1. raSAT: SMT solver for nonlinear constraints Vu Xuan Tung – Mizuhito Ogawa (JAIST) To Van Khanh (UET) 44 th TRS Meeting – 22-23 Feb 2016 - Kanazawa 1

2. (Existential) Non-linear Constraints.  Satisfiability of : where  Example:  Notions: UNSAT for Unsatisfiability, SAT for Satisfiability 2

3. Applications 3 Many applications in Software Verification Invariants Generation Analysis of Round-off and Over-flow Errors Termination Proof of Term Rewriting Systems

4. Related methods 4  CAD: complete DEXP for general quantified formulas. EXP for Quantifier-free formulas with optimizations.  Virtual substitution: degree < 5. EXP.  Grobner basis: Equalities. EXP  Interval Constraint Propagation (ICP): Inequalities. EXP on solvable constraints.  Bit-blasting: Bounded variables and precision.  Linearization: Bounded variables and precision.

5. raSAT – an SMT Solver for Polynomial Constraints  Proposed and developed by Dr. To Van Khanh who received his Ph.D. from JAIST in 2013.  raSAT: ICP + Testing + IVT. ICP = Interval Arithmetic + Constraint Propagation + Box Decomposition Testing: boost SAT detection. IVT: The Intermediate Value Theorem for Equations 5 http://www.jaist.ac.jp/~s1310007/raSAT/

6. Interval Arithmetic (IA) 6 Interval Arithmetic Intervals of Variables Intervals of Polynomials

7. Constraint Propagation (CP)  E.g., x 2 + xy < 4 x ∈ [-3,1], y ∈ [-4,-2] 7 + ** x xxy [-3,1] [-4,-2] [0,9][-4,12] [-4,21] [-4,4] – [-4,12] = [-16,8][-4,4] – [0,9] = [-13,4] [0,8] = [-16,8] ∩ ∩ [-13,4] = [-4,4] ∩ (-∞,∞) = [-4, 2] [-2.8..,1] = [-2.8.., 2.8..]∩ [-4,4] <4 [-4,4) / [-3,1] = (-∞,∞) sqrt([0, 8]) = [-2.8.., 2.8..] [-2,1] Result: x ∈ [-2, 1], y ∈ [-4, 2]

8. ICP vs raSAT loop 8

9. Testing 9 Test-SAT with values for x, y Generate values for each variable based on heuristics from IA find values for variables that satisfy the constraint Test-UNSAT Test-SAT implies SAT while Test-UNSAT implies UNKNOWN

10. Completeness Failure of ICP and raSAT loop 10 SAT DetectionUNSAT Detection Kissing case Convergence

11. Non-constructive Handling of Equations  The Intermediate Value Theorem (IVT) 11

12. Non-constructive Handling of Equations  The Generalized IVT Multiple equations |Variables| ≧ |Equations|  Example: 12 Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Middle East Library, Cambridge University Press (1990)

13. raSAT loop + IVT 13

14. Results  SMT-COMP 2015 QF_NRA: 10184 problems Recently revision on CP + IVT: 8517 QF_NIA: 8475 problems 14 Solver[Z3]Yices2-NLSMT-RATraSATCVC3CVC4 (exp)CVC4 Solved No.1000098548759795235752694 Solver[Z3]AProVEraSAT SMT-RAT (parallel) SMT-RATCVC3CVC4 (exp) Solved No. 84598270791774357309191768277 (1 wrong) http://smtcomp.sourceforge.net/2015/

15. Conclusion  ICP is practically efficient though not complete raSAT shows ability to solve large SAT/UNSAT constraints e.g. matrix-2-3, 2-8,3-5, 4-3 and 4-9 in Zankl which have 57, 17, 81, 139 and 193 variables resp. hong family with UNSAT problems of 1, 2, …, 20 variables where problems with10-20 variables are challenging with CAD-based solvers.  Completeness might be achieved by combining with CAD and Gröbner basis  under investigation. 15

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