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13
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.
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,...
Bridging de Bruijn indices and variable names in explicit substitutions calculi
 Logic Journal of the Interest Group of Pure and Applied Logic (IGPL
, 1996
"... Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ffconversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renew ..."
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Cited by 9 (6 self)
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Calculi of explicit substitutions have almost always been presented using de Bruijn indices with the aim of avoiding ffconversion and being as close to machines as possible. De Bruijn indices however, though very suitable for the machine, are difficult to human users. This is the reason for a renewed interest in systems of explicit substitutions using variable names. Formal systems of explicit substitutions using variable names is a new area however and we believe, it should not develop without being welltied to existing work on explicit substitutions. The aim of this paper is to establish a bridge between explicit substitutions using de Bruijn indices and using variable names. In our aim to do so, we provide the tcalculus: a calculus `a la de Bruijn which can be translated into a calculus with explicit substitutions written with variables names. We present explicitly this translation and use it to obtain preservation of strong normalisation for t. Moreover, we show several prope...
Extending a lambdacalculus with Explicit Substitution which Preserves Strong Normalisation into a Confluent Calculus on Open Terms
, 1993
"... The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oecalculus. In (Kamareddine & R'ios, 1995a), we extended the calculus with explicit substitutions by turning de Bruijn's metaoperators into objectoperators offering a style o ..."
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Cited by 7 (0 self)
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The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oecalculus. In (Kamareddine & R'ios, 1995a), we extended the calculus with explicit substitutions by turning de Bruijn's metaoperators into objectoperators offering a style of explicit substitution that differs from that of oe. The resulting calculus, s, remains as close as possible to the calculus from an intuitive point of view and, while preserving strong normalisation (Kamareddine & R'ios, 1995a), is extended in this paper to a confluent calculus on open terms: the secaculus. Since the establishment of these results, another calculus, i, came into being in (Mu~noz Hurtado, 1996) which preserves strong normalisation and is itself confluent on open terms. However, we believe that se still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fireduction, whereas i is not. To ...
Explicit Substitutions and Reducibility
 Journal of Logic and Computation
, 2001
"... . We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simplytyped callbyname ~calculus enriched with op ..."
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Cited by 7 (1 self)
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. We consider reducibility sets dened not by induction on types but by induction on sequents as a tool to prove strong normalization of systems with explicit substitution. To illustrate this point, we give a proof of strong normalization (SN) for simplytyped callbyname ~calculus enriched with operators of explicit unary substitutions. The ~calculus, dened by Curien & Herbelin, is a variant of calculus with a let operator that exhibits symmetries such as terms/contexts and callbyname /callbyvalue reduction. The ~calculus embeds various standard calculi (and Gentzen's style sequent calculi too) and as an application we derive the strong normalization of Parigot's simplytyped calculus with explicit substitution. Introduction Explicit substitution in calculus The traditional theory of calculus relies on reduction, that is the capture by a function of its argument followed by the process of substituting this argument to the places where it is used. The ...
The Confluence of the ...Calculus Via a Generalized Interpretation Method
, 1996
"... The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oecalculus. In [KR95a], we extended the calculus with explicit substitutions by turning de Bruijn's metaoperators into objectoperators offering a style of explicit substitut ..."
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Cited by 4 (2 self)
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The last fifteen years have seen an explosion in work on explicit substitution, most of which is done in the style of the oecalculus. In [KR95a], we extended the calculus with explicit substitutions by turning de Bruijn's metaoperators into objectoperators offering a style of explicit substitution that differs from that of oe. The resulting calculus, s, remains as close as possible to the calculus from an intuitive point of view and, while preserving strong normalisation ([KR95a]), is extended in this paper to a confluent calculus on open terms: the s e caculus. Since the establishment of the results of this paper 1 , another calculus, i, came into being in [MH95] which preserves strong normalisation and is itself confluent on open terms. However, we believe that s e still deserves attention because, while offering a new style to work with explicit substitutions, it is able to simulate one step of classical fireduction, whereas i is not. To prove confluence we introduce a ge...
Systems for open terms: An overview
, 2001
"... In this paper we make an overview of some existing systems of open (incomplete) terms including ALF, Typelab, OLEG, L, Automath, c and s e. ..."
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Cited by 1 (1 self)
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In this paper we make an overview of some existing systems of open (incomplete) terms including ALF, Typelab, OLEG, L, Automath, c and s e.
Bridging the lambda sigma and lambda sStyles of Explicit Substitutions
, 1997
"... . We present the ! and !e calculi, the twosorted (term and substitution) versions of the s (cf. [KR95a]) and se (cf. [KR96a]) calculi, respectively. We establish an isomorphism between the scalculus and the term restriction of the !calculus, which extends to an isomorphism between se and the te ..."
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. We present the ! and !e calculi, the twosorted (term and substitution) versions of the s (cf. [KR95a]) and se (cf. [KR96a]) calculi, respectively. We establish an isomorphism between the scalculus and the term restriction of the !calculus, which extends to an isomorphism between se and the term restriction of !e . Since the ! and !e calculi are given in the style of the oecalculus (cf. [ACCL91]) they bridge calculi between s and oe and between se and oe and thus we are able to better understand one calculus in terms of the other. We improve our knowledge on the open problem of strong normalisation (SN) of the associated calculus of substitutions se by showing SN for two subcalculi (we use the isomorphism with !e for the proof of SN of one of them). Finally, we present typed versions of all the calculi and check that the above mentioned isomorphism preserves types. As a consequence, the !calculus is a calculus in the oestyle that simulates one step fireduction, is confluent ...
Generalized BetaReduction and Explicit Substitutions
, 1996
"... Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been ..."
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Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently which still has many open problems. Due to this reason, the properties of a calculus combining both generalised reduction and explicit substitutions have never been studied. This paper presents such a calculus sg and shows that it is a desirable extension of the calculus. In particular, we show that sg preserves strong normalisation, is sound and it simulates classical fireduction. Furthermore, we study the simply typed calculus extended with both generalised reduction and explicit substitution and show that welltyped terms are strongly normalising and that other properties such as subtyping and subject reduction hold. 1 Introduction 1.1 The calculus with generalised reduction In (( x : y :N)P )Q, the function starting with x and the argument P result in the redex ( x : y :N)P which when contracted will turn the function starting with y and Q i...