We present new methods for solving the Satisfiability Modulo Theories problem over the theory of QuantifierFree Non-linear Integer Arithmetic, SMT(QF-NIA), which consists of 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 ∃∀ formulas in a fragment of quantified
non-linear arithmetic that appears commonly in verification and synthesis applications.Peer ReviewedPostprint (author's final draft
