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42
Term Rewriting Systems
, 1992
"... Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstra ..."
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Cited by 578 (17 self)
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Term Rewriting Systems play an important role in various areas, such as abstract data type specifications, implementations of functional programming languages and automated deduction. In this chapter we introduce several of the basic comcepts and facts for TRS's. Specifically, we discuss Abstract Reduction Systems
Explicit substitutions
, 1996
"... The λσcalculus is a refinement of the λcalculus where substitutions are manipulated explicitly. The λσcalculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λcalculus and concrete implementatio ..."
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Cited by 410 (11 self)
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The λσcalculus is a refinement of the λcalculus where substitutions are manipulated explicitly. The λσcalculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λcalculus and concrete implementations.
Confluence properties of Weak and Strong Calculi of Explicit Substitutions
 JOURNAL OF THE ACM
, 1996
"... Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence prope ..."
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Cited by 130 (7 self)
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Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oecalculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oecalculus, named the Envcalculus in [23], called here the confluent oecalculus.
A lambdacalculus à la de Bruijn with explicit substitutions
, 1995
"... The aim of this paper is to present the scalculus which is a very simple calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of terms. We shall prove strong normalisation of the corresponding calculus of substitution by tra ..."
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Cited by 90 (35 self)
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The aim of this paper is to present the scalculus which is a very simple calculus with explicit substitutions and to prove its confluence on closed terms and the preservation of strong normalisation of terms. We shall prove strong normalisation of the corresponding calculus of substitution by translating it into the oecalculus [ACCL91], and therefore the relation between both calculi will be made explicit. The confluence of the scalculus is obtained by the "interpretation method" ([Har89], [CHL92]). The proof of the preservation of normalisation follows the lines of an analogous result for the AEcalculus (cf. [BBLRD95]). The relation between s and AE is also studied.
A Notation for Lambda Terms I: A Generalization of Environments
 THEORETICAL COMPUTER SCIENCE
, 1994
"... A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms ..."
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Cited by 33 (12 self)
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A notation for lambda terms is described that is useful in contexts where the intensions of these terms need to be manipulated. This notation uses the scheme of de Bruijn for eliminating variable names, thus obviating ffconversion in comparing terms. This notation also provides for a class of terms that can encode other terms together with substitutions to be performed on them. The notion of an environment is used to realize this `delaying' of substitutions. The precise mechanism employed here is, however, more complex than the usual environment mechanism because it has to support the ability to examine subterms embedded under abstractions. The representation presented permits a ficontraction to be realized via an atomic step that generates a substitution and associated steps that percolate this substitution over the structure of a term. The operations on terms that are described also include ones for combining substitutions so that they might be performed simultaneously. Our notatio...
Explicit Substitutions with de Bruijn’s Levels
 in Rewriting Techniques and Applications, 6th International Conference, Lecture Notes in Computer Science 914
, 1995
"... In the introduction of [7], Curry writes that substitution is the main issue in logic and that λcalculus does not properly answer the problem because substitutions are outside the calculus. He pleads in favor of combinatory logic which offers a full treatment to substitutions through its use of a f ..."
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Cited by 32 (3 self)
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In the introduction of [7], Curry writes that substitution is the main issue in logic and that λcalculus does not properly answer the problem because substitutions are outside the calculus. He pleads in favor of combinatory logic which offers a full treatment to substitutions through its use of a first order rewrite system in
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,...
The scalculus: its typed and its extended versions
, 1995
"... We present in this paper the simply typed version of the scalculus (cf. [KR95]) and prove the strong normalisation of the well typed terms. We also present an extension of the scalculus: the s ecalculus and prove its local con
uence on open terms and the weak normalisation of its corresponding ca ..."
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Cited by 16 (10 self)
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We present in this paper the simply typed version of the scalculus (cf. [KR95]) and prove the strong normalisation of the well typed terms. We also present an extension of the scalculus: the s ecalculus and prove its local con
uence on open terms and the weak normalisation of its corresponding calculus of substitutions s e. The strong normalisation of s e is still an open problem to challenge the rewriting community.
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...