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Explicit Substitution: on the Edge of Strong Normalization
 Theoretical Computer Science
, 1997
"... We use the Recursive Path Ordering (RPO) technique of semantic labelling to show the Preservation of Strong Normalization (PSN) property for several calculi of explicit substitution. Preservation of Strong Normalization states that if a term M is strongly normalizing under ordinary fireduction (us ..."
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Cited by 32 (2 self)
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We use the Recursive Path Ordering (RPO) technique of semantic labelling to show the Preservation of Strong Normalization (PSN) property for several calculi of explicit substitution. Preservation of Strong Normalization states that if a term M is strongly normalizing under ordinary fireduction (using `global' substitutions), then it is strongly normalizing if the substitution is made explicit (`local'). There are different ways of making global substitution explicit and PSN is a quite natural and desirable property for the explicit substitution calculus. Our method for proving PSN is very general and applies to several known systems of explicit substitutions, both with named variables and with De Bruijn indices: AE of Lescanne et al., s of Kamareddine and R'ios and x of Rose and Bloo. We also look at two small extensions of the explicit substitution calculus that allow to permute substitutions. For one of these extensions PSN fails (using the counterexample in [Melli`es 95]). For the...
Confluence and Preservation of Strong Normalisation in an Explicit Substitutions Calculus
, 1996
"... Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The oe calculus of explicit s ..."
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Cited by 20 (4 self)
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Explicit substitutions calculi are formal systems that implement fireduction by means of an internal substitution operator. In that calculi it is possible to delay the application of a substitution to a term or to consider terms with partially applied substitutions. The oe calculus of explicit substitutions, proposed by Abadi, Cardelli, Curien andL evy, is a firstorder rewriting system that implements substitution and renaming mechanism of calculus. However, oe does not preserve strong normalisation of calculus and it is not a confluent system. Typed variants of oe without composition are strongly normalising but not confluent, while variants with composition are confluent but do not preserve strong normalisation. Neither of them enjoys both properties. In this paper we propose the i calculus. This is, as far as we know, the first confluent calculus of explicit substitutions that preserves strong normalisation. 1. Explicit substitutions The calculus is a higherorder theor...
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 16 (8 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 10 (7 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...
Relating the λσ and λsstyles of explicit substitutions
 JOURNAL OF LOGIC AND COMPUTATION
, 2000
"... The aim of this article is to compare two styles of Explicit Substitutions: the  and sstyles. We start by introducing a criterion of adequacy to simulate reduction in calculi of explicit substitutions and we apply it to several calculi: , * , , s, t and u. The latter is presented here for the rs ..."
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Cited by 8 (4 self)
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The aim of this article is to compare two styles of Explicit Substitutions: the  and sstyles. We start by introducing a criterion of adequacy to simulate reduction in calculi of explicit substitutions and we apply it to several calculi: , * , , s, t and u. The latter is presented here for the rst time and may be considered as an adequate variant of s. By doing so, we establish that calculi a la s are usually more adequate at simulatingreduction than calculi in the style. In fact, we prove that t is more adequate than and that u is more adequate than , * and s. We also give counterexamples to show that all other comparisons are impossible according to our criterion. Our next step consists in presenting the ! and !e calculi, the twosorted (term and substitution) versions of the s (cf. [KR95]) and se (cf. [KR97]) 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 calculus (cf. [ACCL91]) they are bridge calculi between s and and between se and and thus we are able to better understand one calculus in terms of the other. 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 style that has the following properties a..g: a) ! simulates one step reduction, b) ! is con
uent (on closed terms), c) ! preserves strong normalisation, d) !'s associated calculus of substitutions is SN, e) the simply typed ! calculus is SN, f) the !calculus possesses an extension !e that is con uent on open terms (terms with eventual metavariables of sort term only), and g) the simply typed !e calculus is weakly normalising (on open term). As far as we know, the !calculus is the rst calculus in the style that has all those properties a..g. However, the open problem of the SN of the associated calculus of substitution of !e remains unsolved and like in the case of , and se, !e does not have PSN.
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 offe ..."
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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 ...
Proof Representation in Type Theory: State of the Art
, 1996
"... In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to repres ..."
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Cited by 6 (0 self)
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In the frame of intuitionistic logic and type theory, it is well known that there is an isomorphism between types and propositions; the CurryHoward Isomorphism. However, it is less clear the relation between terms construction and proofs development. The main difficulty arises when we try to represent incomplete proofs as terms describing a state of knowledge where some part of the proof is built, but another part remains to be built. The pieces of proof terms that are unknown are called placesholders. We present a theoretical approach to placeholders in type theory. In this approach placeholders are represented by metavariables and terms are built incrementally by instantiation of metavariables. We show how an appropriate extension to typed calculus with explicit substitutions and explicit typing of metavariables allows to identify terms construction and proofs development activities.
Pure Type Systems with Explicit Substitution
, 2000
"... We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is ..."
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Cited by 6 (0 self)
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We define an extension of pure type systems with explicit substitution. It is shown that the type systems with explicit substitution are strongly normalizing iff their ordinary counterparts are. Subject reduction is shown to fail in general but a weaker  still useful  subject reduction property is established. A more complicated extension is proposed for which subject reduction does hold in general.
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 subs ..."
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Cited by 5 (3 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...
Calculi of generalised #reduction and explicit substitutions: The type free and simply typed versions
 J. Funct. Logic Programming
, 1998
"... Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently and still has many open problems. This paper is the rst investigation into the properties of a calculus combining both generalised reduction and explicit substitutio ..."
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Cited by 3 (3 self)
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Extending the calculus with either explicit substitution or generalised reduction has been the subject of extensive research recently and still has many open problems. This paper is the rst investigation into the properties of a calculus combining both generalised 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 postponment of work in two dierent but complementary ways. Moreover, gs (and also s) satises desirable properties of calculi of explicit substitutions and generalised reductions. In particular, we show that gs preserves strong normalisation, is a conservative extension of g, and simulates reduction of g and the classical calculus. Furthermore, we study the simply typed versions of s and gs and show that well typed terms are strongly normalising and that other properties such as typing of subterms and subject reduction hold. Our proof of the preservation of strong normalisation (PSN) is based on the minimal derivation method. It is however much simpler because we prove the commutation of arbitrary internal and external reductions. Moreover, we use one proof to show both the preservation of strong normalisation in s and the preservation of gstrong normalisation in gs. We remark that the technique of these proofs is not suitable for calculi without explicit substitutions (e.g. the preservation of strong normalisation in g requires a dierent technique). 1