The task of extracting an unsatisfiable core for a given Boolean formula has been finding more and more applications in recent years. The only existing approach that scales well for large real-world formulas exploits the ability of modern SAT solvers to produce resolution refutations. However, the resulting unsatisfiable cores are suboptimal. We propose a new algorithm for minimal unsatisfiable core extraction, based on a deeper exploration of resolution-refutation properties. We provide experimental results on formal verification benchmarks confirming that our algorithm finds smaller cores than suboptimal algorithms; and that it runs faster than those algorithms that guarantee minimality of the core.

Rarely verification problems originate from bit-level descriptions. Yet, most of the verification technologies are based on bit blasting, i.e., reduction to boolean reasoning. In this paper we advocate reasoning at higher level of abstraction, within the theory of bit vectors (BV), where structural information (e.g. equalities, arithmetic functions) is not blasted into bits. Our approach relies on the lazy Satisfiability Modulo Theories (SMT) paradigm. We developed a satisfiability procedure for reasoning about bit vectors that carefully leverages on the power of boolean SAT solver to deal with components that are more naturally "boolean", and activates bit-vector reasoning whenever possible. The procedure has two distinguishing features. First, it relies on the on-line integration of a SAT solver with an incremental and backtrackable solver for BV that enables dynamical optimization of the reasoning about bit vectors; for instance, this is an improvement over static encoding methods which may generate smaller slices of bit-vector variables. Second, the solver for BV is layered (i.e., it privileges cheaper forms of reasoning), and it is based on a flexible use of term rewriting techniques. We evaluate our approach on a set of realistic industrial benchmarks, and demonstrate substantial improvements with respect to state-of-the-art boolean satisfiability solvers, as well as other decision procedures for SMT(BV).

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Abstract. Interpolation-based model checking (ITP) [14] is an efficient and complete model checking procedure. However, for large problems, interpolants generated by ITP might become extremely large, rendering the procedure slow or even intractable. In this work we present a novel technique for interpolant generation in the context of model checking. The main novelty of our work is that we generate small interpolants in Conjunctive Normal Form (CNF) using a twofold procedure: First we propose an algorithm that exploits resolution refutation properties to compute an interpolant approximation. Then we introduce an algorithm that takes advantage of inductive reasoning to turn the interpolant approximation into an interpolant. Unlike ITP, our approach maintains only the relevant subset of the resolution refutation. In addition, the second part of the procedure exploits the properties of the model checking problem at hand, in contrast to the general-purpose algorithm used in ITP. We developed a new interpolation-based model checking algorithm, called CNF-ITP. Our algorithm takes advantage of the smaller interpolants and exploits the fact that the interpolants are given in CNF. We integrated our method into a SAT-based model checker and experimented with a representative subset of the HWMCC'12 benchmark set. Our experiments show that, overall, the interpolants generated by our method are 42 times smaller than those generated by ITP. Our CNF-ITP algorithm outperforms ITP, and at times solves problems that ITP cannot solve. We also compared CNF-ITP to the successful IC3 [3] algorithm. We found that CNF-ITP outperforms IC3 [3] in a large number of cases.

Abstract. We propose a new decision heuristic for DPLL-based propositional SAT solvers. Its essence is that both the initial and the conflict clauses are arranged in a list and the next decision variable is chosen from the top -most unsatisfied clause. Various methods of initially organizing the list and moving the clauses within it are studied. Our approach is an extension of one used in Berkmin, and adopted by other modern solvers, according to which only conflict clauses are organized in a list, and a literal-scoring-based secondary heuristic is used when there are no more unsatisfied conflict clauses. Our approach, implemented in the 2004 version of zChaff solver and in a generic Chaff-based SAT solver, results in a significant performance boost on hard industrial benchmarks.

Abstract. We show that modern conflict-driven SAT solvers implicitly build and prune a decision tree whose nodes are associated with flipped variables. Practical usefulness of conflict-driven learning schemes, like 1UIP or All UIP, depends on their ability to guide the solver towards refutations associated with compact decision trees. We propose an enhancement of 1UIP that is empirically helpful for real-world industrial benchmarks.