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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.
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.
HigherOrder Unification: A structural relation between Huet’s
"... method and the one based on explicit substitutions ⋆ ..."
A Flexible Framework for Visualisation of Computational Properties of General Explicit Substitutions Calculi
, 2010
"... SUBSEXPL is a system originally developed to visualise reductions, simplifications and normalisations in three important calculi of explicit substitutions and has been applied to understand and explain properties of these calculi and to compare the different styles of making explicit the substitutio ..."
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SUBSEXPL is a system originally developed to visualise reductions, simplifications and normalisations in three important calculi of explicit substitutions and has been applied to understand and explain properties of these calculi and to compare the different styles of making explicit the substitution operation in implementations of the λcalculus in de Bruijn notation. The system was developed in OCaml and now it can be executed inside the Emacs editor within a new mode which allows a very easy interaction. The use of special symbols makes its application very useful for students because the notation on the screen is as close as possible to that on the papers. In addition to λcalculus and explicit substitutions calculi in de Bruijn notation, now it is possible to work with the λcalculus with variables as names and with several calculi of explicit substitutions using also representation of variables with names. Moreover, in contrast to the original version of the system, that was restricted to three specific calculi of explicit substitution, the new version allows the inclusion of new calculi by giving as input their grammatical descriptions. SUBSEXPL has been used with success for teaching basic properties of the λcalculus and for illustrating the computational impact of selecting one kind of representation of variables (either names or indices) and a specific style of making explicit substitutions in real implementations based on the λcalculus. Keywords: Term rewriting systems, calculi of explicit substitutions, λcalculi