty θ = [ ] means an exact match, so subst(θ,p) does nothing Why can t we standardize apart once at the beginning? Remember, backward chaining is a depth- Dirst search procedure Search methods can be used to solve many problems in AI

23 Syntax of AND/OR trees How do we search for backward chaining solutions? Facts are queried Dirst (in the order they appear) Then rules in the order they appear Premises are also solved in the order they appear Alternating node types (except for leaf nodes) A goal/subgoal node OR node representing all the ways the goal can be proved Children are rules or facts A rule node AND node since the premises are conjoined Children are subgoals Fact node leaf node that satisdies a subgoal

24 Example: AND/OR tree for backward chaining process NOTE: Order of the KB is important R1. House(x) Owns(y, x) Cr(y) R2. House(x) Rents(y, x) NoDebts(y) Cr(y) R3. Rich(x) Cr(x) F1. House(H33) F2. House(H21) F3. Owns(John, H21) F4. Rents(Sally, Apt 12) F5. Owns(Max, Tent33) F6. Rich(Mike) F7. Rents(Jim, H33) Query:?Cr(z) who is credit worthy??

25 CR(z) Solutions: [ [z = John], [z = Mike] ] [z=john, x1=h21] OR FAIL [z = Mike] R1 R2 R3 AND AND AND House(x1) Owns(z, x1) OR OR House(x2) Rents(z,x2) NoDebts(z) Rich(z) F1 x1=h33 F2 x1=h21 OR F1 F2 OR OR F6 x2=h33 x2=h21 z = Mike F3 F5 F4 F7 z = John x1 = H21 z = Max x1 = Tent33

26 Search problems Classic search problems Missionaries and cannibals 8- puzzle 4 color map problem Path Dinding Resource allocation Adversarial games extend the basic model Relationship to operations research Factor scheduling (e.g., painting cars) Logistics (e.g., delivering cars to dealers) Finding optimal or good schedules

27 Search framework for problem solving Problem formulation: A set of discrete states (usually Dinite, but often very large) that dedine the problem space A designated start state A subset of states designated as goal states A set of operators or actions the agent can perform at each state A transition function Next: State OP à State Objective: Find a sequence of operators that, if applied in the start state, will lead to a goal state Optimal solution Dind the shortest or least- cost such sequence Cost determined by agent s performance measure

28 Explore the problem space using a search tree State space forms a directed graph or tree structure Root = start state Each node represents a state Actions are branches children are all the possible next- states

29 Graph search (dynamic programming) Graph contains at most one copy of each state Growing tree directly on state- space graph

30 Example: The 8- puzzle States? Actions? Goal test? Path cost? Start State Goal State

31 Example: The 8- puzzle Start State Goal State States? Locations of tiles Actions? Move blank left, right, up, down Goal test? ==Goal State (given) Path cost? 1 per move [NOTE: optimal solution of n- puzzle family is NP- hard]

32 Example: Path finding Sample problem On holiday in Romania; currently in Arad Flight leaves tomorrow from Bucharest Formulate goal: Be in Bucharest Formulate problem: States cities in Romania Actions drive between cities Find solution: Sequence of cities, e.g., Arad, Sibiu, Fagaras, Bucharest

33 Example: Path finding (cont.) 71 Oradea Neamt Arad Zerind 140 Timisoara 151 Sibiu 99 Fagaras 80 Rimnicu Vilcea 87 Iasi 92 Vaslui 111 Lugoj 97 Pitesti Drobeta Mehadia Bucharest 90 Craiova Giurgiu 98 Urziceni Hirsova 86 Eforie

34 Solve path finding with tree search (a) The initial state Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea Arad Lugoj Arad Oradea (b) After expanding Arad Arad

35 Solve path finding with tree search (b) After expanding Arad Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea Arad Lugoj Arad Oradea (c) After expanding Sibiu Arad

36 Solve path finding with tree search (c) After expanding Sibiu Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea Arad Lugoj Arad Oradea

37 Implementa.on: states vs. nodes A state is a (representation of) a physical condiguration A node is a data structure constituting part of a search tree Representative data structure must include state, parent node, action, path cost More than one node can correspond to a single world state PARENT 5 4 Node ACTION = Right PATH-COST = STATE 7 3 2

38 Search strategies Search strategy dedined by picking the order of node expansion Strategies are evaluated along the following dimensions 1. Completeness does it always Dind a solution if one exists? 2. Time complexity number of nodes generated 3. Space complexity maximum number of nodes in memory 4. Optimality does it always Dind a least- cost solution? Time and space complexity are measured in terms of b Maximum branching factor for the search tree d Depth of the least- cost solution m Maximum depth of the state space (may be )

39 Uninformed search strategies Uninformed (blind) search strategies use only the information available in the problem dedinition Search strategies 1. Breadth- Dirst search 2. Uniform- cost search 3. Depth- Dirst search 4. Depth- limited search 5. Iterative deepening search

40 Breadth- first search Expand shallowest unexpanded node All nodes at a given depth expanded before any at the next level Implementation Frontier is a FIFO queue, i.e., new successors go at the end A B C D E F G

41 Breadth- first search Expand shallowest unexpanded node All nodes at a given depth expanded before any at the next level Implementation Frontier is a FIFO queue, i.e., new successors go at the end A B C D E F G

42 Breadth- first search Expand shallowest unexpanded node All nodes at a given depth expanded before any at the next level Implementation Frontier is a FIFO queue, i.e., new successors go at the end A B C D E F G

43 Breadth- first search Expand shallowest unexpanded node All nodes at a given depth expanded before any at the next level Implementation Frontier is a FIFO queue, i.e., new successors go at the end A B C D E F G

44 Proper.es of breadth- first search Complete? Yes (if b is Dinite) Time? 1 + b + b 2 + b b d + b(b d - 1) = O(b d+1 ) Space? O(b d+1 ) keeps every node in memory Optimal? Yes (if cost is non- decreasing function of depth) Space is the bigger problem (more than time) Suppose b = 10, a node uses 1000 bytes, and 1 million nodes per second can be generated Solution at depth 10 takes 3 hours and uses 10 terabytes At level 12 it is 13 days and 1 petabyte; at 14 it is 3.5 years, 99 petabytes Exponential complexity bound search problems cannot be solved by uninformed strategies not scalable

45 Uniform- cost search Expand least- cost unexpanded node Implementation Frontier is a queue ordered by path cost Equivalent to breadth- Dirst search if step costs are all equal Complete? Yes, if step cost ε Time? # of nodes with g cost of optimal solution, O(b ceiling(c*/ ε) ) where C * is the cost of the optimal solution Space? # of nodes with g cost of optimal solution, O(b ceiling(c*/ ε) ) Optimal? Yes nodes are expanded in increasing order of g(n)

46 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A B C D E F G H I J K L M N O

47 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A B C D E F G H I J K L M N O

48 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A B C D E F G H I J K L M N O

49 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A B C D E F G H I J K L M N O

50 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A B C D E F G I J K L M N O

51 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A C E F G J K L M N O

52 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A C E F G J K L M N O

53 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A B C E F G K L M N O

54 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front C F G L M N O

55 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A C F G L M N O

56 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A C F G L M N O

57 Depth- first search Expand deepest unexpanded node Implementation Frontier is a LIFO queue (stack), i.e., new successors go at the front A C F G M N O

58 Proper.es of depth- first search Complete? No fails in indinite depth spaces or spaces with loops (NOTE: our algorithm prevents loops) Modify to avoid repeated states along path complete in Dinite spaces Time? O(b m ) terrible if m is much larger than d Solutions are dense, so may be much faster than breadth- Dirst Space? O(bm) linear space! Optimal? No

59 Depth- limited search Depth- Dirst search with depth limit l Nodes at depth l have no successors

60 Itera.ve deepening search Gradually increase the depth limit in depth- Dirst search Combines benedits of breadth- Dirst and depth- Dirst search

61 Itera.ve deepening search Limit = 0 A A

62 Itera.ve deepening search Limit = 1 A A A A B C B C B C B C Limit = 2

63 Itera.ve deepening search 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 Limit = 3

64 Itera.ve deepening search 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

65 Itera.ve deepening search Number of nodes generated in iterative deepening search to depth d with branching factor b N(IDS) = (d+1)b 0 + (d)b 1 + (d- 1)b (3)b d- 2 +(2)b d- 1 + (1)b d Time complexity = O(b d ) Same as breadth- Dirst search Time complexity comparison to breadth- Dirst search Suppose b = 10 and d = 5 N(IDS) = , , ,000 = 123,450 N(BFS) = , , ,000 = 111,110 Overhead difference = (123, ,110) / 111,110 = 11%

66 Proper.es of itera.ve deepening Complete? Yes Time? (d+1)b 0 + (d)b 1 + (d- 1)b (3)b d- 2 +(2)b d- 1 + (1)b d = O(b d ) Space? O(bd) same as depth- Dirst search Optimal? Yes, if step cost = 1

67 Summary of uninformed search algorithms

68 Informed search Uses problem- speciaic knowledge beyond the problem dedinition to Dind solutions more efdiciently Outline of topics Heuristics Best- Dirst search Greedy best- Dirst search A* search Remember, a search strategy is dedined by picking the order of node expansion

69 Best- first search Use an evaluation function f(n) for each node Estimate of desirability Includes a heuristic function to add more knowledge to the problem h(n) = estimated cost of cheapest path from state at n to the goal Heuristics are problem- specidic Expand most desirable unexpanded node Implementation Order nodes in the frontier in decreasing order of desirability Special cases Greedy best- Dirst search A* search

70 Heuris.c for path- finding Romania example: estimate the straight- line distance from each node to Bucharest (the goal) Arad Oradea 71 Neamt Zerind Sibiu 99 Fagaras 80 Timisoara Rimnicu Vilcea 111 Lugoj 70 Mehadia 75 Drobeta 120 Pitesti Bucharest 90 Craiova Giurgiu 87 Iasi Urziceni Vaslui Hirsova 86 Eforie Straight- line distance to Bucharest Arad 366 Bucharest 0 Craiova 160 Drobeta 242 Eforie 161 Fagaras 176 Giurgiu 77 Hirsova 151 Iasi 226 Lugoj 244 Mehadia 241 Neamt 234 Oradea 380 Pitesti 100 Rimnicu Vilcea 193 Sibiu 253 Timisoara 329 Urziceni 80 Vaslui 199 Zerind 374

71 Greedy best- first search Evaluation function f(n) = h(n) = estimate cost from n to goal e.g., h SLD (n) = straight line distance from n to Bucharest Search strategy Expand the node that appears to be closest to the goal (i.e., has the lowest h(n)

72 Greedy best- first search (a) The initial state (b) After expanding Arad Arad 366 Arad

73 Greedy best- first search 366 (b) After expanding Arad Arad Sibiu Timisoara Zerind (c) After expanding Sibiu Arad

74 Greedy best- first search (c) After expanding Sibiu Arad Sibiu Timisoara 329 Zerind 374 Arad Fagaras Oradea Rimnicu Vilcea (d) After expanding Fagaras

75 Greedy best- first search (d) After expanding Fagaras Arad Sibiu Timisoara Zerind Arad Fagaras Oradea Rimnicu Vilcea Sibiu Bucharest 253 0

76 Proper.es of greedy best- first search Complete? No can get stuck in loops e.g., Iasi à Neamt à Iasi à Neamt à Time? O(b m ), but a good heuristic can give dramatic improvement Space? O(b m ) keeps all nodes in memory Optimal? No

77 A* Search Avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n) g(n) = cost of path so far to reach n h(n) = estimated cost from n to goal f(n) = estimated total cost of path through n to goal

78 A* Search (a) The initial state (b) After expanding Arad Arad 366=0+366 Arad

79 A* Search (b) After expanding Arad Arad Sibiu 393= Timisoara 447= Zerind 449= (c) After expanding Sibiu Arad

80 A* Search (c) After expanding Sibiu Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = = = (d) After expanding Rimnicu Vilcea Arad

81 A* Search (d) After expanding Rimnicu Vilcea Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea 646= = = Rimnicu Vilcea Craiova Pitesti Sibiu 526= = = (e) After expanding Fagaras

82 A* Search (e) After expanding Fagaras Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = Sibiu Bucharest Craiova Pitesti Sibiu 591= = = = = (f) After expanding Pitesti

83 A* Search (f) After expanding Pitesti Arad Sibiu Timisoara Zerind 447= = Arad Fagaras Oradea Rimnicu Vilcea 646= = Sibiu Bucharest Craiova Pitesti Sibiu 591= = = = Bucharest Craiova Rimnicu Vilcea 418= = =

84 Proper.es of A* Complete? Yes (unless there are indinitely many nodes with f(n) < C* Time? Exponential depends on assumptions made about state space Space? Keeps all nodes in memory Optimal? Yes

85 Admissible heuris.cs Heuristic h(n) is admissible if for every node n, h(n) h*(n), where h*(n) is the true cost to reach the goal state from n Never overestimates the cost to reach the goal, i.e., it is optimistic Example h SLD (n) never overestimates the actual road distance by using straight- line distance Theorem If h(n) is admissible, A* using TREE-SEARCH is optimal

86 Op.mality of A* (proof) Suppose some suboptimal goal G 2 is in the frontier. Let n be an unexpanded node in the frontier s.t. n is on a shortest path to an optimal goal G Proof: f(g 2 ) > f(g) from above h(n) h*(n) since h is admissible g(n) + h(n) g(n) + h*(n) f(n) f(g) Hence f(g 2 ) > f(n), and A* will never select G 2 for expansion

87 Consistent heuris.cs Heuristic is consistent if for every node n, every successor n of n generated by action a, h(n) c(n,a,n ) + h(n ) If h is consistent we have f(n ) = g(n ) + h(n ) = g(n) + c(n,a,n ) + h(n ) g(n) + h(n) f(n) i.e., f(n) is non- decreasing along any path Theorem If h(n) is consistent, A* using GRAPH-SEARCH is optimal

88 Genera.ng admissible heuris.cs Possible heuristics for the 8- puzzle Generate from a relaxed problem fewer restrictions than original h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance to the goal position h 1 (S) =? h 2 (S) =? Start State Goal State

89 Genera.ng admissible heuris.cs Possible heuristics for the 8- puzzle Generate from a relaxed problem fewer restrictions than original h 1 (n) = number of misplaced tiles h 2 (n) = total Manhattan distance to the goal position Start State Goal State h 1 (S) =? 8 h 2 (S) =? = 18 h*(s) = 26 so both are admissible heuristics (lower bounds)