A Generalised Branch-and-Bound Approach and Its Application in SAT Modulo Nonlinear Integer Arithmetic

Abstract

The branch-and-bound framework has already been successfully applied in SAT-modulo-theories (SMT) solvers to check the satisfiability of linear integer arithmetic formulas. In this paper we study how it can be used in SMT solvers for non-linear integer arithmetic on top of two real-algebraic decision procedures: the virtual substitution and the cylindrical algebraic decomposition methods. We implemented this approach in our SMT solver SMT-RAT and compared it with the currently best performing SMT solvers for this logic, which are mostly based on bit-blasting. Furthermore, we implemented a combination of our approach with bit-blasting that outperforms the state-of-the-art SMT solvers for most instances.

Experiments with Automated Reasoning in the Class

“…The latest release [37] offers, besides some non-arithmetic components, the CaRL [9] library for arithmetic datatypes and basic computations with them, and solver modules for e.g. the simplex and the Fourier-Motzkin variable elimination methods for linear problems, for non-linear real arithmetic different adaptions of the cylindrical algebraic decomposition method [10,30], the virtual substitution method [48,12,1], subtropical satisfiability [22], methods based on Gröbner bases [47,28], as well as interval constraint propagation [24,25] and the branchand-bound technique [17,31] for finding integer solutions.…”

Modular strategic SMT solving with SMT-RAT

“…• The incomplete branch-and-bound method [41] can be used to extend decision procedures for real arithmetic to check the satisfiability of integer arithmetic constraint sets. It first checks the real relaxation of the input constraints using some decision procedure for the reals, i.e., assuming the variables to be real-valued instead of integer-valued.…”

“…The focus of our research lies however on nonlinear arithmetic. The cylindrical algebraic decomposition method is implemented in the CADModule [16,41,54], the only complete SMT-RAT module for nonlinear real arithmetic, which is complemented by different incomplete solver modules. The GBModule [38] uses a variant Gröbner bases to determine satisfiability over the reals.…”

“…All these modules can also work on nonlinear integer arithmetic problems using the branch-and-bound technique, though all of them are incomplete due to the undecidability of nonlinear integer arithmetic. Additionally the IntBlastModule [42,41] encodes bounded nonlinear integer problems in bit-vector arithmetic, similar to what most other solvers do for this kind of problems.…”

“…Based on the finding that at least our fundamental polynomial procedures are reasonably fast, we want to investigate whether SMT-RAT is competitive with other state-of-the-art solvers. Past publications have shown that SMT-RAT usually performs pretty good, in particular on nonlinear real arithmetic [52,16] and nonlinear integer arithmetic [41,35].…”