Satisfiability Modulo Transcendental Functions via Incremental Linearization

In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted in the abstract space, which is described in terms of the combined theory of linear arithmetic on the rationals with uninterpreted functions, and are incrementally axiomatized by means of upper- and lower-bounding piecewise-linear functions. Suitable numerical techniques are used to ensure that the abstractions of the transcendental functions are sound even in presence of irrationals. Our experimental evaluation on benchmarks from verification and mathematics demonstrates the potential of our approach, showing that it compares favorably with delta-satisfiability /interval propagation and methods based on theorem proving

Incomplete SMT techniques for solving non-linear formulas over the integers

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

SMT-Solving Induction Proofs of Inequalities

This paper accompanies a new dataset of non-linear real arithmetic problems for the SMT-LIB benchmark collection. The problems come from an automated proof procedure of Gerhold--Kauers, which is well suited for solution by SMT. The problems of this type have not been tackled by SMT-solvers before. We describe the proof technique and give one new such proof to illustrate it. We then describe the dataset and the results of benchmarking. The benchmarks on the new dataset are quite different to the existing ones. The benchmarking also brings forward some interesting debate on the use/inclusion of rational functions and algebraic numbers in the SMT-LIB.Comment: Presented at the 2022 SC-Square Worksho

Satisfiability of Non-Linear Transcendental Arithmetic as a Certificate Search Problem

For typical first-order logical theories, satisfying assignments have a straightforward finite representation that can directly serve as a certificate that a given assignment satisfies the given formula. For non-linear real arithmetic with transcendental functions, however, no general finite representation of satisfying assignments is available. Hence, in this paper, we introduce a different form of satisfiability certificate for this theory, formulate the satisfiability verification problem as the problem of searching for such a certificate, and show how to perform this search in a systematic fashion. This does not only ease the independent verification of results, but also allows the systematic design of new, efficient search techniques. Computational experiments document that the resulting method is able to prove satisfiability of a substantially higher number of benchmark problems than existing methods