1,518 research outputs found

    A CNF Analogue to Strengthening

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    A satisfiability procedure for quantified Boolean formulae

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    We present a satisfiability tester QSAT for quantified Boolean formulae and a restriction of QSAT to unquantified conjunctive normal form formulae. QSAT makes use of procedures which replace subformulae of a formula by equivalent formulae. By a sequence of such replacements, the original formula can be simplified to or . It may also be necessary to transform the original formula to generate a subformula to replace. eliminates collections of variables from an unquantified clause form formula until all variables have been eliminated. QSAT and can be applied to hardware verification and symbolic model checking. Results of an implementation of are described, as well as some complexity results for QSAT and . QSAT runs in linear time on a class of quantified Boolean formulae related to symbolic model checking. We present the class of “long and thin” unquantified formulae and give evidence that this class is common in applications. We also give theoretical and empirical evidence that is often faster than Davis and Putnam-type satisfiability checkers and ordered binary decision diagrams (OBDDs) on this class of formulae. We give an example where is exponentially faster than BDDs

    Towards Verifying Nonlinear Integer Arithmetic

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    We eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give n^{O(1)} size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and quasipolynomial- n^{O(\log n)} size proofs for these identities on Wallace tree multipliers.Comment: Expanded and simplified with improved result

    Generating Extended Resolution Proofs with a BDD-Based SAT Solver

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    In 2006, Biere, Jussila, and Sinz made the key observation that the underlying logic behind algorithms for constructing Reduced, Ordered Binary Decision Diagrams (BDDs) can be encoded as steps in a proof in the extended resolution logical framework. Through this, a BDD-based Boolean satisfiability (SAT) solver can generate a checkable proof of unsatisfiability for a set of clauses. Such a proof indicates that the formula is truly unsatisfiable without requiring the user to trust the BDD package or the SAT solver built on top of it. We extend their work to enable arbitrary existential quantification of the formula variables, a critical capability for BDD-based SAT solvers. We demonstrate the utility of this approach by applying a prototype solver to several problems that are very challenging for search-based SAT solvers, obtaining polynomially sized proofs on benchmarks for parity formulas, as well as the Urquhart, mutilated chessboard, and pigeonhole problems.Comment: Extended version of paper published at TACAS 202

    Resolution and binary decision diagrams cannot simulate each other polynomially

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    There are many different ways of proving formulas in proposition logic. Many of these can easily be characterized as forms of resolution. Others use so-called binary decision diagrams (BDDs). Experimental evidence suggests that BDDs and resolution based techniques are fundamentally different, in the sense that their performance can differ very much on benchmarks. In this paper we confirm these findings by mathematical proof. We provide examples that are easy for BDDS and exponentially hard for any form of resolution, and vice versa, examples that ar easy for resolution and exponentially hard for BDDs

    A theory of resolution

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    We review the fundamental resolution-based methods for first-order theorem proving and present them in a uniform framework. We show that these calculi can be viewed as specializations of non-clausal resolution with simplification. Simplification techniques are justified with the help of a rather general notion of redundancy for inferences. As simplification and other techniques for the elimination of redundancy are indispensable for an acceptable behaviour of any practical theorem prover this work is the first uniform treatment of resolution-like techniques in which the avoidance of redundant computations attains the attention it deserves. In many cases our presentation of a resolution method will indicate new ways of how to improve the method over what was known previously. We also give answers to several open problems in the area
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