Integration of Equality Reasoning into the Disconnection Calculus

Letz, Reinhold; Stenz, Gernot

doi:10.1007/3-540-45616-3_13

Cited by 21 publications

(13 citation statements)

References 8 publications

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“…These paramodulation inferences combine (equational) branch literals and clauses into new clauses, where the equation used for paramodulation is added in negated form (as a condition) to the paramodulant. Our main equality inference rules work similarly, and in this sense the calculus of [22] and ME E are conceptually related. But ME E features more powerful and general concepts of redundancy detection and elimination.…”

Section: Introductionmentioning

confidence: 90%

“…More closely related to ME E is an IM based on disconnection tableaux by Letz and Stenz, a successor of Billon's disconnection method [12]. 2 Letz and Stenz discuss various ways of integrating equality reasoning in disconnection tableaux [22], including a variant based on ordered paramodulation. These paramodulation inferences combine (equational) branch literals and clauses into new clauses, where the equation used for paramodulation is added in negated form (as a condition) to the paramodulant.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Model Evolution with equality — Revised and implemented

Baumgartner

Pelzer

Tinelli

2012

Journal of Symbolic Computation

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“…Methods based on ordered equality handling are not compatible with the most successful refinements of tableaux like the connection conditions. In contrast to calculi based on such refinements the disconnection calculus offers a more robust framework for integrating various approaches of equality handling [42]. An approach for integrating equational reasoning in another instantiation-based method, the model evolution calculus, is presented in [12].…”

Section: Equalitymentioning

confidence: 99%

The Disconnection Tableau Calculus

Letz

Stenz

2006

J Autom Reasoning

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In this paper we give a comprehensive presentation of the disconnection tableau calculus, a proof method for formulas in classical first-order clause logic. The distinguishing property of this calculus is that it uses unification in such a manner that important proof-theoretic advantages of the classical (i.e., Smullyan-style) tableau calculus are preserved, specifically the termination and model generation characteristics for certain formula classes. Additionally, the calculus is well suited for fully automated proof search. The calculus is described in detail with soundness and completeness proofs, and a number of important calculus refinements developed over the past years are presented. Referring to the model-finding abilities of the disconnection calculus, we explain the extraction and representation of models. We also describe the integration of paramodulation-based equality handling. Finally, we give an overview of related methods.

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“…The first challenge is the integration of theory reasoning and, in particular, reasoning with real and integer arithmetic. There are results on the integration of equational reasoning [25,16,7] and some initial results on the integration of theory reasoning [17,4], but these should be considerably extended to cover more problems coming from applications.…”

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confidence: 99%

Instantiation-Based Automated Reasoning: From Theory to Practice

Korovin

2009

Automated Deduction – CADE-22

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Instantiation-based automated reasoning aims at combining the efficiency of propositional SAT and SMT technologies with the expressiveness of firstorder logic. Propositional SAT and SMT solvers are probably the most successful reasoners applied to real-world problems, due to extremely efficient propositional methods and optimized implementations. However, the expressiveness of firstorder logic is essential in many applications ranging from formal verification of software and hardware to knowledge representation and querying. Therefore, there is a growing demand to integrate efficient propositional and more generally ground reasoning modulo theories into first-order reasoning.The basic idea behind instantiation-based reasoning is to interleave smart generation of instances of first-order formulae with propositional type reasoning. Instantiation-based methods can be divided into two major categories: (i) finegrained interleaving of instantiation with efficient propositional inference rules, and (ii) modular combination of instantiation and propositional reasoning. Examples from the first category include the disconnection calculus (DCTP) [8,24], which combines instance generation with an efficient tableau data structure, and the model evolution calculus (ME) [6], which interleaves instance generation with DPLL style reasoning. Both DCTP and ME methods have advanced implementations DCTP [33] and Darwin [3], respectively.Our approach to instantiation-based reasoning [15,21] falls into the second category, where propositional reasoning is integrated in a modular fashion and was inspired by work on hyper-linking and its extensions (see [23,31,18]). The main advantage of the modular combination is that it allows one to use off-theshelf SAT/SMT solvers in the context of first-order reasoning. One of our main goals was to develop a flexible theoretical framework, called Inst-Gen, for modular combination of instantiation with propositional reasoning and more generally with ground reasoning modulo theories. This framework provides methods for proving completeness of instantiation calculi, powerful redundancy elimination criteria and flexible saturation strategies. All these ingredients are crucial for developing reasoning systems which can be used in practical applications. We also show that most of the powerful machinery developed in the resolution-based framework can be suitably adapted for the Inst-Gen method.Based on these theoretical results we have developed and implemented an automated reasoning system, called iProver [22]. iProver features state-of-theart implementation techniques such as unification and simplification indexes;Supported by The Royal Society

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Integration of Equality Reasoning into the Disconnection Calculus

Cited by 21 publications

References 8 publications

Model Evolution with equality — Revised and implemented

Model Evolution with equality — Revised and implemented

The Disconnection Tableau Calculus

Instantiation-Based Automated Reasoning: From Theory to Practice

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