6,717 research outputs found

    Deduction modulo theory

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    This paper is a survey on Deduction modulo theor

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

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    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

    A semantic method to prove strong normalization from weak normalization

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    Deduction modulo is an extension of first-order predicate logic where axioms are replaced by rewrite rules and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. In a previous paper we proposed a refinement of the notion of model for theories expressed in deduction modulo, in a way allowing not only to prove soundness, but also completeness: a theory has the strong normalization property if and only if it has a model of this form. In this paper, we present how we can use these techniques to prove that all weakly normalizing theories expressed in minimal deduction modulo, are strongly normalizing

    Proof Certification in Zenon Modulo: When Achilles Uses Deduction Modulo to Outrun the Tortoise with Shorter Steps

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    International audienceWe present the certifying part of the Zenon Modulo automated theorem prover, which is an extension of the Zenon tableau-based first order automated theorem prover to deduction modulo. The theory of deduction modulo is an extension of predicate calculus, which allows us to rewrite terms as well as propositions, and which is well suited for proof search in axiomatic theories, as it turns axioms into rewrite rules. In addition, deduction modulo allows Zenon Modulo to compress proofs by making computations implicit in proofs. To certify these proofs, we use Dedukti, an external proof checker for the λΠ-calculus modulo, which can deal natively with proofs in deduction modulo. To assess our approach, we rely on some experimental results obtained on the benchmarks provided by the TPTP library

    Specifying programs with propositions and with congruences

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    We give a presentation of Krivine and Parigot's Second-order functional arithmetic in Deduction modulo. Expressing this theory in Deduction modulo sheds light on an original aspect of this theory: the fact that programs are specified, not with propositions, but with congruences

    Automated Deduction in the B Set Theory using Typed Proof Search and Deduction Modulo

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    International audienceWe introduce an encoding of the set theory of the B method using polymorphic types and deduction modulo, which is used for the automated verication of proof obligations in the framework of theBWare project. Deduction modulo is an extension of predicate calculus with rewriting both on terms and propositions. It is well suited for proof search in theories because it turns many axioms into rewrite rules. We also present the associated automated theorem prover Zenon Modulo, an extension of Zenon to polymorphic types and deduction modulo, along with its backend to the Dedukti universal proof checker, which also relies on types and deduction modulo, and which allows us to verify the proofs produced by Zenon Modulo. Finally, we assess our approach over the proof obligation benchmark of BWare

    The Stratified Foundations as a theory modulo

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    The Stratified Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratifiable propositions. It is known that this theory is consistent and that proofs strongly normalize in this theory. Deduction modulo is a formulation of first-order logic with a general notion of cut. It is known that proofs normalize in a theory modulo if it has some kind of many-valued model called a pre-model. We show in this paper that the Stratified Foundations can be presented in deduction modulo and that the method used in the original normalization proof can be adapted to construct a pre-model for this theory

    Semantic A-translation and Super-consistency entail Classical Cut Elimination

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    We show that if a theory R defined by a rewrite system is super-consistent, the classical sequent calculus modulo R enjoys the cut elimination property, which was an open question. For such theories it was already known that proofs strongly normalize in natural deduction modulo R, and that cut elimination holds in the intuitionistic sequent calculus modulo R. We first define a syntactic and a semantic version of Friedman's A-translation, showing that it preserves the structure of pseudo-Heyting algebra, our semantic framework. Then we relate the interpretation of a theory in the A-translated algebra and its A-translation in the original algebra. This allows to show the stability of the super-consistency criterion and the cut elimination theorem

    Cut elimination for Zermelo set theory

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    We show how to express intuitionistic Zermelo set theory in deduction modulo (i.e. by replacing its axioms by rewrite rules) in such a way that the corresponding notion of proof enjoys the normalization property. To do so, we first rephrase set theory as a theory of pointed graphs (following a paradigm due to P. Aczel) by interpreting set-theoretic equality as bisimilarity, and show that in this setting, Zermelo's axioms can be decomposed into graph-theoretic primitives that can be turned into rewrite rules. We then show that the theory we obtain in deduction modulo is a conservative extension of (a minor extension of) Zermelo set theory. Finally, we prove the normalization of the intuitionistic fragment of the theory
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