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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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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.
Pattern Matching as Cut Elimination
 In Logic in Computer Science
, 1999
"... We present typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the CurryHoward Isomorphism, ..."
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We present typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for minimal logic. Our calculus is inspired by the CurryHoward Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to sequent proof normalization performed by cut elimination. The calculus enjoys subject reduction, confluence, preservation of strong normalization w.r.t a system with metalevel substitutions, and strong normalization for welltyped terms, and, as a consequence, can be seen as an implementation calculus for functional formalisms using metalevel operations for pattern matching and substitutions.
From HigherOrder to FirstOrder Rewriting
 In Proceedings of the 12th International Conference on Rewriting Techniques and Applications (RTA’01
, 2001
"... . We show how higherorder rewriting may be encoded into ..."
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. We show how higherorder rewriting may be encoded into
SUBSEXPL: A Framework for Simulating and Comparing Explicit Substitutions Calculi A Tutorial
, 2005
"... In this paper we present a framework, called SUBSEXPL, for simulating and comparing explicit substitutions calculi. This framework was developed in Ocaml, a language of the ML family, and it allows the manipulation of expressions of the λcalculus and of several styles of explicit substitutions calc ..."
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In this paper we present a framework, called SUBSEXPL, for simulating and comparing explicit substitutions calculi. This framework was developed in Ocaml, a language of the ML family, and it allows the manipulation of expressions of the λcalculus and of several styles of explicit substitutions calculi. Applications of this framework include: the visualisation of the contractions of the λcalculus, and of guided onestep reductions and normalisation via each of the associated substitution calculi. Many useful facilities are available: reductions can be easily recorded and stored into files, latex output and useful examples for dealing with, among other things, arithmetic operations and computational operators such as conditionals and repetitions in the λcalculus. The current implementation of SUBSEXPL includes treatment of three different calculi of explicit substitutions: the λσ, the λse and the suspension calculus; other explicit substitutions calculi can be easily incorporated into the system. An implementation of the ηreduction is provided for each of these explicit substitutions calculi. This system has been of great help for systematically comparing explicit substitutions calculi, as well as for understanding properties of explicit substitutions such as the Preservation of Strong Normalisation. In addition, it has been used for teaching basic properties of the λcalculus such as: computational adequacy, the importance of de Bruijn’s notation and of making explicit substitutions in real implementations based on the λcalculus. Keywords: λCalculus, Explicit Substitutions, Visualisation of β and ηContraction and Normalisation. 1
SUBSEXPL: A Tool for Simulating and Comparing Explicit Substitutions Calculi ⋆
"... Abstract. We present the system SUBSEXPL used for simulating and comparing explicit substitutions calculi. The system allows the manipulation of expressions of the λcalculus and of three different styles of explicit substitutions: the λσ, the λse and the suspension calculus. Implementations of the ..."
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Abstract. We present the system SUBSEXPL used for simulating and comparing explicit substitutions calculi. The system allows the manipulation of expressions of the λcalculus and of three different styles of explicit substitutions: the λσ, the λse and the suspension calculus. Implementations of the ηreduction are provided for each calculi. Other explicit substitutions calculi can be incorporated into the system easily due to its modular structure. Its applications include: the visualisation of the contractions of the λcalculus, and of guided onestep reductions as well as normalisation via each of the associated substitution calculi. Many useful facilities are available: reductions can be easily recorded and stored into files or Latex outputs and several examples for dealing with arithmetic operations and computational operators such as conditionals and repetitions in the λcalculus are available. The system has been of great help for systematically comparing explicit substitutions calculi, as well as for understanding properties of explicit substitutions such as the Preservation of Strong Normalisation. In addition, it has been used for teaching basic properties of the λcalculus such as: computational adequacy, the importance of de Bruijn’s notation and of making explicit substitutions in real implementations.
Confluence of PatternBased Calculi
"... Abstract Different pattern calculi integrate the functional mechanisms from the λcalculus and the matching capabilities from rewriting. Several approaches are used to obtain the confluence but in practice the proof methods share the same structure and each variation on the way patternabstractions a ..."
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Abstract Different pattern calculi integrate the functional mechanisms from the λcalculus and the matching capabilities from rewriting. Several approaches are used to obtain the confluence but in practice the proof methods share the same structure and each variation on the way patternabstractions are applied needs another proof of confluence. We propose here a generic confluence proof where the way patternabstractions are applied is axiomatized. Intuitively, the conditions guarantee that the matching is stable by substitution and by reduction. We show that our approach directly applies to different pattern calculi, namely the lambda calculus with patterns, the pure pattern calculus and the rewriting calculus. We also characterize a class of matching algorithms and consequently of patterncalculi that are not confluent.
A Leftlinear Variant of λσ
, 1997
"... In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcal ..."
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In this paper we consider calculi of explicit substitutions that admit open expressions, i.e. expressions with metavariables. In particular, we propose a variant of the oecalculus that we call L . For this calculus and its simplytyped version, we study its metatheoretical properties. The Lcalculus enjoys the same general characteristics as oe, i.e. a simple and finitary firstorder presentation, confluent on expressions with metavariables of terms and weakly normalizing on typed expressions. Moreover, L does not have the nonleftlinear surjective pairing rule of oe which raises technical problems in some frameworks.
Explicit Substitutions Calculi with One Step Etareduction Decided Explicitly
"... It has long been argued that the notion of substitution in the λcalculus needs to be made explicit. This resulted in many calculi have been developed in which the computational steps of the substitution operation involved in βcontractions have been atomised. In contrast to the great variety of dev ..."
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It has long been argued that the notion of substitution in the λcalculus needs to be made explicit. This resulted in many calculi have been developed in which the computational steps of the substitution operation involved in βcontractions have been atomised. In contrast to the great variety of developments for making explicit formalisations of the Beta rule, less work has been done for giving explicit definitions of the conditional Eta rule. In this paper constructive Eta rules are proposed for both the λσ and the λsecalculi of explicit substitutions. Our results can be summarised as follows: 1) we introduce constructive and explicit definitions of the Eta rule in the λσ and the λsecalculi, 2) we prove that these definitions are correct and preserve basic properties such as subject reduction. In particular, we show that the explicit definitions of the eta rules coincide with the Eta rule for pure λterms and that moreover, their application is decidable in the sense that Eta redices are effectively detected (and contracted). The formalisation of these Eta rules involves the development of specific calculi for explicitly checking the condition of the proposed Eta rules while constructing the Eta contractum.