Decomposing Farkas Interpolants

Abstract

Modern verification commonly models software with Boolean logic and a system of linear inequalities over reals and overapproximates the reachable states of the model with Craig interpolation to obtain, for example, candidates for inductive invariants. Interpolants for the linear system can be efficiently constructed from a Simplex refutation by applying the Farkas' lemma. However, Farkas interpolants do not always suit the verification task and in the worst case they may even be the cause of divergence of the verification algorithm. This work introduces the decomposed interpolants, a fundamental extension of the Farkas interpolants obtained by identifying and separating independent components from the interpolant structure using methods from linear algebra. We integrate our approach to the model checker Sally and show experimentally that a portfolio of decomposed interpolants results in immediate convergence on instances where state-of-the-art approaches diverge. Being based on the efficient Simplex method, the approach is very competitive also outside these diverging cases.

Competition Report: CHC-COMP-20

“…It relies on an SMT solver for answering satisfiability queries, generalization and interpolation. Our specialized interpolation procedure computes stronger interpolants than traditional interpolation algorithm and this has been shown to be useful in model checking scenarios [3]. The parallel version on the other hand leverages a portfolio of interpolation algorithms for discovering useful facts about the system under analysis, as well as a cooperative framework for sharing the discovered information [4].…”

“…The competition entry uses Sally's PD-KIND reasoning engine, with the SMT solvers Yices2 [9] and OpenSMT [16] for generalization and interpolation, respectively. OpenSMT uses an interpolation algorithm that computes decomposed LRA interpolants [3]. The parallel version uses SMTS framework [21] for managing multiple instances and their communication.…”

“…Martin Blicha 3 Università della Svizzera italiana, Switzerland Algorithm. Our competition entry is derived from the model checker Sally [17], which we have enhanced in two ways: First, we have supplied our interpolating SMT solver OpenSMT with a specialized interpolation procedure for Linear Real Arithmetic (LRA) as part of Sally's backend.…”

Global Guidance for Local Generalization in Model Checking

“…The limitations of local reasoning in SMT-based infinite state model checking are well known. Most commonly, they are addressed with either (a) different strategies for local generalization in interpolation (e.g., [1,6,19,23]), or (b) shifting the focus to global invariant inference by learning an invariant of a restricted shape (e.g., [9,[14][15][16]28]).…”

“…1(b)). On the contrary, Blicha et al [6] decompose interpolants to be numerically simpler (but with more literals), which helps with excessive, but not with myopic, generalizations. Deciding locally between these two techniques or on their combination (i.e., some parts of an interpolant might need to be split while others combined) seems impossible.…”