The Wumpus Game. Stench Gold. Start. Cao Hoang Tru CSE Faculty - HCMUT

Transcription

1 The Wumpus Game Stench Stench Gold Stench Start 1

2 The Wumpus Game Stench in the square containing the wumpus and in the directly adjacent squares in the squares directly adjacent to a pit Glitter in the square where the gold is Bump when an agent walks into a wall Scream when the wumpus is killed 2

3 The Wumpus Game Agent s percept: [Stench,, Glitter, Bump, Scream] Agent s actions: Go forward, turn right 90 o, turn left 90 o Grab to pick up an object in the same square Fire 1 arrow in a straight line in the faced direction Climb to leave the cave Death if entering a square of a pit or a live wumpus Agent s goal: find and bring the gold back to the start 3

4 The Wumpus Game Stench A = Agent B = Stench Gold G = Glitter/Gold OK = Safe square P = Pit Stench OK S = Stench V = Visited Start W = Wumpus A OK 4

5 The Wumpus Game Stench A = Agent B = Stench Gold G = Glitter/Gold OK = Safe square P = Pit Stench S = Stench OK P? V = Visited Start W = Wumpus V A P? 5

6 The Wumpus Game Stench A = Agent B = Stench Gold G = Glitter/Gold OK = Safe square P = Pit Stench S = Stench A OK V = Visited Start W = Wumpus V V 6

7 The Wumpus Game Stench A = Agent B = Stench Gold G = Glitter/Gold OK = Safe square P = Pit Stench S = Stench V A V = Visited Start W = Wumpus V V 7

8 The Wumpus Game Stench A = Agent B = Stench Gold A G = Glitter/Gold OK = Safe square P = Pit Stench S = Stench V V V = Visited Start W = Wumpus V V 8

9 Propositional Logic Syntax Logical constants:, Propositional symbols: P, Q, Logical connectives:,,,, Sentences (formulas): Logical constants Proposition symbols If α is a sentence, then so are α and (α) If α and β are sentences, then so are α β, α β, α β, and α β 9

10 Propositional Logic Semantics Interpretation: propositional symbol / The truth value of a sentence is defined by the truth table P Q P P Q P Q P Q P Q 10

11 Propositional Logic Semantics Satisfiable: under an interpretation Valid: under all interpretations P Q P P (P Q) Q ((P Q) Q) P unsatisfiable satisfiable valid 11

12 Propositional Logic Semantics Model: an interpretation under which the sentence is P Q P Q P Q P Q P Q P Q P Q P Q 12

13 Propositional Logic Semantics Entailment: KB = α iff every model of KB is a model of α α is a logical consequence of KB P Q P Q P Q, P {P Q, P} = Q 13

14 Propositional Logic Semantics Equivalence: α β iff α = β and β = α P Q P Q P Q P Q P Q 14

15 Propositional Logic Semantics Theorems: α = β iff α β is valid KB = α can be proved by validity of KB α α = β iff α β is unsatisfiable KB = α can be proved by refutation of KB α 15

16 Inference Rules Rule R: Premises: α 1, α 2,..., α n Conclusion: β Soundness: R is sound iff {α 1, α 2,..., α n } = β 16

17 Inference Rules Modus Ponens: α β, α β -Elimination: α 1 α 2... α n α i 17

18 Inference Rules -Introduction: α 1, α 2,, α n α 1 α 2 α n -Introduction: α i α 1 α 2 α n 18

19 Inference Rules Double-negation elimination: α α 19

20 Resolution Unit resolution: P 1 P i P n, Q P i Q (all P i and Q are literals) P 1 P i-1 P i+1 P n Full resolution: P i Q j P 1 P i P n, Q 1 Q j Q m P 1 P i-1 P i+1 P n Q 1 Q j-1 Q j+1 Q m 20

21 Resolution Conjunctive normal form (CNF): conjunction of disjunctions of literals (L 11 L 1k ) (L n1 L nm ) clause k-cnf: each clause contains at most k literals Every sentence can be written in CNF 21

22 Resolution Conjunctive normal form (CNF): conjunction of disjunctions of literals (P Q) R (P Q) R P Q R (S T) Q (S T) Q ( S Q) ( T Q) 22

23 Inference Algorithms To derive/assert a sentence given a knowledge base (i.e., a set of sentences) Inference algorithm = Inference rules + a searching algorithm 23

24 Inference Algorithms Algorithm PL-Resolution(KB, α) input: KB and α are PL sentences output: (to assert KB = α) or (otherwise) Clauses = set of clauses in the CNF representation of KB α New = {} loop for each C i, C j in Clauses do Resolvents = PL-Resolve(C i, C j ) if Resolvents contains an empty clause then return New = New Resolvents if New Clauses then return Clauses = Clauses New 24

25 Example: Inference Algorithms KB = {P, (P Q) R, (S T) Q, T} α = R 25

26 Inference Algorithms Soundness: every deduced answer is correct 26

27 Inference Algorithms Completeness: every correct answer is deducible 27

28 Inference Algorithms Example: Google search 28

29 Inference Algorithms Soundness Completeness Complexity 29

30 Inference Algorithms Soundness: every deduced answer is correct if PL-Resolution returns then KB = α 30

31 Inference Algorithms Completeness: every correct answer is deducible if KB = α then PL-Resolution returns 31

32 Inference Algorithms PL-Resolution is sound and complete 32

33 Inference Algorithms Modus Ponens with any searching algorithm is not complete KB = {P Q, Q R} α = P R 33

34 Inference Algorithms Horn clause: disjunction of literals of which at most one is positive or P 1 P 2 P n Q P 1 P 2 P n Q Forward or backward chaining can be used for inference on Horn clauses 34

35 Satisfiability Problem SAT problem: to determine if a sentence is satisfiable (to determine if the symbols in the sentence can be assigned / as to make it evaluate to ) ( D B C) (B A C) ( C B E) (E D B) (B E C) 35

36 Satisfiability Problem SAT problem in propositional logic is NP-complete This does not mean all instances of PL inference has the exponential complexity Polynomial-time inference procedures for Horn clauses 36

37 Satisfiability Problem Many combinatorial problems in computer science can be reduced to SAT 37

38 Satisfiability Problem Backtracking algorithms Davis and Putnam (1960) Davis, Logemann, Loveland (1962) DPLL 38

39 DPLL algorithm: Satisfiability Problem Recursive, depth-first enumeration of possible models 39

40 DPLL algorithm: Satisfiability Problem Early termination: a clause is if any literal is (A B) (A C) is if A is regardless of B and C. 40

41 DPLL algorithm: Satisfiability Problem Pure symbol: has the same sign in all clauses (A B) ( B C) (C A) A model would have the literals of pure symbols assigned. Satisfied clause removal makes new pure symbols. 41

42 DPLL algorithm: Satisfiability Problem Unit clause: has just one literal (A B) B Unit clause satisfaction makes new unit clauses (unit propagation). 42

43 Statisfiability Problem Algorithm DPLL(Clauses, Symbols, Model) if every clause in Clauses is in Model then return if some clause in Clauses is in Model then return (P, value) = PURE-SYMBOL(Clauses, Symbols, Model) if P null then return DPLL(Clauses, Symbols - P, EXT(P, value, Model)) (P, value) = UNIT-CLAUSE(Clauses, Symbols, Model) if P null then return DPLL(Clauses, Symbols - P, EXT(P, value, Model)) choose a P in Symbols return DPLL(Clauses, Symbols - P, EXT(P,, Model)) or DPLL(Clauses, Symbols - P, EXT(P,, Model)) 43

44 DPLL algorithm: Satisfiability Problem ( D B C) (B A C) ( C B E) (E D B) (B E C) 44

45 Satisfiability Problem Local search algorithms Evaluation function counts the number of unsatisfied clauses 45

46 Satisfiability Problem Local search algorithms Evaluation function counts the number of unsatisfied clauses Randomness to escape local minima Flipping the truth value of one symbol at a time 46

47 Statisfiability Problem Algorithm WalkSAT(Clauses, p, max_flips) /*1996*/ input: p is flipping probability, max_flips is number of flips allowed output: a model or failure Model = a random assignment of truth values to the symbols for i = 1 to max_flips do if Model satisfies Clauses then return Model C = a randomly selected clause from Clauses that is in Model with probability p flip the value in Model of a symbol from C else flip whichever symbol in C to maximizes no. of satisfied clauses return failure 47

48 Satisfiability Problem WalkSAT algorithm: ( D B C) (B A C) ( C B E) (E D B) ( A E C) Initial model: A = B = C = D = E = 48

49 Satisfiability Problem The larger the ratio clause/symbol is, the less likely the clause is satisfiable. 49

50 Satisfiability Problem The probability of satisfiability drops sharply around the ratio of

51 Satisfiability Problem That is actually when the problem is hard. 51

52 Satisfiability Problem WalkSAT is much faster than DPLL. 52

53 Inference Monotonicity Monotonicity: the set of entailed sentences can only increase when information is added to the KB if KB 1 = α then (KB 1 KB 2 ) = α 53

54 Inference Monotonicity Propositional logic is monotone 54

55 An Agent for the Wumpus Game Knowledge Base: X ij = column i, row j S 11 B 11 4 Stench S B S B Stench Gold Stench 2 Start

56 An Agent for the Wumpus Game Knowledge Base: X ij = column i, row j R 1 : S 11 W 11 W 12 W 21 R 2 : S 21 W 11 W 21 W 22 W 31 R 3 : S 12 W 11 W 12 W 22 W 13 R 4 : S 12 W 13 W 12 W 22 W 11 56

57 Agent for the Wumpus Game Finding the wumpus: Modus Ponens ( S 11 + R 1 ): W 11 W 12 W 21 -Elimination: W 11 W 12 W 21 Modus Ponens ( S 21 + R 2 ): W 11 W 21 W 22 W 31 -Elimination: W 11 W 21 W 22 W 31 Modus Ponens (S 12 + R 4 ): W 13 W 12 W 22 W 11 Unit resolution ( W 11 + W 13 W 12 W 22 W 11 ): W 13 W 12 W 22 Unit resolution ( W 22 + W 13 W 12 W 22 ): W 13 W 12 Unit resolution ( W 12 + W 13 W 12 ): W 13 57

58 Agent for the Wumpus Game Translating knowledge into action: A 11 East A W 21 Forward 58

59 Agent for the Wumpus Game Problems with the propositional agent: Too many propositions to handle: "Don t go forward if the wumpus is in front of you!" requires 16 x 4 = 64 propositions. Hard to deal with time and change Not expressive enough to represent or answer a question like "What action should the agent take?", 59

60 Homework In Russell & Norvig s AIMA (2 nd ed.): Exercises of Chapter 7. 60