Author name cluster

Guillaume Burel

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

5 papers

2 author rows

MFCS Conference 2018 Conference Paper

Linking Focusing and Resolution with Selection

Guillaume Burel

Focusing and selection are techniques that shrink the proof search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atom can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof search space.

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CSL Conference 2010 Conference Paper

Embedding Deduction Modulo into a Prover

Guillaume Burel

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.

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I&C Journal 2010 Journal Article

Regaining cut admissibility in deduction modulo using abstract completion

Guillaume Burel
Claude Kirchner

Deduction modulo is a way to combine computation and deduction in proofs, by applying the inference rules of a deductive system (e. g. natural deduction or sequent calculus) modulo some congruence that we assume here to be presented by a set of rewrite rules. Using deduction modulo is equivalent to proving in a theory corresponding to the rewrite rules, and leads to proofs that are often shorter and more readable. However, cuts may be not admissible anymore. We define a new system, the unfolding sequent calculus, and prove its equivalence with the sequent calculus modulo, especially w. r. t. cut-free proofs. It permits to show that it is even undecidable to know if cuts can be eliminated in the sequent calculus modulo a given rewrite system. Then, to recover the cut admissibility, we propose a procedure to complete the rewrite system such that the sequent calculus modulo the resulting system admits cuts. This is done by generalizing the Knuth–Bendix completion in a non-trivial way, using the framework of abstract canonical systems. These results enlighten the entanglement between computation and deduction, and the power of abstract completion procedures. They also provide an effective way to obtain systems admitting cuts, therefore extending the applicability of deduction modulo in automated theorem proving.

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LFMTP Workshop 2009 Invited Paper

How can we prove that a proof search method is not an instance of another?

Guillaume Burel
Gilles Dowek

We introduce a method to prove that a proof search method is not an instance of another. As an example of application, we show that Polarized resolution modulo, a method that mixes clause selection restrictions and literal selection restrictions, is not an instance of Ordered resolution with selection.

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CSL Conference 2007 Conference Paper

Unbounded Proof-Length Speed-Up in Deduction Modulo

Guillaume Burel

Abstract In 1973, Parikh proved a speed-up theorem conjectured by Gödel 37 years before: there exist arithmetical formulæ that are provable in first order arithmetic, but whose shorter proof in second order arithmetic is arbitrarily smaller than any proof in first order. On the other hand, resolution for higher order logic can be simulated step by step in a first order narrowing and resolution method based on deduction modulo, whose paradigm is to separate deduction and computation to make proofs clearer and shorter. We prove that i + 1-th order arithmetic can be linearly simulated into i -th order arithmetic modulo some confluent and terminating rewrite system. We also show that there exists a speed-up between i -th order arithmetic modulo this system and i -th order arithmetic without modulo. All this allows us to prove that the speed-up conjectured by Gödel does not come from the deductive part of the proofs, but can be expressed as simple computation, therefore justifying the use of deduction modulo as an efficient first order setting simulating higher order.

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