Principles of Superdeduction

International audienceIn predicate logic, the proof that a theorem P holds in a theory Th is typically conducted in natural deduction or in the sequent calculus using all the information contained in the theory in a uniform way. Introduced ten years ago, Deduction modulo allows us to make use of the computational part of the theory Th for true computations modulo which deductions are performed. Focussing on the sequent calculus, this paper presents and studies the dual concept where the theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. We call such a new deduction system "superdeduction''. We introduce a proof-term language and a cut-elimination procedure both based on Christian Urban's work on classical sequent calculus. Strong normalisation is proven under appropriate and natural hypothesis, therefore ensuring the consistency of the embedded theory and of the deduction system. The proofs obtained in such a new system are much closer to the human intuition and practice. We consequently show how superdeduction along with deduction modulo can be used to ground the formal foundations of new extendible proof assistants. We finally present lemuridae, our current implementation of superdeduction modulo

Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems---such as for instance natural deduction---are applied. Therefore, the reasoning that is intrinsic of the theory does not appear in the length of proofs. In general, the congruence is defined through a rewrite system over terms and propositions. We define a rigorous framework to study proof lengths in deduction modulo, where the congruence must be computed in polynomial time. We show that even very simple rewrite systems lead to arbitrary proof-length speed-ups in deduction modulo, compared to using axioms. As higher-order logic can be encoded as a first-order theory in deduction modulo, we also study how to reinterpret, thanks to deduction modulo, the speed-ups between higher-order and first-order arithmetics that were stated by G\"odel. We define a first-order rewrite system with a congruence decidable in polynomial time such that proofs of higher-order arithmetic can be linearly translated into first-order arithmetic modulo that system. We also present the whole higher-order arithmetic as a first-order system without resorting to any axiom, where proofs have the same length as in the axiomatic presentation

Axiom directed Focusing

Long versionInternational audienceSuperdeduction and deduction modulo are methods specially designed to ease the use of first-order theories in predicate logic. Superdeduction modulo, which combines both, enables the user to make a distinct use of computational and reasoning axioms. Although soundness is ensured, using superdeduction and deduction modulo to extend deduction with awkward theories can jeopardize essential properties of the extended system such as cut-elimination or completeness \wrt~predicate logic. Therefore one has to design criteria for theories which can safely be used through superdeduction and deduction modulo. In this paper we revisit the superdeduction paradigm by comparing it with the focusing approach. In particular we prove a focalization theorem for cut-free superdeduction modulo: we show that permutations of inference rules can transform any cut-free proof in deduction modulo into a cut-free proof in superdeduction modulo and conversely, provided that some hypotheses on the synchrony of reasoning axioms are verified. It implies that cut-elimination for deduction modulo and for superdeduction modulo are equivalent. Since several criteria have already been proposed for theories that do not break cut-elimination of the corresponding deduction modulo system, these criteria also imply cut-elimination of the superdeduction modulo system, provided our synchrony hypotheses hold. Finally we design a tableaux method for superdeduction modulo which is sound and complete provided cut-elimination holds

Unbounded proof-length speed-up in deduction modulo

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

Automatisation des preuves pour la vérification des règles de l'Atelier B

Cette thèse porte sur la vérification des règles ajoutées de l'Atelier B en utilisant une plate-forme appelée BCARe qui repose sur un plongement de la théorie sous-jacente à la méthode B (théorie de B) dans l'assistant à la preuve Coq. En particulier, nous proposons trois approches pour prouver la validité d'une règle, ce qui revient à prouver une formule exprimée dans la théorie de B. Ces trois approches ont été évaluées sur les règles de la base de règles de SIEMENS IC-MOL. La première approche dite autarcique est développée avec le langage de tactiques de Coq Ltac. Elle repose sur une première étape qui consiste à déplier tous les opérateurs ensemblistes pour obtenir une formule de la logique du premier ordre. Puis nous appliquons une procédure de décision qui met en oeuvre une heuristique naïve en ce qui concerne les instanciations. La deuxième approche, dite sceptique,appelle le prouveur automatique de théorèmes Zenon après avoir effectué l'étape de normalisation précédente. Nous vérifions ensuite les preuves trouvées par Zenon dans le plongement profond de B en Coq. La troisième approche évite l'étape de normalisation précédente grâce à une extension de Zenon utilisant des règles d'inférence spécifiques à la théorie de B. Ces règles sont obtenues grâce à la technique de superdéduction. Cette dernière approche est généralisée en une extension de Zenon à toute théorie grâce à un calcul dynamique des règles de superdéduction. Ce nouvel outil, appelé Super Zenon, peut par exemple prouver des problèmes issus de la bibliothèque de problèmes TPTP.The purpose of this thesis is the verification of Atelier B added rules using the framework named BCARe which relies on a deep embedding of the B theory within the logic of the Coq proof assistant. We propose especially three approaches in order to prove the validity of a rule, which amounts to prove a formula expressed in the B theory. These three approaches have been assessed on the rules coming from the rule database maintained by Siemens IC-MOL. To do so, the first approach, so-called autarkic approach, is developed thanks to the Coq tactic language, Ltac. It rests upon a first step which consists in unfolding the set operators so as to obtain a first order formula. A decision procedure which implements an heuristic is applied afterwards to deal with instantiation. We propose a second approach, so-called skeptic approach, which uses the automated first order theorem prover Zenon, after the previous normalization step has been applied. Then we verify the Zenon proofs in the deep embedding of B in Coq. A third approach consists in using anextension of Zenon to the B method thanks to the superdeduction. Superdeduction allows us to add the axioms of the B theory by means of deduction rules in the proof mechanism of Zenon. This last approach is generalized in an extension of Zenon to every theory thanks to a dynamic calculus of the superdeduction rules. This new tool, named Super Zenon, is able to prove problems coming from the problem library TPTP, for example.PARIS-CNAM (751032301) / SudocSudocFranceF