450 research outputs found

    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

    An Approach to Abductive Reasoning in Equational Logic

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    http://ijcai.org/papers13/contents.php - Posters: Constraints, Satisfiability, and Search (ijcai13.org)International audienceAbduction has been extensively studied in propositional logic because of its many applications in artificial intelligence. However, its intrinsic complexity has been a limitation to the implementation of abductive reasoning tools in more expressive logics. We have devised such a tool in ground flat equational logic, in which literals are equations or disequations between constants. Our tool is based on the computation of prime implicates. It uses a relaxed paramodulation calculus, designed to generate all prime implicates of a formula, together with a carefully defined data structure storing the implicates and able to efficiently detect, and remove, redundancies. In addition to a detailed description of this method, we present an analysis of some experimental results

    Redundancy in Logic II: 2CNF and Horn Propositional Formulae

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    We report complexity results about redundancy of formulae in 2CNF form. We first consider the problem of checking redundancy and show some algorithms that are slightly better than the trivial one. We then analyze problems related to finding irredundant equivalent subsets (I.E.S.) of a given set. The concept of cyclicity proved to be relevant to the complexity of these problems. Some results about Horn formulae are also shown.Comment: Corrected figures on Theorem 10; added and modified some reference

    An analysis and implementation of linear derivation strategies

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    This study examines the efficacy of six linear derivation strategies: (i) s-linear resolution, (ii) the ME procedure; (iii) t-linear resolution, (iv) SL -resolution, (v) the GC procedure, and (vi) SLM. The analysis is focused on the different restrictions and operations employed in each derivation strategy. The selection function, restrictive ancestor resolution, compulsory ancestor resolution on literals having atoms which are or become identical, compulsory merging operations, reuse of truncated literals, spreading of FALSE literals, no-tautologies resection, no two non-B-literals having identical atoms restriction, and the use of semantic information to trim irrelevant derivations from the search tree are the major features found In these six derivation strategies. Detecting loops and minimizing irrelevant derivations are the identified weak points of SLM. Two variations of SLM are suggested to rectify these problems. The ME procedure, SL-resolution, the GC procedure, SLM and one of the suggested variations of SLM were implemented using the Arity/Prolog compiler to produce the ME -TP, SL-TP, GC-TP, SLM-TP and SLM5-TP theorem provers respectively. In addition to the original features of each derivation strategy, the following search strategies were included in the implementations : the modified consecutively bounded depth-first search unit preference strategy, set of support strategy, pure literal elimination, tautologous clause elimination, selection function based on the computed weight of a literal, and a match check. The extension operation used by each theorem prover was extended to include subsumed unit extension and paramodulation. The performance of each theorem prover was determined. Experimental results were obtained using twenty four selected problems. The performance was measured in terms of the memory use and the execution time. A comparison of results between the five theorem provers using the, ME-TP as the basis was done. The results show that none of the theorem provers, consistently perform better than the others. Two of the selected problems were not proved by SL-TP and one problem was not proved by SLM-TP due to memory problems. The ME-TP, GC-TP and SLM5-TP proved all the selected problems. In some problems, the ME-TP and GC-TP performed better than SLM5-TP. However, the ME-TP and GC-TP had difficulties in some problems in which SLM5-TP performed well

    Towards Next Generation Sequential and Parallel SAT Solvers

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    This thesis focuses on improving the SAT solving technology. The improvements focus on two major subjects: sequential SAT solving and parallel SAT solving. To better understand sequential SAT algorithms, the abstract reduction system Generic CDCL is introduced. With Generic CDCL, the soundness of solving techniques can be modeled. Next, the conflict driven clause learning algorithm is extended with the three techniques local look-ahead, local probing and all UIP learning that allow more global reasoning during search. These techniques improve the performance of the sequential SAT solver Riss. Then, the formula simplification techniques bounded variable addition, covered literal elimination and an advanced cardinality constraint extraction are introduced. By using these techniques, the reasoning of the overall SAT solving tool chain becomes stronger than plain resolution. When using these three techniques in the formula simplification tool Coprocessor before using Riss to solve a formula, the performance can be improved further. Due to the increasing number of cores in CPUs, the scalable parallel SAT solving approach iterative partitioning has been implemented in Pcasso for the multi-core architecture. Related work on parallel SAT solving has been studied to extract main ideas that can improve Pcasso. Besides parallel formula simplification with bounded variable elimination, the major extension is the extended clause sharing level based clause tagging, which builds the basis for conflict driven node killing. The latter allows to better identify unsatisfiable search space partitions. Another improvement is to combine scattering and look-ahead as a superior search space partitioning function. In combination with Coprocessor, the introduced extensions increase the performance of the parallel solver Pcasso. The implemented system turns out to be scalable for the multi-core architecture. Hence iterative partitioning is interesting for future parallel SAT solvers. The implemented solvers participated in international SAT competitions. In 2013 and 2014 Pcasso showed a good performance. Riss in combination with Copro- cessor won several first, second and third prices, including two Kurt-Gödel-Medals. Hence, the introduced algorithms improved modern SAT solving technology

    A Generic Framework for Implicate Generation Modulo Theories

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    International audienceThe clausal logical consequences of a formula are called its implicates. The generation of these implicates has several applications, such as the identification of missing hypotheses in a logical specification. We present a procedure that generates the implicates of a quantifier-free formula modulo a theory. No assumption is made on the considered theory, other than the existence of a decision procedure. The algorithm has been implemented (using the solvers MiniSAT, CVC4 and Z3) and experimental results show evidence of the practical relevance of the proposed approach
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