Regaining cut admissibility in deduction modulo using abstract completion

Burel, Guillaume; Kirchner, Claude

doi:10.1016/j.ic.2009.10.005

Cited by 10 publications

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“…We also need a sequent calculus modulo which is more adapted to our purpose. Following the ideas of [10], we introduce the one-sided polarized unfolding sequent calculus (short 1PUSC R ) where all formulae are put in the left-hand side of the sequents, instantiations are ground, rewrite steps are explicit, and rewrit- ing and axioms can be applied to literals only. Its inference rules are presented in Figure 5.…”

Section: Refining Polarized Resolution Modulomentioning

confidence: 99%

“…We need to prove that weakening is admissible, that we can make the rewriting explicits and that we can restrict − , ↑ − − and ↑ + − to literals. The proof is the same as for [10,Proposition 7], except that we are here in a onesided sequent calculus, which is not problematic since all negations are put down on the literal level.…”

Section: Refining Polarized Resolution Modulomentioning

confidence: 99%

“…Examples of theories that can be used in deduction modulo include arithmetic [5], Zermelo's set theory [6], simple type theory (a.k.a. higher-order logic) [7] or pure type systems [8,9], and there exists a procedure to present any first-order classical theory as a rewrite system usable in deduction modulo [10].…”

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confidence: 99%

“…Due to Gödel's theorem, this proof has to be done in a stronger theory than the one defined by the rewrite system. A bunch of techniques exists to prove cut admissibility in deduction modulo [13][14][15] (in particular, they can be applied to the theories cited before, for which proof search methods modulo are therefore complete) and a completion procedure was designed to transform a rewrite system so that cut admissibility holds [10].…”

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Embedding Deduction Modulo into a Prover

Burel¹

2010

Computer Science Logic

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Abstract. Deduction modulo consists in presenting a theory through rewrite rules to support automatic and interactive proof search. It induces proof search methods based on narrowing, such as the polarized resolution modulo. We show how to combine this method with more traditional ordering restrictions. Interestingly, no compatibility between the rewriting and the ordering is requested to ensure completeness. We also show that some simplification rules, such as strict subsumption eliminations and demodulations, preserve completeness. For this purpose, we use a new framework based on a proof ordering. These results show that polarized resolution modulo can be integrated into existing provers, where these restrictions and simplifications are present. We also discuss how this integration can actually be done by diverting the main algorithm of state-of-the-art provers.Whatever their applications, proofs are rarely searched for without context: mathematical proofs rely on set theory, or Euclidean geometry, or arithmetic, etc.; proofs of program correctness are done using e.g. pointer arithmetic and/or theories defining data structures (chained lists, trees, . . . ); concerning security, theories are used for instance to model properties of encryption algorithms. It is therefore essential to have theoretical foundations and practical methods that handle theories conveniently and efficiently. For this purpose, there are two directions: to develop methods that are really specific to a particular theory; or to develop a generic framework that can handle all theories. The first option is appealing for efficiency reasons: for instance, combining a SAT solver with the Simplex method leads to very powerful SMT solvers for linear arithmetic. However, developing methods for new theories is hard. Even the combination of such specific methods is not trivial, although there have been a lot of interesting results in that direction in the recent years. In this paper, we are more interested in the second option: having a generic way to handle theories efficiently. A naive way to do so would be to use an axiomatization of the theory, but in general, this approach would be really inefficient for automated proving. Somehow, we need to present the theory so as to take advantage of its properties.A first idea is to use the consistency of the theory. When proving a goal in a consistent theory by refutation, resolving the clauses of the theory is useless, since it will not bring out a contradiction. This idea defines the set-of-support

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Section: Refining Polarized Resolution Modulomentioning

confidence: 99%

Section: Refining Polarized Resolution Modulomentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Embedding Deduction Modulo into a Prover

Burel¹

2010

Computer Science Logic

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“…This paper studies the automation of the transformation of the presentation of an intuitionistic first-order theory into a rewrite system that is applied modulo. In a submitted paper [7], we proposed a complete solution in the case of classical logic: First, we have shown how to transform any presentation of a theory into a compatible rewrite system; Then, we have defined a completion procedure that transforms the resulting rewrite system to ensure that the sequent calculus modulo the final rewrite system admits cut. In intuitionistic logic, however, there are theories that cannot be transformed into a compatible rewrite system, as we will soon show, and we cannot separate the production of the rewrite system and its completion that ensures the cut admissibility.…”

Section: Introductionmentioning

confidence: 99%

Automating Theories in Intuitionistic Logic

Burel

2009

Frontiers of Combining Systems

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Abstract. Deduction modulo consists in applying the inference rules of a deductive system modulo a rewrite system over terms and formulae. This is equivalent to proving within a so-called compatible theory. Conversely, given a first-order theory, one may want to internalize it into a rewrite system that can be used in deduction modulo, in order to get an analytic deductive system for that theory. In a recent paper, we have shown how this can be done in classical logic. In intuitionistic logic, however, we show here not only that this may be impossible, but also that the set of theories that can be transformed into a rewrite system with an analytic sequent calculus modulo is not co-recursively enumerable. We nonetheless propose a procedure to transform a large class of theories into compatible rewrite systems. We then extend this class by working in conservative extensions, in particular using Skolemization.

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Experimenting with Deduction Modulo

Burel¹

2011

Lecture Notes in Computer Science

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Regaining cut admissibility in deduction modulo using abstract completion

Cited by 10 publications

References 12 publications

Embedding Deduction Modulo into a Prover

Embedding Deduction Modulo into a Prover

Automating Theories in Intuitionistic Logic

Experimenting with Deduction Modulo

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