Superposition Modulo Non-linear Arithmetic

ACM Trans. Comput. Logic

et al. 2018

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;

Section: Deductive Methods the Metitarskimentioning

confidence: 99%

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

ACM Trans. Comput. Logic

et al. 2018

“…used by Z3 and SMT-RAT). Moreover, in the context of theorem proving, two remarkable deductive methods are described in [33] (the METITARSKI theorem prover) and in [34].…”

Section: Related Workmentioning

confidence: 99%

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

2018 20th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC)

et al. 2018

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. In this extended abstract, we overview our recent approach called Incremental Linearization, which successfully tackles the problems of SMT over the theories of nonlinear arithmetic over the reals (NRA), nonlinear arithmetic over the integers (NIA) and their combination, and of NRA augmented with transcendental (exponential and trigonometric) functions (NTA). Moreover, we showcase some of the experimental results and outline interesting future directions.

“…The approach presented in [10], where the NTA theory is referred to as NLA, is similar in spirit to METITARSKI in that it combines the SPASS theorem prover [27] with the iSAT3 SMT solver. The approach relies on the SUP(NLA) calculus that combines superposition-based first-order logic reasoning with SMT(NTA).…”

Section: Related Workmentioning

confidence: 99%

Satisfiability Modulo Transcendental Functions via Incremental Linearization

Automated Deduction – CADE 26

et al. 2017

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.