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24
Nominal Logic: A First Order Theory of Names and Binding
 Information and Computation
, 2001
"... This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal L ..."
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Cited by 161 (15 self)
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This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of firstorder manysorted logic with equality containing primitives for renaming via nameswapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...
Theorem Proving Modulo
 Journal of Automated Reasoning
"... Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first ..."
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Cited by 75 (14 self)
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Abstract. Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of much wider interest because it permits to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof theoretic account of the combination of computations and deductions. The congruence on propositions is handled via rewrite rules and equational axioms. Rewrite rules apply to terms and also directly to atomic propositions. The second contribution is to give a complete proof search method, called Extended Narrowing and Resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higherorder logic can be presented as a theory modulo. Applying the Extended Narrowing and Resolution method to this presentation of higherorder logic subsumes full higherorder resolution.
Embedding pure type systems in the lambdaPicalculus modulo
 TLCA
, 2007
"... The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all fu ..."
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Cited by 19 (5 self)
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The lambdaPicalculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambdaPicalculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.
Polarized Resolution Modulo
"... Abstract. We present a restriction of Resolution modulo where the rewrite rules are such that a clause always rewrites to a clause. This way, the reduct of a clause needs not be further transformed into clause form. Restricting Resolution modulo this way requires to extend it in another way and dist ..."
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Cited by 7 (2 self)
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Abstract. We present a restriction of Resolution modulo where the rewrite rules are such that a clause always rewrites to a clause. This way, the reduct of a clause needs not be further transformed into clause form. Restricting Resolution modulo this way requires to extend it in another way and distinguish the rules that apply to negative and to positive atomic propositions. As an example, we show how this method applies to a firstorder presentation of Simple type theory. Finally, we show that this method can be seen as a restriction of Equational resolution that mixes clause selection restrictions and literal selection restrictions, but unlike many restrictions of Resolution, it is not an instance of Ordered resolution. 1
A completeness theorem for strong normalization in minimal deduction modulo
, 2009
"... Abstract. Deduction modulo is an extension of firstorder 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 ..."
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Cited by 6 (2 self)
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Abstract. Deduction modulo is an extension of firstorder 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. Dowek and Werner have given a semantic sufficient condition for a theory to have the strong normalization property: they have proved a ”soundness ” theorem of the form: if a theory has a model (of a particular form) then it has the strong normalization property. In this paper, we refine their notion of model in a way allowing not only to prove soundness, but also completeness: if a theory has the strong normalization property, then it has a model of this form. The key idea of our model construction is a refinement of Girard’s notion of reducibility candidates. By providing a sound and complete semantics for theories having the strong normalization property, this paper contributes to explore the idea
Truth value algebras and proof normalization
 In TYPES 2006
, 2006
"... Abstract. We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a Bvalued model for some non trivial truth values algebra B. A theory that has a Bvalued model for all truth values algebras B is said to be superconsi ..."
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Cited by 5 (3 self)
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Abstract. We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a Bvalued model for some non trivial truth values algebra B. A theory that has a Bvalued model for all truth values algebras B is said to be superconsistent. We prove that superconsistency is a modeltheoretic sufficient condition for strong normalization. 1
Proof Search and Proof Check for Equational and Inductive Theorems
 Conference on Automated Deduction  CADE19
, 2003
"... Abstract. This paper presents ongoing researches on theoretical and practical issues of combining rewriting based automated theorem proving and userguided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof a ..."
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Cited by 5 (0 self)
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Abstract. This paper presents ongoing researches on theoretical and practical issues of combining rewriting based automated theorem proving and userguided proof development, with the strong constraint of safe cooperation of both. In practice, we instantiate the theoretical study on the Coq proof assistant and the ELAN rewriting based system, focusing first on equational and then on inductive proofs. Different concepts, especially rewriting calculus and deduction modulo, contribute to define and to relate proof search, proof representation and proof check.
From HigherOrder to FirstOrder Rewriting
 In Proceedings of the 12th International Conference on Rewriting Techniques and Applications (RTA’01
, 2001
"... . We show how higherorder rewriting may be encoded into ..."
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Cited by 5 (0 self)
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. We show how higherorder rewriting may be encoded into
Unbounded prooflength speedup in deduction modulo
 CSL 2007, VOLUME 4646 OF LNCS
, 2007
"... In 1973, Parikh proved a speedup 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 h ..."
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Cited by 3 (2 self)
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In 1973, Parikh proved a speedup 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+1th order arithmetic can be linearly simulated into ith order arithmetic modulo some confluent and terminating rewrite system. We also show that there exists a speedup between ith order arithmetic modulo this system and ith order arithmetic without modulo. All this allows us to prove that the speedup 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.
Cut Elimination in Deduction Modulo by Abstract Completion
 in "FoSSaCS 2007, 24/03/2007, Braga/Portugal
"... ..."