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Cubicle is a new model checker for verifying safety properties of parameterized systems. It implements a parallel symbolic backward reachability procedure using Satisfiabilty Modulo Theories. Experiments done on classic and challenging mutual exclusion algorithms and cache coherence protocols show that Cubicle is effective and competitive with state-of-the-art model checkers. 2 System Description Language Cubicle's input language is a typed version of Murϕ [8] similar to the one of Uclid [6], rudimentary at the moment, but more user-friendly than mcmt and sufficiently expressive for typical parameterized systems. A system is described in Cubicle by: (1) a set of type, variable, and array declarations; (2) a formula for the initial states; and (3) a set of transitions. It is parametrized by a set of process identifiers, denoted by the built-in type proc. Standard types int, real, and bool are also built in. Additionally, the user

Abstract. SMT solvers have been used successfully as reasoning engines for automated verification. Current techniques for dealing with quantified formulas in SMT are generally incomplete, forcing SMT solvers to report "unknown" when they fail to prove the unsatisfiability of a formula with quantifiers. This inability to return counter-models limits their usefulness in applications that produce quantified verification conditions. We present a novel finite model finding method that reduces these limitations in the case of quantifiers ranging over free sorts. Our method contrasts with previous approaches for finite model finding in first-order logic by not relying on the introduction of domain constants for the free sorts and by being fully integrated into the general architecture used by most SMT solvers. This integration is achieved through the addition of a novel solver for sort cardinality constraints and a module for quantifier instantiation over finite domains. Initial experiments with verification conditions generated from a deductive verification tool developed at Intel Corp. show that our approach compares quite favorably with the state of the art in SMT.

Abstract. SMT-based applications increasingly rely on SMT solvers being able to deal with quantified formulas. Current work shows that for formulas with quantifiers over uninterpreted sorts counter-models can be obtained by integrating a finite model finding capability into the architecture of a modern SMT solver. We examine various strategies for on-demand quantifier instantiation in this setting. Here, completeness can be achieved by considering all ground instances over the finite domain of each quantifier. However, exhaustive instantiation quickly becomes unfeasible with larger domain sizes. We propose instantiation strategies to identify and consider only a selection of ground instances that suffices to determine the satisfiability of the input formula. We also examine heuristic quantifier instantiation techniques such as E-matching for the purpose of accelerating the search. We give experimental evidence that our approach is practical for use in industrial applications and is competitive with other approaches.

Abstract. We give a method for modular generation of ground interpolants in modern SMT solvers supporting multiple theories. Our method uses a novel algorithm to modify the proof tree obtained from an unsatifiability run of the solver into a proof tree without occurrences of troublesome "uncolorable" literals. An interpolant can then be readily generated using existing procedures. The principal advantage of our method is that it places few restrictions (none for convex theories) on the search strategy of the solver. Consequently, it is straightforward to implement and enables more efficient interpolating SMT solvers. In the presence of non-convex theories our method is incomplete, but still more general than previous methods.

Abstract. Given a theory T and two formulas A and B jointly unsatisfiable in T , a theory interpolant of A and B is a formula I such that (i) its non-theory symbols are shared by A and B, (ii) it is entailed by A in T , and (iii) it is unsatisfiable with B in T . Theory interpolants are used in model checking to accelerate the computation of reachability relations. We present a novel method for computing ground interpolants for ground formulas in the theory of equality. Our algorithm computes interpolants from colored congruence graphs representing derivations in the theory of equality. These graphs can be produced by conventional congruence closure algorithms in a straightforward manner. By working with graphs, rather than at the level of individual proof steps, we are able to derive interpolants that are pleasingly simple (conjunctions of Horn clauses) and smaller than those generated by other tools.

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