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Solving bitvectors with MCSAT: explanations from bits and pieces Stphane Graham-Lengrand, Dejan Jovanovi, Bruno Dutertre SRI International IJCAR, July 2020 1/32 tl;dl (Too Long; Didnt Listen) MCSAT (Model-Constructing

  1. Solving bitvectors with MCSAT: explanations from bits and pieces Stéphane Graham-Lengrand, Dejan Jovanović, Bruno Dutertre SRI International IJCAR, July 2020 1/32

  2. tl;dl (Too Long; Didn’t Listen) ◮ MCSAT (Model-Constructing Satisfiability) is a scheme for SMT-solving (Satisfiability-Modulo-Theories), alternative to DPLL( T ). ◮ To apply the scheme to a particular theory T , you need a form of interpolation mechanism for T . ◮ Designing an efficient mechanism for the full theory of bitvectors is difficult. So we do it for 2 fragments of the theory: ◮ Equality + concatenation and extraction of bitvectors ◮ A fragment of bitvector arithmetic Outside these fragments we use a less efficient, but generic, procedure. ◮ The approach is implemented in SRI’s SMT-solver Yices. ◮ We experimented it on the SMTLib benchmarks. 2/32

  3. Overview of MCSAT The bitvector theory in MCSAT Experimentation on the SMTLib benchmarks Conclusion 3/32

  4. 1. Overview of MCSAT 4/32

  5. The model-constructing approach to SMT-solving 1/2 MCSAT introduced in [dMJ13, JBdM13, Jov17], inspired by multiple contributions including Conflict Resolution [KTV09] and specific decision procedures for theories such as non-linear arithmetic [JdM12]. 5/32

  6. The model-constructing approach to SMT-solving 1/2 MCSAT introduced in [dMJ13, JBdM13, Jov17], inspired by multiple contributions including Conflict Resolution [KTV09] and specific decision procedures for theories such as non-linear arithmetic [JdM12]. MCSAT tailored to theories with a standard model used for evaluating constraints (example: arithmetic) Evaluation is a key aspect of MCSAT 5/32

  7. The model-constructing approach to SMT-solving 1/2 MCSAT introduced in [dMJ13, JBdM13, Jov17], inspired by multiple contributions including Conflict Resolution [KTV09] and specific decision procedures for theories such as non-linear arithmetic [JdM12]. MCSAT tailored to theories with a standard model used for evaluating constraints (example: arithmetic) Evaluation is a key aspect of MCSAT Solving satisfiability problem (set of constraints on variables x 1 , . . . , x n ) = finding values for variables x 1 , . . . , x n (so that constraints evaluate to true) 5/32

  8. The model-constructing approach to SMT-solving 1/2 MCSAT introduced in [dMJ13, JBdM13, Jov17], inspired by multiple contributions including Conflict Resolution [KTV09] and specific decision procedures for theories such as non-linear arithmetic [JdM12]. MCSAT tailored to theories with a standard model used for evaluating constraints (example: arithmetic) Evaluation is a key aspect of MCSAT Solving satisfiability problem (set of constraints on variables x 1 , . . . , x n ) = finding values for variables x 1 , . . . , x n (so that constraints evaluate to true) MCSAT offers: ◮ a template for decision procedures ◮ an integration of such procedures with Boolean reasoning ◮ new possibilities for combining theories [JBdM13, BGLS19] 5/32

  9. The model-constructing approach to SMT-solving 1/2 MCSAT introduced in [dMJ13, JBdM13, Jov17], inspired by multiple contributions including Conflict Resolution [KTV09] and specific decision procedures for theories such as non-linear arithmetic [JdM12]. MCSAT tailored to theories with a standard model used for evaluating constraints (example: arithmetic) Evaluation is a key aspect of MCSAT Solving satisfiability problem (set of constraints on variables x 1 , . . . , x n ) = finding values for variables x 1 , . . . , x n (so that constraints evaluate to true) MCSAT offers: ◮ a template for decision procedures ◮ an integration of such procedures with Boolean reasoning ◮ new possibilities for combining theories [JBdM13, BGLS19] The template is a generalisation of how CDCL works, the core calculus of SAT-solvers. Run = alternation of search phases and conflict analysis phases 5/32

  10. The model-constructing approach to SMT-solving 2/2 ◮ Like CDCL’s trail assigns Boolean values to Boolean variables, MCSAT’s trail assigns ◮ Boolean values to theory atoms; these constitute theory contraints ◮ model values to first-order variables (e.g., x ← 3 / 4) 6/32

  11. The model-constructing approach to SMT-solving 2/2 ◮ Like CDCL’s trail assigns Boolean values to Boolean variables, MCSAT’s trail assigns ◮ Boolean values to theory atoms; these constitute theory contraints ◮ model values to first-order variables (e.g., x ← 3 / 4) ◮ As in CDCL, MCSAT successively guesses assignments. . . . . . while maintaining the invariant that no constraint evaluates to false according to the assignments ; 6/32

  12. The model-constructing approach to SMT-solving 2/2 ◮ Like CDCL’s trail assigns Boolean values to Boolean variables, MCSAT’s trail assigns ◮ Boolean values to theory atoms; these constitute theory contraints ◮ model values to first-order variables (e.g., x ← 3 / 4) ◮ As in CDCL, MCSAT successively guesses assignments. . . . . . while maintaining the invariant that no constraint evaluates to false according to the assignments ; ◮ To pick a value for variable y after x 1 , . . . , x n were assigned values v 1 , . . . , v n , simply worry about constraints over variables x 1 , . . . , x n , y (i.e. constraints that have become unit in y ) 6/32

  13. The model-constructing approach to SMT-solving 2/2 ◮ Like CDCL’s trail assigns Boolean values to Boolean variables, MCSAT’s trail assigns ◮ Boolean values to theory atoms; these constitute theory contraints ◮ model values to first-order variables (e.g., x ← 3 / 4) ◮ As in CDCL, MCSAT successively guesses assignments. . . . . . while maintaining the invariant that no constraint evaluates to false according to the assignments ; ◮ To pick a value for variable y after x 1 , . . . , x n were assigned values v 1 , . . . , v n , simply worry about constraints over variables x 1 , . . . , x n , y (i.e. constraints that have become unit in y ) ◮ If all variables get values while maintaining invariant: SAT . illustration on the next slide. 6/32

  14. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  15. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  16. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  17. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  18. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  19. Search phase (satisfiable case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i SAT 7/32

  20. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  21. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  22. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

  23. Search phase (conflict case) Free var within Constraints (unit ones in red) Feasible set Var C 1 1 , . . . , C 1 { x 1 } j , . . . x 1 C 2 1 , C 2 2 , . . . , C 2 { x 1 , x 2 } j , . . . x 2 C 3 1 , C 3 2 , . . . , C 3 { x 1 , x 2 , x 3 } j , . . . x 3 . . . C i 1 , C i 2 , . . . , C i 42 , . . . , C i { x 1 , . . . , x i } j , . . . x i 7/32

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