Nikolaj Bjørner Senior Researcher Microsoft Research Redmond

1. Nikolaj Bjørner Senior Researcher Microsoft Research Redmond Modern Satisfiability Modulo Theories Solvers in Program Analysis

2. Lectures Wednesday10:45–12:15An Introduction to Z3 with Applications Thursday August 30th15:45–17:15 Introduction to SAT and SMT Friday 10:30–10:45 Theories and Solving Algorithms Friday 15:45–17:15 Advanced: Quantifiers, Arrays, Fixed-points

3. Plan • Logic: Calculus of Computation • SAT, First-order Theorem Proving, SMT • Introduction to Propositional Logic(SAT): • Language, Satisfiability, Validity, • Normal forms, CNF, • Search, Resolution, DPLL search • Introduction to SMT by example • SMT solving, Combining theories

4. Takeaways The syntax and semantics of propositional and predicate logic Algorithmic principles of modern SAT solvers: • DPLL algorithm • Conflict Directed Clause Learning (CDCL) • Two-watch literal indexing Principles of modern SMT solvers

5. Background Reading

6. Background Reading September 2011

7. Logic: Calculus of Computation Formal Logic: Language: Precise syntax of well-formed formulas Examples: propositional logic, equational logic, first-order logic, higher-order logic, and modal logics. Model theory: Precise semantics of truth and valid consequences Proof theory: Axioms and inference rules for truth and consequences Calculus: Basis for specifications and reasoning about computation Mechanized Formal Logic: Symbolic: The art and science of effective symbolic reasoning Automatic: Computers can automate symbolic calculations

8. Symbolic Engines: SAT, FTP and SMT SAT: Propositional Satisfiability. (Tie  Shirt)  (Tie Shirt)  (Tie  Shirt) FTP: First-order Theorem Proving. X,Y,Z [X*(Y*Z) = (X*Y)*Z] X [X*inv(X) = e] X [X*e = e] SMT: Satisfiability Modulo background Theoriesb + 2 = c  A[3]≠ A[c-b+1]

9. SAT - Milestones Problems impossible 10 years ago are trivial today Concept 2002 2010 Millions of variables from HW designs Courtesy Daniel le Berre

10. FTP - Milestones • Some successstories: • Open Problems (of 25 years):XCB: X  ((X  Y)  (Z  Y))  Z)is a single axiom for equivalence • Knowledge Ontologies GBs of formulas Courtesy Andrei Voronkov, U of Manchester

11. SMT - Milestones Z3 (of ’07) Time On BoogieRegression 1sec Simplify (of ’01) time Z3 Time On VCC Regression Includes progress from SAT: 15KLOC + 285KLOC = Z3 Nov 08 March 09

12. Propositional Logic Syntax, Semantics and Normal Forms Resolution, Classical DPLL, Modern DPLL

13. Propositional Logic: Language Logical symbols: , ,  ,true, false,  - fixed interpretation Non-logical symbols: p, q, r - interpretations vary Well-formed formulas: built by combining logical and non-logical symbols

14. Propositional Logic: Language Formulas:  := p | 12| 12|1 | 1 2 Examples: p  q  q  p p  q  (p  q) We say pand qare propositional variables.

15. Propositional Logic Syntax, Semantics and Normal Forms Resolution, Classical DPLL, Modern DPLL

16. Interpretation An interpretation M assigns values {true, false} to propositional variables. Let F and G range over PL formulas.

17. Satisfiability & Validity A formula is: • satisfiable if it has an interpretation that makes it logically true. In this case, we say the interpretation is a model. • unsatisfiable if it does not have any model. • valid if it is logically true in any interpretation. • A propositional formula is valid if and only if its negation is unsatisfiable.

18. Satisfiability & Validity: examples p  q  q  p p  q  q p  q  (p  q)

19. Satisfiability & Validity: examples p  q  q  p VALID p  q  q SATISFIABLE p  q  (p  q) UNSATISFIABLE

20. Equivalence We say two formulas F and G are equivalent if and only if they evaluate to the same value (true or false) in every interpretation

21. Equisatisfiable We say formulas A and B are equisatisfiable if and only if A is satisfiable if and only if B is. equisat. During this tutorial, we describe transformations that preserve equivalence and equisatisfiability.

22. Propositional Logic Syntax, Semantics, Normal Forms Resolution, Classical DPLL, Modern DPLL

23. Normal Forms Literal – either a propositional atom or its negation NNF – Negation Normal Form Formula with negation only used for literals CNF – Conjunctive Normal Form Conjunction of disjunctions of literals DNF – Disjunctive Normal Form Disjunction of conjunctions of literals

24. Normal Forms Conditional normal form Formula with only collectives BDD – (reduced ordered) Binary Decision Diagram Formula with only collectives test only uses atomsAtoms are ordered such that DAG: Share common sub-expressions Exercises: • Show that every propositional formula is equivalent to a formula in • (1) NNF, (2) CNF, (3) DNF, (4)CondNF, (5) BDD • In each case, what is the size overhead of the conversion? • Show that every n-ary Boolean function can be expressed using and

25. Normal Forms NNF? (p  q)  (q  (r  p))

26. Normal Forms NNF? NO

27. Normal Forms NNF? NO

28. Normal Forms NNF? NO 

29. Normal Forms NNF? NO

30. Normal Forms CNF? ((p  s) (q r))  (q  p s)  (r s)

31. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s)

32. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

33. Normal Forms CNF? NO ((p  s) (q  r))  (q  p  s)  (r  s)  ((p  s) q)) ((p  s) r))  (q  p  s)  (r  s) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

34. Normal Forms CNF? NO Distributivity

35. Normal Forms CNF? .. yes ((p  s) (q  r))  (q  p  s)  (r  s)  ((p  s) q)) ((p  s) r))  (q  p  s)  (r  s)  (p  q)  (s q) ((p  s) r))  (q  p  s)  (r  s)  (p  q)  (s  q)  (p  r) (s  r)  (q  p  s)  (r  s)

36. Normal Forms DNF? p  (p  q)  (q  r)

37. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r)

38. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

39. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r)  ((p  p) (p q))  (q  r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC)

40. Normal Forms DNF? NO, actually this formula is in CNF p  (p  q)  (q  r)  ((p  p) (p q))  (q  r)  (p  q)  (q  r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC) Other Rules AA   A  A

41. Normal Forms DNF? … yes p  (p  q)  (q  r)  ((p  p) (p q))  (q  r)  (p  q)  (q  r)  ((p  q) q) ((p  q)  r)  (pq)  (p r) (q r) Distributivity 1. A(BC)  (AB)(AC) 2. A(BC)  (AB)(AC) Other Rules AA   A  A

42. Efficient CNF Translation CNF translation using distributivity rule is too expensive (exponential blowup). Linear time/space translation produces equisatisfiable formula: where is a fresh variable. Exercise: show that each transformation preserves satisfiability. Exercise: finish the transformation for:

43. CNF translation (example)

44. Propositional Logic Syntax, Semantics, Normal Forms Resolution, Classical DPLL, Modern DPLL

45. Resolution Formula must be in CNF Resolution rule: Example: The result of resolution is the resolvent(clause). Original clauses are kept (not deleted). Duplicate literals are deleted from the resolvent. Note: No branching. Termination: Only finite number of possible derived clauses.

46. Resolution (example)

47. Unit & Input Resolution Unit resolution: (is subsumed by Input resolution: ( member of input F). Exercise: Set of clauses F: F has an input refutation iffF has a unit refutation.

48. Propositional Logic Syntax, Semantics, Normal Forms Resolution, Classical DPLL, Modern DPLL

49. DPLL DPLL: David Putnam Logeman Loveland = Unit resolution + split rule. split unit Ingredient of most efficient SAT solvers

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