Model-Driven Decision Procedures for Arithmetic SYNASC 2013

1. Model-Driven Decision Proceduresfor ArithmeticSYNASC 2013 Leonardo de Moura Microsoft Research

2. Logic Engines as a Service SAGE SecGuru

3. Satisfiability Solution/Model sat, unsat, Proof Is execution path P feasible? Is assertion X violated? WI TNESS SAGE Is Formula FSatisfiable?

4. The RISE of Model-Driven Techniques

5. Saturation x Search Proof-finding Model-finding Proofs Conflict Resolution Models

6. SAT CNF is a set (conjunction) set of clauses Clause is a disjunction of literals Literal is an atom or the negation of an atom

7. Two procedures

8. Resolution Improvements Delete tautologies Ordered Resolution Subsumption (delete redundant clauses) …

9. Resolution: Example

10. Resolution: Example

11. Resolution: Example

12. Resolution: Example

13. Resolution: Example

14. Resolution: Problem Exponential time and space

15. Unit Resolution subsumes

16. DPLL Split rule DPLL = Unit Resolution + Split rule

17. DPLL

18. DPLL

19. DPLL

20. DPLL

21. DPLL

22. DPLL

23. CDCL: Conflict Driven Clause Learning DPLL Conflict Resolution Resolution Model Proof

24. Linear Arithmetic

25. Fourier-Motzkin Very similar to Resolution Exponential time and space

26. Polynomial Constraints AKA Existential Theory of the Reals R

27. Applications

28. CAD “Big Picture” 1. Project/Saturate set of polynomials 2. Lift/Search: Incrementally build assignment Isolate roots of polynomials Select a feasible cell , and assign some If there is no feasible cell, then backtrack

29. CAD “Big Picture” 1. Saturate 2. Search

30. CAD “Big Picture” 1. Saturate 2. Search

31. CAD “Big Picture” 1. Saturate CONFLICT 2. Search

32. NLSAT: Model-Based Search Static x Dynamic Optimistic approach Key ideas Proofs Conflict Resolution Models Start the Search before Saturate/Project We saturate on demand Model guides the saturation

33. NLSAT (1) Two kinds of decision 1. case-analysis (Boolean) 2. model construction (CAD lifting)

34. NLSAT (1) Two kinds of decision 1. case-analysis (Boolean) 2. model construction (CAD lifting) Parametric calculus: Finite basis explanation function Explanations may contain new literals They evaluate to false in the current state

35. NLSAT (2) Key ideas: Use partial solution to guide the search Feasible Region Starting search Partial solution: What is the core? Can we extend it to ?

36. NLSAT (2) Key ideas: Use partial solution to guide the search Feasible Region Starting search Partial solution: What is the core? Can we extend it to ?

37. NLSAT (3) Key ideas: Solution based Project/Saturate Standard project operators are pessimistic. Coefficients can vanish!

38. NLSAT (4) Key ideas: Lemma Learning Prevent a Conflict from happening again. Current assignment Conflict Current assignments does not satisfy new constraint. Lemma

39. NLSAT (5) Key ideas: Nonchronological Backtracking Conflict = 1 The values chosen for and are irrelevant.

40. Machinery Multivariate & univariate Polynomials Basic operations, Pseudo-division, GCD, Resultant, PSC, Factorization, Root isolation algorithms, Sturm sequences Binary rationals Real Algebraic Numbers

41. Experimental Results (1) OUR NEW ENGINE

42. Experimental Results (2) OUR NEW ENGINE

43. Other examples(for linear arithmetic) Generalizing DPLL to richer logics [McMillan et al 2009] Fourier-Motzkin X Conflict Resolution [Korovin et al 2009]

44. Other examples Lemmas on Demand For Theory of Array [Brummayer-Biere 2009] Array Theory by Axiom Instantiation X

45. Saturation: successful instances Polynomial time procedures Gaussian Elimination Congruence Closure

46. MCSat Model-Driven SMT Lift ideas from CDCL to SMT Generalize ideas found in model-driven approaches Easier to implement Model construction is explicit

47. MCSat

48. MCSat Propagations

49. MCSat Propagations

50. MCSat Propagations