Introduction to Artificial Intelligence Propositional Logic & SAT Solving. UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010

Transcription

1 Introduction to Artificial Intelligence Propositional Logic & SAT Solving UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010

2 Today Representation in Propositional Logic Semantics & Deduction Axioms and Facts Boolean Algebras Generate and Test Checking Satisfiability (SAT) using DPLL

3 Representing Knowledge Propositional symbols represent facts under consideration: there_is_rain, there_are_clouds, door1_open, robot_in_pos_56_210 Not propositions: is_there_rain? location_of_robot Dan_Roth

4 Truth Values Logical formulas: atoms, connectives, sentences Truth Assignments Evaluating the truth value of a formula Truth tables

5 Propositional Logic Semantics: Truth assignments that satisfy KB/formula I 1 I 2 I 3 I 4 -a -b a -b -a b a b Interpretations: I 1 [a]=false I 1 [b]=false assign truth values to propositional symbols

6 Propositional Logic Semantics: Truth assignments that satisfy KB/formula I 1 -a -b I 2 a -b M 1 = I 3 -a b M 2 = I 4 a b Models of f: Interpretations that satisfy f

7 Propositional Logic Semantics: Truth assignments that satisfy KB/formula M 1 = I 3 M 2 = I 4 Models of f: Interpretations that satisfy f

8 Propositional Logic Semantics: Truth assignments that satisfy KB/formula M 1 M 2 = I 4 Models of f: Interpretations that satisfy f

9 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment M 1

10 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment M 1

11 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment M 1

12 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment M 1

13 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment M 1

14 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment M 1

15 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment M 1

16 Examples Double negation, double disjunction De-Morgan s law Distributivity of conjunction and disjunction

17 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment

18 Propositional Logic Semantics: Truth assignments that satisfy KB/formula Logical Entailment Deduction (inference)

19 Notations Interpretations ~ Models Axioms formulae that are assumed Signature the symbols used by a KB Theory ~ KB (a set of axioms), or Theory ~ the complete set of sentences entailed by the axioms The value that symbol p takes in model M: M[p] Clauses: {lit1, lit2, lit3, } or lit1 lit2 lit3...

20 Representing Knowledge Knowledge bases are sets of formulae There_is_rain there_are_clouds Robot_in_pos_3_1 Position_3_1_empty Has_drink coffee tea

21 Knowledge Engineering Select a language: set of features Examine cases Decide on dependencies between features Write dependencies formally Test

22 Clausal Form Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). Examples (by Distributivity):

23 Clausal Form Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). Examples (by Distributivity): ( a a)

24 Clausal Form Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). Examples (by Distributivity): ( a a) ( a b)

25 Clausal Form Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). Examples (by Distributivity): ( a a) ( a b) ( b a)

26 Clausal Form Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). Examples (by Distributivity): ( a a) ( a b) ( b a) ( b b)

27 Clausal Form Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). Examples: Face_door_t1 ^ turn_cl_90_t1 -> ~face_door_t2 ~Face_door_t1 v ~turn_cl_90_t1 v ~face_door_t2

28 Clausal Form Every formula can be reformulated into an equivalent CNF formula (conjunction of clauses). Examples: face_door_t1 ^ move_fwd_t1 -> at_corridor_t2 ^ ~face_door_t2 ~face_door_t1 v ~move_fwd_t1 v ~at_corridor_t2 ~face_door_t1 v ~move_fwd_t1 v ~face_door_t2

29 Application: Hardware Verification x 1 x 2 AND f 1 not f 3 not f 2 AND f 5 x 3 OR f 4

30 Application: Hardware Verification x 1 x 2 AND f 1 not f 3 not f 2 AND f 5 x 3 OR f 4 Question: Can we set this boolean cirtuit to TRUE?

31 Application: Hardware Verification x 1 x 2 AND f 1 not f 3 not f 2 AND f 5 x 3 OR f 4 Question: Can we set this boolean cirtuit to TRUE? f 5 (x 1,x 2,x 3 ) = a function of the input signal

32 Application: Hardware Verification x 1 x 2 AND f 1 not f 3 not f 2 AND f 5 x 3 OR f 4 SAT(f 5 )? Question: Can we set this boolean cirtuit to TRUE? f 5 (x 1,x 2,x 3 ) = f 3 f 4 = f 1 (f 2 x 3 ) = (x 1 x 2 ) ( x 2 x 3 ) M[x 1 ]=FALSE M[x 2 ]=FALSE M[x 3 ]=FALSE

33 Hardware Verification Questions in logical circuit verification Equivalence of circuits Arrival of the circuit to a state (required a temporal model, which gets propositionalized) Achieving an output from the circuit

34 SATisfiability of Logical Formulas

35 SATisfiability of Logical Formulas Given KB in CNF If we have a truth table of KB, then we can check that KB satisfiable by looking at it.

36 SATisfiability of Logical Formulas Given KB in CNF If we have a truth table of KB, then we can check that KB satisfiable by looking at it. Problem: n propositional symbols 2 n rows in truth table Checking truth table with KB takes time O( KB ) Generating table is expensive: O(2 n KB ) time

37 SATisfiability of Logical Formulas Given KB in CNF If we have a truth table of KB, then we can check that KB satisfiable by looking at it. Problem: n propositional symbols 2 n rows in truth table Checking truth table with KB takes time O( KB ) Generating table is expensive: O(2 n KB ) time Observation: SAT requires us to look only for one model

38 DPLL (1960) Procedure for CNF 1. If KB is empty: True 2. If KB has empty clause: False 3. If KB has a unit clause ( a literal ) propagate literal to simplify clauses 4. else: Choose a variable heuristically Set var True, Recurse Set var False, Recurse Return value of recursion Simplify if possible. Search for satisfying assignment. Just a recursive search.

39 Propagating a Truth-Value KB in CNF, and we observe a=true Then, removing clauses with a positive from KB gives an equivalent theory. Example: KB

40 Propagating a Truth-Value KB in CNF, and we observe a=true Then, removing clauses with a positive from KB gives an equivalent theory. Example: KB Observe

41 Propagating a Truth-Value KB in CNF, and we observe a=true Then, removing clauses with a positive from KB gives an equivalent theory. Example: KB Observe

42 Propagating a Truth-Value KB in CNF, and we observe a=true Then, removing clauses with a positive from KB gives an equivalent theory. Example: KB Observe

43 Propagating a Truth-Value KB in CNF, and we observe a=true Then, removing negative instances of a from KB gives an equivalent theory. Example: KB Observe

44 Propagating a Truth-Value KB in CNF, and we observe a=true Then, removing negative instances of a from KB gives an equivalent theory. Example: KB Observe

45 DPLL Example

46 DPLL Example p XOR q = (p v q) & (-p v -q)

47 DPLL Example p XOR q = (p v q) & (-p v -q) p IFF q = (p XOR q)

48 DPLL Example p XOR q = (p v q) & (-p v -q) p IFF q = (p XOR q) Not SAT

49 DPLL Example p XOR q = (p v q) & (-p v -q) p IFF q = (p XOR q) (p XOR q) & (p XOR -q) Not SAT

50 DPLL Example p XOR q = (p v q) & (-p v -q) p IFF q = (p XOR q) (p XOR q) & (p XOR -q) Not SAT ( ((p1 XOR p2) XOR p3) XOR p4 ) &

51 DPLL Example p XOR q = (p v q) & (-p v -q) p IFF q = (p XOR q) (p XOR q) & (p XOR -q) Not SAT ( ((p1 XOR p2) XOR p3) XOR p4 ) & ( ((p1 XOR p2) XOR p3) XOR -p4 )

52 DPLL Example p XOR q = (p v q) & (-p v -q) p IFF q = (p XOR q) (p XOR q) & (p XOR -q) Not SAT ( ((p1 XOR p2) XOR p3) XOR p4 ) & ( ((p1 XOR p2) XOR p3) XOR -p4 ) DPLL takes O(2 n ) time, sometimes

53 SAT Solving Topics Order of selection of variables Stochastic local search Binary Decision Diagrams Strategies other than unit 2-SAT is solvable in linear time Smart backtracking Clauses/Vars in Random SAT SAT via probabilistic models