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21
Cut Rules and Explicit Substitutions
, 2000
"... this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the l ..."
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this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via leftintroduction and rightintroduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them Lsystems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in Lsystems. The advantage of Nsystems is that they seem closer to actual reasoning, while Lsystems on the other hand seem to have an easier proof theory. Lsystems are often extended with a "cut" rule as part of showing that for a given Lsystem and Nsystem, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider
The theory of calculi with explicit substitutions revisited
 CSL 2007
, 2007
"... Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we fi ..."
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Calculi with explicit substitutions (ES) are widely used in different areas of computer science. Complex systems with ES were developed these last 15 years to capture the good computational behaviour of the original systems (with metalevel substitutions) they were implementing. In this paper we first survey previous work in the domain by pointing out the motivations and challenges that guided the development of such calculi. Then we use very simple technology to establish a general theory of explicit substitutions for the lambdacalculus which enjoys fundamental properties such as simulation of onestep betareduction, confluence on metaterms, preservation of betastrong normalisation, strong normalisation of typed terms and full composition. The calculus also admits a natural translation into Linear Logic’s proofnets.
A Systematic Study of Functional Language Implementations
 ACM Transactions on Programming Languages and Systems
, 1998
"... : We introduce a unified framework to describe, relate, compare and classify functional language implementations. The compilation process is expressed as a succession of program transformations in the common framework. At each step, different transformations model fundamental choices. A benefit of t ..."
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: We introduce a unified framework to describe, relate, compare and classify functional language implementations. The compilation process is expressed as a succession of program transformations in the common framework. At each step, different transformations model fundamental choices. A benefit of this approach is to structure and decompose the implementation process. The correctness proofs can be tackled independently for each step and amount to proving program transformations in the functional world. This approach also paves the way to formal comparisons by making it possible to estimate the complexity of individual transformations or compositions of them. Our study aims at covering the whole known design space of sequential functional languages implementations. In particular, we consider callbyvalue, callbyname and callbyneed reduction strategies as well as environment and graphbased implementations. We describe for each compilation step the diverse alternatives as program tr...
A Finite FirstOrder Theory of Classes
"... Abstract. We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. This allows us to give finite firstorder axiomatizations of arithmetic and real analysis, and a presentation of arithmetic in deduction modulo that has a finite n ..."
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Abstract. We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. This allows us to give finite firstorder axiomatizations of arithmetic and real analysis, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to provide a simple and finite framework. 1
A Theory of Explicit Substitutions with Safe and Full Composition
 Logical Methods in Computer Science
"... Vol. 5 (3:1) 2009, pp. 1–29 ..."
Axiomatic Rewriting Theory I  A Diagrammatic Standardization Theorem
, 2001
"... Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation. ..."
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Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation.
Resource operators for λcalculus
 INFORM. AND COMPUT
, 2007
"... We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties ..."
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We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simplytyped terms, step by step simulation of βreduction and full composition.
An abstract machine based on linear logic and explicit substitutions
, 1997
"... a mis hermanas, Patricia y Paula, y a mi sobrino y ahijado, Nicol'as. Acknowledgements First of all, I would like to express my gratitude to my supervisor, Eike Ritter, for his wisdom, insight, uncountably many discussions, and invaluable friendship. I am indebted to my tutor, Valeria de Paiva, ..."
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a mis hermanas, Patricia y Paula, y a mi sobrino y ahijado, Nicol'as. Acknowledgements First of all, I would like to express my gratitude to my supervisor, Eike Ritter, for his wisdom, insight, uncountably many discussions, and invaluable friendship. I am indebted to my tutor, Valeria de Paiva, who also believed in me from the very beginning, encouraged me to work in this area, showed me the beauty of logic, and, above all, honoured me with her friendship. This thesis would not exist if it were not for their constant support. Thanks to my old friends, Cecilia C. Crespo, Santiago M. Peric'as, and, especially, Mat'ias Giovannini, for being always a wonderful critic of my work. Many thanks to Mathias Kegelmann for showing me the thrill of theorem proving; and to my former supervisor, Achim Jung, for introducing me to semantics.
Explaining the lazy Krivine machine using explicit substitution and addresses
 HIGHER ORDER AND SYMBOLIC COMPUTATION
, 2007
"... Abstract. In a previous paper, Benaissa, Lescanne, and Rose, have extended the weak lambdacalculus of explicit substitution λσw with addresses, so that it gives an account of the sharing implemented by lazy functional language interpreters. We show in this paper that their calculus, called λσ a w, ..."
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Abstract. In a previous paper, Benaissa, Lescanne, and Rose, have extended the weak lambdacalculus of explicit substitution λσw with addresses, so that it gives an account of the sharing implemented by lazy functional language interpreters. We show in this paper that their calculus, called λσ a w, fits well to the lazy Krivine machine, which describes the core of a lazy (callbyneed) functional programming language implementation. The lazy Krivine machine implements term evaluation sharing, that is essential for efficiency of such languages. The originality of our proof is that it gives a very detailed account of the implemented strategy.
Games and WeakHead Reduction for Classical PCF
 Proceedings of TLCA 97, LNCS 1210
, 1997
"... . We present a game model for classical PCF, a finite version of PCF extended by a catch/throw mechanism. This model is build from Edialogues, a kind of twoplayers game defined by Lorenzen. In the Edialogues for classical PCF, the strategies of the first player are isomorphic to the Bohm trees of ..."
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. We present a game model for classical PCF, a finite version of PCF extended by a catch/throw mechanism. This model is build from Edialogues, a kind of twoplayers game defined by Lorenzen. In the Edialogues for classical PCF, the strategies of the first player are isomorphic to the Bohm trees of the language. We define an interaction in Edialogues and show that it models the weakhead reduction in classical PCF. The interaction is a variant of Coquand's debate and the weakhead reduction is a variant of the reduction in Krivine's Abstract Machine. We then extend Edialogues to a kind of games similar to HylandOng's games. Interaction in these games also models weakhead reduction. In the intuitionistic case (i.e. without the catch/throw mechanism), the extended Edialogues are HylandOng's games where the innocence condition on strategies is now a rule. Our model for classical PCF is different from Ong's model of Parigot's lambdamucalculus. His model works by adding new moves t...