Incomplete SMT Techniques for Solving Non-Linear Formulas over the Integers

TL;DR

This paper introduces novel SMT solving techniques for non-linear integer formulas by leveraging optimization methods to guide domain enlargement, enabling efficient solving of complex quantified formulas in verification and synthesis.

Contribution

It presents new optimization-guided domain expansion strategies for SMT(QF-NIA), extending to Max-SMT and quantified formulas, improving solving efficiency over previous methods.

Findings

Effective domain enlargement guided by Max-SMT reduces solving time.

Optimization techniques enable solving Max-SMT over non-linear integer arithmetic.

Application to quantified formulas benefits verification and synthesis tasks.

Abstract

We present new methods for solving the Satisfiability Modulo Theories problem over the theory of Quantifier-Free Non-linear Integer Arithmetic, SMT(QF-NIA), which consists in deciding the satisfiability of ground formulas with integer polynomial constraints. Following previous work, we propose to solve SMT(QF-NIA) instances by reducing them to linear arithmetic: non-linear monomials are linearized by abstracting them with fresh variables and by performing case splitting on integer variables with finite domain. For variables that do not have a finite domain, we can artificially introduce one by imposing a lower and an upper bound, and iteratively enlarge it until a solution is found (or the procedure times out). The key for the success of the approach is to determine, at each iteration, which domains have to be enlarged. Previously, unsatisfiable cores were used to identify the domains to be changed, but no clue was obtained as to how large the new domains should be. Here we explain two novel ways to guide this process by analyzing solutions to optimization problems: (i) to minimize the number of violated artificial domain bounds, solved via a Max-SMT solver, and (ii) to minimize the distance with respect to the artificial domains, solved via an Optimization Modulo Theories (OMT) solver. Using this SMT-based optimization technology allows smoothly extending the method to also solve Max-SMT problems over non-linear integer arithmetic. Finally we leverage the resulting Max-SMT(QF-NIA) techniques to solve $\exists \forall$ formulas in a fragment of quantified non-linear arithmetic that appears commonly in verification and synthesis applications.