SMT-based Verification of LTL Specifications with Integer Constraints and its Application to Runtime Checking of Service Substitutability

An important problem that arises during the execution of service-based applications concerns the ability to determine whether a running service can be substituted with one with a different interface, for example if the former is no longer available. Standard Bounded Model Checking techniques can be used to perform this check, but they must be able to provide answers very quickly, lest the check hampers the operativeness of the application, instead of aiding it. The problem becomes even more complex when conversational services are considered, i.e., services that expose operations that have Input/Output data dependencies among them. In this paper we introduce a formal verification technique for an extension of Linear Temporal Logic that allows users to include in formulae constraints on integer variables. This technique applied to the substitutability problem for conversational services is shown to be considerably faster and with smaller memory footprint than existing ones

Pushing the envelope of Optimization Modulo Theories with Linear-Arithmetic Cost Functions

In the last decade we have witnessed an impressive progress in the expressiveness and efficiency of Satisfiability Modulo Theories (SMT) solving techniques. This has brought previously-intractable problems at the reach of state-of-the-art SMT solvers, in particular in the domain of SW and HW verification. Many SMT-encodable problems of interest, however, require also the capability of finding models that are optimal wrt. some cost functions. In previous work, namely "Optimization Modulo Theory with Linear Rational Cost Functions -- OMT(LAR U T )", we have leveraged SMT solving to handle the minimization of cost functions on linear arithmetic over the rationals, by means of a combination of SMT and LP minimization techniques. In this paper we push the envelope of our OMT approach along three directions: first, we extend it to work also with linear arithmetic on the mixed integer/rational domain, by means of a combination of SMT, LP and ILP minimization techniques; second, we develop a multi-objective version of OMT, so that to handle many cost functions simultaneously; third, we develop an incremental version of OMT, so that to exploit the incrementality of some OMT-encodable problems. An empirical evaluation performed on OMT-encoded verification problems demonstrates the usefulness and efficiency of these extensions.Comment: A slightly-shorter version of this paper is published at TACAS 2015 conferenc

Delta-Complete Decision Procedures for Satisfiability over the Reals

We introduce the notion of "\delta-complete decision procedures" for solving SMT problems over the real numbers, with the aim of handling a wide range of nonlinear functions including transcendental functions and solutions of Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational number \delta, a \delta-complete decision procedure determines either that \varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that allows \delta-bounded numerical perturbations on \varphi. We prove the existence of \delta-complete decision procedures for bounded SMT over reals with functions mentioned above. For functions in Type 2 complexity class C, under mild assumptions, the bounded \delta-SMT problem is in NP^C. \delta-Complete decision procedures can exploit scalable numerical methods for handling nonlinearity, and we propose to use this notion as an ideal requirement for numerically-driven decision procedures. As a concrete example, we formally analyze the DPLL framework, which integrates Interval Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient conditions for its \delta-completeness. We discuss practical applications of \delta-complete decision procedures for correctness-critical applications including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201

Instantiation of SMT problems modulo Integers

Many decision procedures for SMT problems rely more or less implicitly on an instantiation of the axioms of the theories under consideration, and differ by making use of the additional properties of each theory, in order to increase efficiency. We present a new technique for devising complete instantiation schemes on SMT problems over a combination of linear arithmetic with another theory T. The method consists in first instantiating the arithmetic part of the formula, and then getting rid of the remaining variables in the problem by using an instantiation strategy which is complete for T. We provide examples evidencing that not only is this technique generic (in the sense that it applies to a wide range of theories) but it is also efficient, even compared to state-of-the-art instantiation schemes for specific theories.Comment: Research report, long version of our AISC 2010 pape

Decision procedures for linear arithmetic

In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit präsentieren wir neue Entscheidungsprozeduren für lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollständiger Kalkül für ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der Größte-Würfeltest und der Einheitswürfeltest, zwei korrekte (wenn auch unvollständige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollständig unbegrenzten Systemen, welche für viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren präsentieren wir mehrere Anwendungsmöglichkeiten für diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfüllbarkeitsäquivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollständige und effiziente Entscheidungsprozedur für unbeschränkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver für lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf präsentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestätigen

A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

We present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over-)approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system