Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, cross-fertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms that are most successful at solving both kinds of problems. They also exhibit differences in the way they are used to state and solve problems, since SAT’s approach is in general a black-box approach, while CP aims at being tunable and programmable. This survey overviews the two areas in a comparative way, emphasising the similarities and differences between the two and the points where we feel that one technology can benefit from ideas or experience acquired

Abstract. Lazy algorithms for Satisfiability Modulo Theories (SMT) combine a generic DPLL-based SAT engine with a theory solver for the given theory T that can decide the T-consistency of conjunctions of ground literals. For many theories of interest, theory solvers need to reason by performing internal case splits. Here we argue that it is more convenient to delegate these case splits to the DPLL engine instead. The delegation can be done on demand for solvers that can encode their internal case splits into one or more clauses, possibly including new constants and literals. This results in drastically simpler theory solvers. We present this idea in an improved version of DPLL(T), a general SMT architecture for the lazy approach, and formalize and prove it correct in an extension of Abstract DPLL Modulo Theories, a framework for modeling and reasoning about lazy algorithms for SMT. A remarkable additional feature of the architecture, also discussed in the paper, is that it naturally includes an efficient Nelson-Oppen-like combination of multiple theories and their solvers. 1

Abstract. Predicate abstraction is a technique for automatically extracting finite-state abstractions for systems with potentially infinite state space. The fundamental operation in predicate abstraction is to compute the best approximation of a Boolean formula ϕ over a set of predicates P. In this work, we demonstrate the use for this operation of a decision procedure based on the DPLL(T) framework for SAT Modulo Theories (SMT). The new algorithm is based on a careful generation of the set of all satisfying assignments over a set of predicates. It consistently outperforms previous methods by a factor of at least 20, on a diverse set of hardware and software verification benchmarks. We report detailed analysis of the results and the impact of a number of variations of the techniques. We also propose and evaluate a scheme for incremental refinement of approximations for predicate abstraction in the above framework. 1

Abstract. Solvers for SAT Modulo Theories (SMT) can nowadays handle large industrial (e.g., formal hardware and software verification) problems over theories such as the integers, arrays, or equality. Here we show that SMT approaches can also efficiently solve problems that, at first sight, do not have a typical SMT flavor. In particular, here we deal with SAT and SMT problems where models M are sought such that a given cost function f(M) is minimized. For this purpose, we introduce a variant of SMT where the theory T DPLL Modulo Theories framework. We discuss two different examples of applications of this SMT variant: weighted Max-SAT and weighted Max-SMT. We show how, with relatively little effort, one can obtain a competitive system that, in the case of weighted Max-SMT in the theory of Difference Logic, can even handle well-known hard radio frequency assignment problems without any tailored heuristics. These results seem to indicate that Max-SAT/SMT techniques can already be used for realistic applications. 1

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Satisfiability Modulo Theories (SMT(T)) is the problem of deciding the satisfiability of a formula with respect to a given background theory T. When T is the combination of two simpler theories T1 and T2 (SMT(T1 ∪ T2)), a standard and general approach is to handle the integration of T1 and T2 by performing some form of search on the equalities between the shared variables. A frequent and very relevant sub-case of SMT(T1 ∪ T2) is when T1 is the theory of Equality and Uninterpreted Functions (EUF). For this case, an alternative approach is to eliminate first all uninterpreted function symbols by means of Ackermann’s expansion, and then to solve the resulting SMT (T2) problem. In this paper we build on the empirical observation that there is no absolute winner between these two alternative approaches, and that the performance gaps between them are often dramatic, in either direction. We propose a simple technique for estimating a priori the costs and benefits, in terms of the size of the search space of an SMT tool, of applying Ackermann’s expansion to all or part of the function symbols. We have implemented a preprocessor which analyzes the input formula, decides autonomously which functions to expand, performs such expansions and gives the resulting formula as input to an SMT tool. A thorough experimental analysis, including the benchmarks of the SMT’05 competition, shows that our preprocessor performs the best choice(s) nearly always, and that the proposed technique is extremely effective in improving the overall performance of the SMT tool.

We present a new efficient algorithm for obtaining utilitarian optimal solutions to Disjunctive Temporal Problems with Preferences (DTPPs). The previous state-of-the-art system achieves temporal preference optimization using a SAT formulation, with its creators attributing its performance to advances in SAT solving techniques. We depart from the SAT encoding and instead introduce the Valued DTP (VDTP). In contrast to the traditional semiring-based formalism that annotates legal tuples of a constraint with preferences, our framework instead assigns elementary costs to the constraints themselves. After proving that the VDTP can express the same set of utilitarian optimal solutions as the DTPP with piecewise-constant preference functions, we develop a method for achieving weighted constraint satisfaction within a meta-CSP search space that has traditionally been used to solve DTPs without preferences. This allows us to directly incorporate several powerful techniques developed in previous decision-based DTP literature. Finally, we present empirical results demonstrating that an implementation of our approach consistently outperforms the SAT-based solver by orders of magnitude.

competitions In 2005 there were numerous “competitive events ” in the area of automated reasoning (in the broad sense): The Tenth CASC competition[23], the First Satisfiability Modulo Theory Competition[3], the First CSP competition, the Fourth SAT competition, the Third QBF evaluation and the First Pseudo Boolean evaluation, etc. Starting such an event is suitable for promoting a common input format and to build a repository of benchmarks in that format: this was the motivation behind the SMT and CSP competitions and the PB evaluation. The First QBF evaluation was also organized two years ago in that spirit and the renewal since then allowed to impose the QDimacs input format and to increase both the number of solvers and benchmarks available each year. More mature events, such as the CASC competitions, are a bit different: both the pool of benchmarks (TPTP) and the pool of solvers are quite stable and the competitions allow to track the progress of new versions of those solvers on a well studied set of benchmarks. The SAT competitions are again different: the first SAT competition is older than

Abstract. Theorem provers, model checkers, static analyzers, test generators... all of these and many other kinds of formal methods tools can contribute to the analysis and development of computer systems and software. It is already quite common to use several kinds of tools in a loose combination: for example, we might use static analysis and then model checking to help find and eliminate design flaws prior to undertaking formal verification with a theorem prover. And some modern tools, such as test generators, are built using model checkers, predicate abstractors, decision procedures and constraint solvers as components in tight combination. But we can foresee a different kind of combination where many tools and methods are used in ad hoc combination within a single analysis. For example, static analysis might yield invariants that enable decision procedures to build a predicate abstraction whose reachable states are calculated as a BDD and then concretized to yield a strong invariant for the original system; the invariant then enables properties of the original system to be verified by highly automated theorem proving. This sort of combination clearly requires an integrating platform—a tool bus—to connect the various tools together; but the capabilities required go beyond those of platforms such as Eclipse. The entities exchanged among clients of the bus—proofs, counterexamples, specifications, theorems, counterexamples, abstractions—have logical content, and the overall purpose of the bus is to gather and integrate evidence for verification or refutation. In this paper I propose requirements for such an “evidential tool bus,” and sketch a possible architecture. 1

Abstract. In this paper we present a new progressive cooperating simplifier for deciding the satisfiability of a quantifier-free formula in the first-order theory of integers involving combinations of sublogics, referred to as Satisfiability Modulo Theories (SMT). Our approach, given an SMT problem, replaces each non-propositional theory atom with a Boolean indicator variable yielding a purely propositional formula to be decided by a SAT solver. Starting with the most abstract representation (the Boolean formula), the solver gradually integrates more complex theory solvers into the working decision procedure. Additionally, we propose a method to simplify “expensive ” atoms into suitable conjunctions of “cheaper ” theory atoms when conflicts occur. This process considerably increases the efficiency of the overall procedure by reducing the number of calls to the slower theory solvers. This is made possible by adopting our novel inter-logic implication framework, as proposed in this paper. We have implemented these methods in our Ario SMT solver by combining three different theory solvers within a DPLL-style SAT solver: a Unit-Two-Variable-Per-Inequality (UTVPI) solver, an integer linear programming (ILP) solver, and a solver for systems of equalities with uninterpreted functions. The efficiencies of our proposed algorithms are demonstrated and exhaustively investigated on a wide range of benchmarks in hardware and software verification domain. Empirical results are also presented showing the advantages/limitations of our methods over other modern techniques for solving these SMT problems. 1