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Preservation of Strong Normalisation in Named Lambda Calculi with Explicit Substitution and Garbage Collection
 IN CSN95: COMPUTER SCIENCE IN THE NETHERLANDS
, 1995
"... In this paper we introduce and study a new lambdacalculus with explicit substitution, lambdaxgc, which has two distinguishing features: first, it retains the use of traditional variable names, specifying terms modulo renaming; this simplifies the reduction system. Second, it includes reduction rul ..."
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Cited by 65 (7 self)
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In this paper we introduce and study a new lambdacalculus with explicit substitution, lambdaxgc, which has two distinguishing features: first, it retains the use of traditional variable names, specifying terms modulo renaming; this simplifies the reduction system. Second, it includes reduction rules for explicit garbage collection; this simplifies several proofs. We show that lambdaxgc is a conservative extension which preserves strong normalisation (PSN) of the untyped lambdacalculus. The result is obtained in a modular way by first proving it for garbagefree reduction and then extending to `reductions in garbage'. This provides insight into the counterexample to PSN for lambdasigma of Melliès (1995); we exploit the abstract nature of lambdaxgc to show how PSN is in conflict with any reasonable substitution composition rule (except for trivial composition rules of which we mention one). Key words: lambda calculus, explicit substitution, strong normalisation, garbage collection.
lambdacalculi with explicit substitutions and composition which preserve beta strong normalization (Extended Abstract)
, 1996
"... ) Maria C. F. Ferreira 1 and Delia Kesner 2 and Laurence Puel 2 1 Dep. de Inform'atica, Fac. de Ciencias e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal, cf@fct.unl.pt. 2 CNRS & Lab. de Rech. en Informatique, Bat 490, Univ. de ParisSud, 91405 Orsay Cede ..."
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Cited by 27 (3 self)
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) Maria C. F. Ferreira 1 and Delia Kesner 2 and Laurence Puel 2 1 Dep. de Inform'atica, Fac. de Ciencias e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal, cf@fct.unl.pt. 2 CNRS & Lab. de Rech. en Informatique, Bat 490, Univ. de ParisSud, 91405 Orsay Cedex, France, fkesner,puelg@lri.fr. Abstract. We study preservation of fistrong normalization by d and dn , two confluent calculi with explicit substitutions defined in [10]; the particularity of these calculi is that both have a composition operator for substitutions. We develop an abstract simulation technique allowing to reduce preservation of fistrong normalization of one calculus to that of another one, and apply said technique to reduce preservation of fistrong normalization of d and dn to that of f , another calculus having no composition operator. Then, preservation of fistrong normalization of f is shown using the same technique as in [2]. As a consequence, d and dn become the fir...
Confluence Properties of Extensional and NonExtensional lambdaCalculi with Explicit Substitutions (Extended Abstract)
 in Proceedings of the Seventh International Conference on Rewriting Techniques and Applications
, 1996
"... ) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For ..."
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Cited by 22 (5 self)
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) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many wellknown calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and nonextensional versions. 1 Introduction The #calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #reduction. The...
Strong Normalization of Explicit Substitutions via Cut Elimination in Proof Nets
, 1997
"... In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #xcalculus [30, 29] is strongly normalizing, as ..."
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Cited by 22 (4 self)
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In this paper, we show the correspondence existing between normalization in calculi with explicit substitution and cut elimination in sequent calculus for Linear Logic, via Proof Nets. This correspondence allows us to prove that a typed version of the #xcalculus [30, 29] is strongly normalizing, as well as of all the calculi isomorphic to it such as # # [24], # s [19], # d [21], and # f [11]. In order to achieve this result, we introduce a new notion of reduction in Proof Nets: this extended reduction is still confluent and strongly normalizing, and is of interest of its own, as it correspond to more identifications of proofs in Linear Logic that differ by inessential details. These results show that calculi with explicit substitutions are really an intermediate formalism between lambda calculus and proof nets, and suggest a completely new way to look at the problems still open in the field of explicit substitutions.
Preservation of Strong Normalisation for Explicit Substitution
, 1995
"... this paper is different and has been invented independently of the proofs in [Kamareddine & Rios 95] and [BBLR 95]. We show by means of a counterexample that an extension of exp with certain interaction between substitutions does not preserve strong normalisation. In appendix A we use a more common ..."
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Cited by 15 (2 self)
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this paper is different and has been invented independently of the proofs in [Kamareddine & Rios 95] and [BBLR 95]. We show by means of a counterexample that an extension of exp with certain interaction between substitutions does not preserve strong normalisation. In appendix A we use a more common notation trying to determine the borderline between preservation of strong normalisation and interaction of substitutions. 2 The calculus
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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Cited by 15 (0 self)
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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
Calculi of Generalised βReduction and Explicit Substitutions: The TypeFree and Simply Typed Versions
, 1998
"... Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substit ..."
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Cited by 14 (7 self)
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Extending the λcalculus with either explicit substitution or generalized reduction has been the subject of extensive research recently, and still has many open problems. This paper is the first investigation into the properties of a calculus combining both generalized reduction and explicit substitutions. We present a calculus, gs, that combines a calculus of explicit substitution, s, and a calculus with generalized reduction, g. We believe that gs is a useful extension of the  calculus, because it allows postponement of work in two different but complementary ways. Moreover, gs (and also s) satisfies properties desirable for calculi of explicit substitutions and generalized reductions. In particular, we show that gs preserves strong normalization, is a conservative extension of g, and simulates fireduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs, and show that welltyped terms are strongly normalizing and that other properties,...
Confluence of Extensional and NonExtensional λcalculi with Explicit Substitutions
 Theoretical Computer Science
"... This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. O ..."
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Cited by 12 (2 self)
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This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many wellknown calculi such as oe , oe * , OE , s , AE , f , d and dn . Keywords: functional programming, calculi, explicit substitutions, confluence, extensionality. 1 Introduction The calculus is a convenient framework to study functional programming, where the evaluation process is modeled by fireduction. The main mechanism used to perform fireduction is substitution, which consists of the replacement of formal parameters by actual arguments. The correctness of substitution is guaranteed by a systematic renaming of bound variables, inconvenient which can be simply avoided in the calculus `a la de Bruijn by using natur...
Comparing and Implementing Calculi of Explicit Substitutions with Eta Reduction
 Annals of Pure and Applied Logic
, 2005
"... The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that e ..."
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Cited by 10 (8 self)
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The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the etareductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of betacontraction.