Refutational theorem proving for hierarchic first-order theories

Bachmair, Leo; Ganzinger, Harald; Waldmann, Uwe

doi:10.1007/bf01190829

Cited by 91 publications

(183 citation statements)

References 3 publications

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“…As said before, we plan to extend the calculus directly with B-sorted (non-constant) function symbols. This could be done along the lines in [3]. Another pressing issue is to strengthen ME E (T)'s model-building capabilities.…”

Section: Discussionmentioning

confidence: 99%

“…This is a true restriction for non-constant function symbols. 3 For example, if Σ is the signature of lists of integers, with T being again LIA and F being the list sort, our logic allows formulas like cdr(cons(x, y)) ≈ y but not car(cons(x, y)) ≈ x, as car would be integer-sorted. To overcome this limitation somewhat, one could turn car into a predicate symbol and use car(cons(x, y), x) instead, together with the (universal) functionality constraint ¬car(…”

Section: Preliminariesmentioning

confidence: 99%

“…Yet, we think ME E (T) is relevant through its combination of powerful techniques for first-order equational logic with equality, based on an adaptation of the Bachmair-Ganzinger theory of superposition, with a black-box theory reasoner, which greatly enhances its versatility. ME E (T) is thus similarly positioned as the hierarchic superposition calculus [1,3].…”

Section: Introductionmentioning

confidence: 99%

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Model Evolution with Equality Modulo Built-in Theories

Baumgartner

Tinelli

2011

Lecture Notes in Computer Science

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Many applications of automated deduction require reasoning modulo background theories, in particular some form of integer arithmetic. Developing corresponding automated reasoning systems that are also able to deal with quantified formulas has recently been an active area of research. We contribute to this line of research and propose a novel instantiation-based method for a large fragment of first-order logic with equality modulo a given complete background theory, such as linear integer arithmetic. The new calculus is an extension of the Model Evolution Calculus with Equality, a firstorder logic version of the propositional DPLL procedure, including its ordering-based redundancy criteria. We present a basic version of the calculus and prove it sound and (refutationally) complete under certain conditions.

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Section: Discussionmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Model Evolution with Equality Modulo Built-in Theories

Baumgartner

Tinelli

2011

Lecture Notes in Computer Science

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“…It has been shown that the Scan algorithm is complete and terminates for modal axioms belonging to the famous Sahlqvist class [9]. In 1994, the hierarchical theorem proving method was developed by Bachmair et al [2] and it has been shown that it can be used to solve second-order quantification problems. Around the same time, in 1995, Sza las [27] described a different algorithm for the second-order quantifier elimination problem, which exploits Ackermann's Lemma.…”

Section: Related Workmentioning

confidence: 99%

Concept Forgetting in $$\mathcal {ALCOI}$$-Ontologies Using an Ackermann Approach

Zhao

Schmidt

2015

The Semantic Web - ISWC 2015

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Abstract. We present a method for forgetting concept symbols in ontologies specified in the description logic ALCOI. The method is an adaptation and improvement of a second-order quantifier elimination method developed for modal logics and used for computing correspondence properties for modal axioms. It follows an approach exploiting a result of Ackermann adapted to description logics. An important feature inherited from the modal approach is that the inference rules are guided by an ordering compatible with the elimination order of the concept symbols. This provides more control over the inference process and reduces non-determinism, resulting in a smaller search space. The method is extended with a new case splitting inference rule, and several simplification rules. Compared to related forgetting and uniform interpolation methods for description logics, the method can handle inverse roles, nominals and ABoxes. Compared to the modal approach on which it is based, it is more efficient in time and improves the success rates. The method has been implemented in Java using the OWL API. Experimental results show that the order in which the concept symbols are eliminated significantly affects the success rate and efficiency.

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“…In that case, the calculus addresses the problem by adding to Λ, if possible, (a variant of) the complement K of some literal K in Γ. 3 This typically ensures that the new context satisfies some (possibly, all) ground instances of K and, as a consequence, some ground instances of D. If some instances of D remain falsified, a new empty clause reflecting that will be derivable later.…”

Section: Main Ideasmentioning

confidence: 99%

Model Evolution with equality — Revised and implemented

Baumgartner

Pelzer

Tinelli

2012

Journal of Symbolic Computation

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In many theorem proving applications, a proper treatment of equational theories or equality is mandatory. In this paper we show how to integrate a modern treatment of equality in the Model Evolution calculus (ME), a first-order version of the propositional DPLL procedure. The new calculus, ME E , is a proper extension of the ME calculus without equality. Like ME it maintains an explicit candidate model, which is searched for by DPLL-style splitting. For equational reasoning ME E uses an adapted version of the superposition inference rule, where equations used for superposition are drawn (only) from the candidate model. The calculus also features a generic, semantically justified simplification rule which covers many simplification techniques known from superposition-style theorem proving. Our main theoretical result is the correctness of the ME E calculus in the presence of very general redundancy elimination criteria. We also describe our implementation of the calculus, the E-Darwin system, and we report on practical experiments with it on the TPTP problems library.

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Refutational theorem proving for hierarchic first-order theories

Cited by 91 publications

References 3 publications

Model Evolution with Equality Modulo Built-in Theories

Model Evolution with Equality Modulo Built-in Theories

Concept Forgetting in $$\mathcal {ALCOI}$$-Ontologies Using an Ackermann Approach

Model Evolution with equality — Revised and implemented

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