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Abstract. We consider the problem of bounded model checking (BMC) for linear temporal logic (LTL). We present several efficient encodings that have size linear in the bound. Furthermore, we show how the encodings can be extended to LTL with past operators (PLTL). The generalised encoding is still of linear size, but cannot detect minimal length counterexamples. By using the virtual unrolling technique minimal length counterexamples can be captured, however, the size of the encoding is quadratic in the specification. We also extend virtual unrolling to Büchi automata, enabling them to accept minimal length counterexamples.Our BMC encodings can be made incremental in order to benefit from incremental SAT technology. With fairly small modifications the incremental encoding can be further enhanced with a termination check, allowing us to prove properties with BMC.An analysis of the liveness-to-safety transformation reveals many similarities to the BMC encodings in this paper. We conduct experiments to determine the advantage of employing dedicated BMC encodings for PLTL over combining more general but potentially less efficient approaches with BMC: the liveness-to-safety transformation with invariant checking and Büchi automata with fair cycle detection.Experiments clearly show that our new encodings improve performance of BMC considerably, particularly in the case of the incremental encoding, and that they are very competitive for finding bugs. Dedicated encodings seem to have an advantage over using more general methods with BMC. Using the liveness-to-safety translation with BDD-based invariant checking results in an efficient method to find shortest counterexamples that complements the BMC-based approach. For proving complex properties BDD-based methods still tend to perform better.

Many approaches to deciding the satisfiability of quantifier-free formulae with respect to a background theory T-also known as Satisfiability Modulo Theory, or SMT(T)-rely on the integration between an enumerator of truth assignments and a decision procedure for conjunction of literals in T. When the background theory T is the combination T 1 ∪ T 2 of two simpler theories, the approach is typically instantiated by means of a theory combination schema (e.g. Nelson-Oppen, Shostak). In this paper we propose a new approach to SMT(T 1 ∪ T 2), where the enumerator of truth assignments is integrated with two decision procedures, one for T 1 and one for T 2 , acting independently from each other. The key idea is to search for a truth assignment not only to the atoms occurring in the formula, but also to all the equalities between variables which are shared between the theories. This approach is simple and expressive: for instance, no modification is required to handle non-convex theories (as opposed to traditional Nelson-Oppen combinations which require a mechanism for splitting). Furthermore, it can be made practical by leveraging on state-of-the-art boolean and SMT

Abstract. In this paper we present a new decision procedure for the satisfiability of Linear Arithmetic Logic (LAL), i.e. boolean combinations of propositional variables and linear constraints over numerical variables. Our approach is based on the well known integration of a propositional SAT procedure with theory deciders, enhanced in the following ways.First, our procedure relies on an incremental solver for linear arithmetic, that is able to exploit the fact that it is repeatedly called to analyze sequences of increasingly large sets of constraints. Reasoning in the theory of LA interacts with the boolean top level by means of a stack-based interface, that enables the top level to add constraints, set points of backtracking, and backjump, without restarting the procedure from scratch at every call. Sets of inconsistent constraints are found and used to drive backjumping and learning at the boolean level, and theory atoms that are consequences of the current partial assignment are inferred.Second, the solver is layered: a satisfying assignment is constructed by reasoning at different levels of abstractions (logic of equality, real values, and integer solutions). Cheaper, more abstract solvers are called first, and unsatisfiability at higher levels is used to prune the search. In addition, theory reasoning is partitioned in different clusters, and tightly integrated with boolean reasoning.We demonstrate the effectiveness of our approach by means of a thorough experimental evaluation: our approach is competitive with and often superior to several state-of-the-art decision procedures. Motivations and GoalsMany practical domains require a degree of expressiveness beyond propositional logic. For instance, timed and hybrid systems have a discrete component as well as a dynamic evolution of real variables; proof obligations arising in software verification are often boolean combinations of constraints over integer variables; circuits described at Register Transfer Level, even though expressible via booleanization, might be easier to analyze at a higher level of abstraction (see e.g. [15]). Many of the verification problems arising in such domains can be naturally modeled as satisfiability in Linear Arithmetic Logic (LAL), i.e., the boolean combination of propositional variables and linear constraints over numerical variables. For its practical relevance, LAL has been devoted a lot of interest, and several decision procedures exist that are able to deal with it (e.g., SVC [17], ICS [24,19], CVCLITE [17,10], UCLID [36,33], HDPLL [30]).In this paper, we propose a new decision procedure for the satisfiability of LAL, both for the real-valued and integer-valued case. We start from a well known approach, previously applied in MATHSAT [26,4] and in several other systems [24,19,17,10, 35,3,21]: a propositional SAT procedure, modified to enumerate propositional assignments for the propositional abstraction of the problem, is integrated with dedicated theory deciders, used to check consistency of propositional assignments wit...

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