Subtropical Satisfiability

Abstract

Abstract. Quantifier-free nonlinear arithmetic (QF NRA) appears in many applications of satisfiability modulo theories solving (SMT). Accordingly, efficient reasoning for corresponding constraints in SMT theory solvers is highly relevant. We propose a new incomplete but efficient and terminating method to identify satisfiable instances. The method is derived from the subtropical method recently introduced in the context of symbolic computation for computing real zeros of single very large multivariate polynomials. Our method takes as input conjunctions of strict polynomial inequalities, which represent more than 40% of the QF NRA section of the SMT-LIB library of benchmarks. The method takes an abstraction of polynomials as exponent vectors over the natural numbers tagged with the signs of the corresponding coefficients. It then uses, in turn, SMT to solve linear problems over the reals to heuristically find suitable points that translate back to satisfying points for the original problem. Systematic experiments on the SMT-LIB demonstrate that our method is not a sufficiently strong decision procedure by itself but a valuable heuristic to use within a portfolio of techniques.

Positive Solutions of Systems of Signed Parametric Polynomial Inequalities

“…Our main result also shines a new light on recent heuristic subtropical methods [24,12]: We provide a precise characterization of their incompleteness in terms of the existence of parametric positive solutions for the originally nonparametric input problems considered there.…”

“…We have noted already in [12, p.192] that an inspection of the proof shows that this is not necessary. Therefore we discuss here a slightly improved and simpler version without that special treatment, which has been explicitly stated as [12,Lemma 2]. The proof of the loop invariant (I 1 ) in [24,Theorem 5(ii)] shows that the positive support need not be considered in the conjunction:…”

“…(i) The incomplete subtropical satisfiability checking method for several inequalities over QF_NRA (quantifier-free nonlinear real arithmetic) introduced in [12] succeeds on f > 0.…”

“…For instance, unknowns in the biological and chemical context of [28,11,5,10] are typically positive concentrations of species or reaction rates, where the direction of the reaction is known. In SMT solving, positivity is often not required but, in the satisfiable case, benchmarks typically have also positive solutions; comprehensive statistical data for several thousand benchmarks can be found in [12,Sect. 6].…”

“…Then we present and prove the main result of this paper, which shows how to check the existence of a parametric positive solution and, in the positive case, how to find one. In Section 4, we apply use our framework and our result to re-analyze and improve the above-mentioned subtropical methods [24,12].…”