Satisfiability Modulo Theories

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1 Satisfiability Modulo Theories Bruno Dutertre SRI International Leonardo de Moura Microsoft Research

2 Satisfiability a > b + 2, a = 2c + 10, c + b 1000 SAT a = 0, b = 3, c = 5 Model 0 > 3 + 2, 0 = , 5 + ( 3) 1000

3 Satisfiability b + 2 = c, f(read(write(a,b,3), c-2)) f(c-b+1)

4 Satisfiability b + 2 = c, f(read(write(a,b,3), c-2)) f(c-b+1) Arithmetic

5 Satisfiability b + 2 = c, f(read(write(a,b,3), c-2)) f(c-b+1) Array Theory

6 Satisfiability b + 2 = c, f(read(write(a,b,3), c-2)) f(c-b+1) Uninterpreted Functions

7 Satisfiability b + 2 = c, f(read(write(a,b,3), c-2)) f(c-b+1)

8 Satisfiability b + 2 = c, f(read(write(a,b,3), b+2-2)) f(b+2-b+1)

9 Satisfiability b + 2 = c, f(read(write(a,b,3), b+2-2)) f(b+2-b+1)

10 Satisfiability b + 2 = c, f(read(write(a,b,3), b)) f(b+2-b+1)

11 Satisfiability b + 2 = c, f(read(write(a,b,3), b)) f(3) Array Theory Axiom a,i,v : read(write(a, i, v), i) = v

12 Satisfiability b + 2 = c, f(3) f(3) UNSAT

13 Applications Is execution path P feasible? Is assertion X violated? Test case Generator W I T N E S S Program Verifier Is Formula F Satisfiable?

14 Test Case Generation unsigned GCD(x, y) { requires(y > 0); while (true) { unsigned m = x % y; if (m == 0) return y; x = y; y = m; SSA (y 0 > 0) and (m 0 = x 0 % y 0 ) and not (m 0 = 0) and (x 1 = y 0 ) and (y 1 = m 0 ) and (m 1 = x 1 % y 1 ) and } (m 1 = 0) } We want a trace where the loop is executed twice. Solver x 0 = 2 y 0 = 4 m 0 = 2 x 1 = 4 y 1 = 2 m 1 = 0

15 More Applications Planning Scheduling Constraint Solving Systems Biology Invariant Generation Type Checking Model Based Testing Termination

16 Some Applications at Microsoft SAGE HAVOC Vigilante

17 Validity F is VALID iff not F is UNSATISFIABLE Prove that x 1, y 1 x + y 2 Is x 1, y 1, not x + y 2 UNSAT?

18 Download Yices Z3 Available for Windows, OSX and Linux

19 SMT-Lib Online Tutorials

20 Roadmap Lecture 1: Introduction, SAT Lecture 2: SMT, EUF, Linear Arithmetic Lecture 3: Quantifiers Lecture 4: Applications and Challenges

21 SAT Propositional Logic

22 CNF p 1 p 2, p 1 p 2 p 3, p 3 p 1 = true, p 2 = true, p 3 = true CNF is a set (conjunction) set of clauses Clause is a disjunction of literals Literal is an atom or the negation of an atom

23 Conversion to CNF p l 1 l 2

24 Conversion to CNF

25 Conversion to CNF CNF q 2 q 3, = p 1, p 1 q 2 q 3, q 2 p 1, q 3 p 1 CNF q 1, = q 1, p 1 q 2 q 3

26 Conversion to CNF

27 Conversion to CNF: Improvements

28 Two procedures Resolution Proof-finder Saturation DPLL Model-finder Search

29 Resolution C l, D l C D l, l unsat Improvements Delete tautologies l l C Ordered Resolution Subsumption (delete redundant clauses) C subsumes C D

30 Resolution: Example

31 Resolution: Example

32 Resolution: Example

33 Resolution: Example

34 Resolution: Example

35 Resolution: Correctness

36 Resolution: Correctness Suppose C and D are false in M i Let j = i + 1 p j is maximal in p j C, pj D C D is false in M i

37 Resolution: Problem Exponential time and space

38 Unit Resolution C l, l C C subsumes C l Complete for Horn Clauses q 1 qn p

39 DPLL Split rule S S, p S, p DPLL = Unit Resolution + Split rule

40 Pure Literals A literal is pure if only occurs positively or negatively.

41 DPLL

42 DPLL : Example

43 DPLL : Example

44 DPLL : Example

45 DPLL : Example

46 DPLL : Example

47 DPLL : Example

48 CDCL: Conflict Driven Clause Learning DPLL Resolution Model Proof

49 CDCL: Conflict Driven Clause Learning Modern SAT solvers are based on CDCL Backjumping Learning Restarts Indexing

50 Abstract CDCL/DPLL Partial model Set of clauses

51 Abstract CDCL/DPLL

52 Abstract CDCL : Example

53 Abstract CDCL : Example

54 Abstract CDCL : Example

55 Abstract CDCL : Example

56 Abstract CDCL : Example

57 Abstract CDCL : Example

58 Abstract CDCL : Example

59 Abstract CDCL : Example

60 Abstract CDCL : Example

61 Abstract CDCL : Example

62 Abstract CDCL : Example p 5 p 6 p 6 p 5 p 2 p 5 p 2

63 Abstract CDCL : Example

64 Abstract CDCL : Example

65 Abstract CDCL : Example

66 Abstract CDCL

67 Abstract CDCL : Strategy

68 Abstract CDCL : Decision Strategy

69 Abstract CDCL : Phase Selection

70 Abstract CDCL : Extra Rules

71 Abstract CDCL : Restart Strategies

72 Abstract CDCL : Indexing

73 Abstract CDCL : Indexing

74 Indexing : Two watch literal

75 Indexing : Two watch literal

76 CDCL: Conflict Driven Clause Learning DPLL Resolution Model Proof

77 Preprocessing & Inprocessing

78 Homework Install Yices & Z3 in your notebook Yices input language: Online tutorials: SMT Python

Satisfiability Modulo Theories

Satisfiability Modulo Theories Summer School on Formal Methods Menlo College, 2011 Bruno Dutertre and Leonardo de Moura bruno@csl.sri.com, leonardo@microsoft.com SRI International, Microsoft Research SAT/SMT