Extending resolution for model construction

Caferra, Ricardo; Zabel, Nicolas

doi:10.1007/bfb0018439

Cited by 27 publications

(14 citation statements)

References 10 publications

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“…this theoretical result for devising an algorithm which is, in general, considerably more efficient than previously known ones, e.g. the partial saturation method in Fermüller and Leitsch (1996) and the transformation into equational problems in Caferra and Zabel (1991). The main results of this work are algorithms for solving the model equivalence problem and the clause evaluation problem for AR's, where the time complexity is nonpolynomial only in the number of atoms.…”

Section: Introductionmentioning

confidence: 91%

“…The main results of this work are algorithms for solving the model equivalence problem and the clause evaluation problem for AR's, where the time complexity is nonpolynomial only in the number of atoms. In contrast to the methods of Fermüller and Leitsch (1996) and Caferra and Zabel (1991), the complexity of the expressions involved (e.g. the arity of the predicate symbols and, in particular, the term depth of the arguments) only has polynomial influence on the complexity of our algorithms.…”

Section: Introductionmentioning

confidence: 96%

“…Nevertheless, what we can do is find out the real "source" of complexity and make use of this theoretical result for devising an algorithm which, in general, has a considerably smaller upper bound on the complexity than previously known algorithms, e.g. the partial saturation method in Fermüller and Leitsch (1996) and the transformation into equational problems in Caferra and Zabel (1991).The main results of this paper are algorithms for these two decision problems, where the complexity depends non-polynomially on the number of atoms (rather than on the total size) of the input model equivalence problem or clause evaluation problem, respectively. Hence, in contrast to the above-mentioned algorithms, the complexity of the expressions involved (e.g.…”

mentioning

confidence: 98%

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Speeding up Algorithms on Atomic Representations of Herbrand Models via New Redundancy Criteria

Pichler

2000

Journal of Symbolic Computation

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The importance of models within automated deduction is generally acknowledged both in constructing countermodels (rather than just giving the answer "NO", if a given formula is found to be not a theorem) and in speeding up the deduction process itself (e.g. by semantic resolution refinement).However, so far little attention has been paid to the efficiency of algorithms to actually work with models. There are two fundamental decision problems as far as models are concerned, namely: the equivalence of two models and the truth evaluation of an arbitrary clause within a given model. This paper focuses on the efficiency of algorithms for these problems in the case of Herbrand models given through atomic representations.Both problems have been shown to be coNP-hard by Gottlob and Pichler (1999), so there is a certain limit to the efficiency that we can possibly expect. Nevertheless, what we can do is find out the real "source" of complexity and make use of this theoretical result for devising an algorithm which, in general, has a considerably smaller upper bound on the complexity than previously known algorithms, e.g. the partial saturation method in Fermüller and Leitsch (1996) and the transformation into equational problems in Caferra and Zabel (1991).The main results of this paper are algorithms for these two decision problems, where the complexity depends non-polynomially on the number of atoms (rather than on the total size) of the input model equivalence problem or clause evaluation problem, respectively. Hence, in contrast to the above-mentioned algorithms, the complexity of the expressions involved (e.g. the arity of the predicate symbols and, in particular, the term depth of the arguments) only has polynomial influence on the overall complexity of the algorithms.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 96%

mentioning

confidence: 98%

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Speeding up Algorithms on Atomic Representations of Herbrand Models via New Redundancy Criteria

Pichler

2000

Journal of Symbolic Computation

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“…These interpretations are called eq-interpretations (eq as equational). Eq-interpretations were introduced in Caferra and Zabel (1991). Roughly speaking, a partial interpretation is said to be an eq-interpretation if it can be represented by an equational formula, interpreted on the Herbrand universe.…”

Section: Introductionmentioning

confidence: 99%

Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models

Caferra¹,

Peltier²

2000

Journal of Symbolic Computation

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A new method for building infinite models for first-order formulae is presented. The method combines enumeration techniques with existing deductive (in a broad sense) ones. Its soundness and completeness w.r.t. the class of models that can be represented by equational constraints are proven. This shows that the use of enumeration techniques strictly increases the power of existing methods for building Herbrand models that are not complete in this sense. Some strategies are proposed to reduce the search space. We give examples and show how to use this approach for building interactively a model of a formula introduced by Goldfarb in his proof of the undecidability of the Gödel class with identity. This formula is satisfiable but has no finite model.

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“…In automated model building, equational constraints play a crucial role for model representation (e.g. [5,11]). On the other hand, equational constraint solving may be applied for several purposes in the areas of functional/logic programming and databases.…”

Section: Introductionmentioning

confidence: 99%

Equational Constraint Solving Via a Restricted Form of Universal Quantification

Àlvez

Lucio

2006

Lecture Notes in Computer Science

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Abstract. In this paper, we present a syntactic method for solving firstorder equational constraints over term algebras. The presented method exploits a novel notion of quasi-solved form that we call answer. By allowing a restricted form of universal quantification, answers provide a more compact way to represent solutions than the purely existential solved forms found in the literature. Answers have been carefully designed to make satisfiability test feasible and also to allow for boolean operations, while maintaining expressiveness and user-friendliness. We present detailed algorithms for (1) satisfiability checking and for performing the boolean operations of (2) negation of one answer and (3) conjunction of n answers. Based on these three basic operations, our solver turns any equational constraint into a disjunction of answers. We have implemented a prototype that is available on the web.

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Extending resolution for model construction

Cited by 27 publications

References 10 publications

Speeding up Algorithms on Atomic Representations of Herbrand Models via New Redundancy Criteria

Speeding up Algorithms on Atomic Representations of Herbrand Models via New Redundancy Criteria

Combining Enumeration and Deductive Techniques in order to Increase the Class of Constructible Infinite Models

Equational Constraint Solving Via a Restricted Form of Universal Quantification

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