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Type theories
 In STACS ’02: Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
, 1995
"... Abstract. Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones. ..."
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Abstract. Deduction modulo is a way to express a theory using computation rules instead of axioms. We present in this paper an extension of deduction modulo, called Polarized deduction modulo, where some rules can only be used at positive occurrences, while others can only be used at negative ones. We show that all theories in propositional calculus can be expressed in this framework and that cuts can always be eliminated with such theories. Mathematical proofs are almost never built in pure logic, but besides the deduction rules and the logical axioms that express the meaning of the connectors and quantifiers, they use something else a theory that expresses the meaning of the other symbols of the language. Examples of theories are equational theories, arithmetic, type theory, set theory,... The usual definition of a theory, as a set of axioms, is sufficient when one is interested in the provability relation, but, as wellknown, it is not when one is interested in the structure of proofs and in the theorem proving process. For
CoqInE: Translating the Calculus of Inductive Constructions into the λΠcalculus Modulo
 in "Second International Workshop on Proof Exchange for Theorem Proving
, 2012
"... We show how to translate the Calculus of Inductive Constructions (CIC) as implemented by Coq into the λΠcalculus modulo, a proposed common backend proof format for heterogeneous proof assistants. 1 ..."
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We show how to translate the Calculus of Inductive Constructions (CIC) as implemented by Coq into the λΠcalculus modulo, a proposed common backend proof format for heterogeneous proof assistants. 1
A Shallow Embedding of Resolution and Superposition Proofs into the λΠCalculus Modulo
 PXTP
, 2013
"... The λΠcalculus modulo is a proof language that has been proposed as a proof standard for (re)checking and interoperability. Resolution and superposition are proofsearch methods that are used in stateoftheart firstorder automated theorem provers. We provide a shallow embedding of resolution an ..."
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The λΠcalculus modulo is a proof language that has been proposed as a proof standard for (re)checking and interoperability. Resolution and superposition are proofsearch methods that are used in stateoftheart firstorder automated theorem provers. We provide a shallow embedding of resolution and superposition proofs in the λΠcalculus modulo, thus offering a way to check these proofs in a trusted setting, and to combine them with other proofs. We implement this embedding as a backend of the prover iProver Modulo.
Checking foundational proof certificates for firstorder logic
"... We present the design philosophy of a proof checker based on a notion of foundational proof certificates. This checker provides a semantics of proof evidence using recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higherorder) log ..."
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We present the design philosophy of a proof checker based on a notion of foundational proof certificates. This checker provides a semantics of proof evidence using recent advances in the theory of proofs for classical and intuitionistic logic. That semantics is then performed by a (higherorder) logic program: successful performance means that a formal proof of a theorem has been found. We describe how the λProlog programming language provides several features that help guarantee such a soundness claim. Some of these features (such as strong typing, abstract datatypes, and higherorder programming) were features of the ML programming language when it was first proposed as a proof checker for LCF. Other features of λProlog (such as support for bindings, substitution, and backtracking search) turn out to be equally important for describing and checking the proof evidence encoded in proof certificates. Since trusting our proof checker requires trusting a programming language implementation, we discuss various avenues for enhancing one’s trust of such a checker. 1
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 7 (3 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.
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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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
Embedding deduction modulo into a prover
 CSL. Lecture Notes in Computer Science
, 2010
"... 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 restric ..."
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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 stateoftheart provers. Whatever their applications, proofs are rarely searched for without context: mathematical proofs rely on set theory, or Euclidean geometry, or arithmetic,
Efficiently Simulating HigherOrder Arithmetic by a FirstOrder Theory Modulo
"... Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. Ho ..."
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Deduction modulo is a paradigm which consists in applying the inference rules of a deductive system—such as for instance natural deduction—modulo a rewrite system over terms and propositions. It has been shown that higherorder logic can be simulated into the firstorder natural deduction modulo. However, a theorem stated by Gödel and proved by Parikh expresses that proofs in secondorder arithmetic may be unboundedly shorter than proofs in firstorder arithmetic, even when considering only formulæ provable in firstorder arithmetic. We investigate how deduction modulo can be used to translate proofs of higherorder arithmetic into firstorder proofs without inflating their length. First we show how higher orders can be encoded through a quite simple (finite, terminating, confluent, leftlinear) rewrite system. Then, a proof in higherorder arithmetic can be linearly translated into a proof in firstorder arithmetic modulo this system. Second, in the continuation of a work of Dowek and Werner, we show how to express the whole higherorder arithmetic as a rewrite system. Then, proofs of higherorder arithmetic can be linearly translated into proofs in the empty theory modulo this rewrite system. These results show that the speedup between firstand secondorder arithmetic, and more generally between i th and i +1 storder arithmetic, can in fact be expressed as computation, and does not lie in the really deductive part of the proofs.
Strategic Computation and Deduction
, 2009
"... I'd like to conclude by emphasizing what a wonderful eld this is to work in. Logical reasoning plays such a fundamental role in the spectrum of intellectual activities that advances in automating logic will inevitably have a profound impact in many intellectual disciplines. Of course, these thi ..."
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I'd like to conclude by emphasizing what a wonderful eld this is to work in. Logical reasoning plays such a fundamental role in the spectrum of intellectual activities that advances in automating logic will inevitably have a profound impact in many intellectual disciplines. Of course, these things take time. We tend to be impatient, but we need some historical perspective. The study of logic has a very long history, going back at least as far as Aristotle. During some of this time not very much progress was made. It's gratifying to realize how much has been accomplished in the less than fty years since serious e orts to mechanize logic began.
Automation of HigherOrder Logic
 THE HANDBOOK OF THE HISTORY OF LOGIC, EDS. D. GABBAY & J. WOODS; VOLUME 9: LOGIC AND COMPUTATION, EDITOR JÖRG SIEKMANN
, 2014
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