Incremental Linearization for Satisfiability and Verification Modulo Nonlinear Arithmetic and Transcendental Functions

Abstract

Satisfiability Modulo Theories (SMT) is the problem of deciding the satisfiability of a first-order formula with respect to some theory or combination of theories; Verification Modulo Theories (VMT) is the problem of analyzing the reachability for transition systems represented in terms of SMT formulae. In this article, we tackle the problems of SMT and VMT over the theories of nonlinear arithmetic over the reals (NRA) and of NRA augmented with transcendental (exponential and trigonometric) functions (NTA). We propose a new abstraction-refinement approach for SMT and VMT on NRA or NTA, called Incremental Linearization. The idea is to abstract nonlinear multiplication and transcendental functions as uninterpreted functions in an abstract space limited to linear arithmetic on the rationals with uninterpreted functions. The uninterpreted functions are incrementally axiomatized by means of upper-and lower-bounding piecewiselinear constraints. In the case of transcendental functions, particular care is required to ensure the soundness of the abstraction. The method has been implemented in the MathSAT SMT solver and in the nuXmv model checker. An extensive experimental evaluation on a wide set of benchmarks from verification and mathematics demonstrates the generality and the effectiveness of our approach. CCS Concepts: • Theory of computation → Automated reasoning; Constraint and logic programming; Verification by model checking; Logic and verification; • Mathematics of computing → Solvers;

Incremental linearization: A practical approach to satisfiability modulo nonlinear arithmetic and transcendental functions

“…The technique is surprisingly effective, even compared to other complete (when available) and more mature approaches. Moreover, in [4], [7], we have successfully applied it to the verification of invariant properties of infinite transition systems with nonlinear constraints Figure 9. Survival plots for SMT(NTA) -bounded benchmarks (the approach has been implemented within the NUXMV model checker [39]).…”

“…If the value of TF(x) in µ is not included in the interval [QL, QU], we generate (piecewise) linear constraints that remove the point ( µ[x], µ[TF(x)]) (and possibly many others) from the graph of f TF , thus refining the abstraction. First, we attempts to exclude the bad point by invoking basic lemmas, which instantiates some basic constraint schemata describing very general properties of the transcendental function TF under consideration (see [7] for details). These constraints encode some simple properties of transcendental functions (such as sign and monotonicity conditions, or bounds at noteworthy values) via linear relations.…”

“…Remark: It is important to notice that we describe the refinement strategy which is currently implemented, which is only one of the many alternative strategies by which refinement can be performed. Moreover, we skipped the method to conclude the existence of an NTA-model, the details can be found in [7].…”

“…Recently, we have proposed a conceptually-simple yet effective practical approach for dealing with the quantifierfree theory of nonlinear arithmetic over the reals, over the reals extended with transcendental functions, and over integers, called Incremental Linearization [4], [5], [7], [8]. Its underlying idea is that of trading the use of expensive, exact solvers for nonlinear arithmetic for an abstraction-refinement loop on top of much less expensive solvers for linear arithmetic and uninterpreted functions.…”

Counterexample-Guided Prophecy for Model Checking Modulo the Theory of Arrays

“…However, their technique utilizes interpolants for finding violated axioms and cannot infer universally quantified invariants. The work of [18] also uses lazy axiombased refinement, abstracting non-linear arithmetic with uninterpreted functions. We differ in the domain and the use of auxiliary variables.…”