M. Abadi, L. Cardelli, P.-L. Curien, and J.-J.L'evy. Explicit substitutions. In ACM Symp. on Principles of Programming Languages, 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....gets stuck on a free variable if the initial expression is closed. 7. RELATED WORK Cyclic explicit substitutions Rose [14] defines a calculus with mutually recursive definitions, where recursion is introduced by explicit cyclic substitutions, extending the explicit substitutions of Abadi et al. [1]. Instead of lifting recursive bindings to the top of terms like we do, Rose s calculus pushes them inside terms, as usual with explicit substitutions. This results in the loss of sharing information. Any term is allowed in recursive bindings, but inside a recursive binding, when computing a ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit substitutions. J. Func. Progr., 1(4):375--416, 1991.

....the corresponding reductions. We also extend the usual notion of fi reduction, an extension which is an evident consequence of local substitution. The framework for the description of terms, as explained before, is very adequate for this matter. Finally in section 4, we interpret the approach of [1] in our framework concluding that ours is more general. In fact, we believe that our account of substitution is the most refined and general one to date. 2 The Calculus In this section, we start by introducing the reader to the lambda calculus augmented with de Bruijn s indices. We will explain ....

....We note that our updating is less complicated, but also less general than in the original treatment of de Bruijn indices (see [5] where the usual fi reduction is applied (the global relation) and substitution is not presented as a step wise process. In explicit substitution procedures as in [1], the more general, but complicated update functions are used. Our loss of generality has the following cause. A oe item (toe ) is supposed to be cut off from the rest of the term. Variables in t may have lost their reference value; in case a variable x in t is bound by a outside t, then ....

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M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. L'evy, "Explicit Substitutions", Functional Programming 1 (4) (1991) 375-416.

....cases to using implicit rather than explicit substitution. Implementations of the calculus provide their own explicit substitution procedures as in HOL [GM 93] Nuprl [Con 86] and Authomath [NGdV 94] Furthermore, research on theories of explicit substitution has been striving lately ( HL 89] ACCL 91] KN 93] Mel 95] BBLR 95] and [KR 95] In this paper, we extend the calculus of [KN 93] which is influenced by Authomath) giving B, a calculus which uses de Bruijn indices and where reduction and substitution are step wise and explicit. The species of variable names is cultivated and ....

....the classical one does not. Work on explicit substitution with de Bruijn indices has been first done in depth by Curien (in his PhD thesis, 1983) and was based on categorical combinators. Curien s original work was pursued by applications such as the categorical abstract machine of [CCM 87] ACCL 91] provides an algebraic syntax and semantics for explicit substitution where de Bruijn s indices are used. The connection with the classical calculus is not investigated. HL 89] proposes confluent systems of substitution based on the study of categorical combinators and [Field 90] provides an ....

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Abadi, M., Cardelli, L., Curien, P.-L. and L'evy, J.-J., (1991) Explicit substitutions, Functional Programming 1 (4), 375-416.

....as a great improvement over Definition 1.3. But just imagine that in the calculus you had not only and ffi as internal operators but also oe for substitution, for typing and so on. In fact, internalising substitution (i.e. making it explicit) has been a topic of research in the last decade (see [1], 8] 7] 9] Now, internalising extra operators means that in classical notation, in Definition 1.3, two extra rules are added for each new operator. In item notation on the other hand, Definition 2.3 does not depend on the number of operators. Simply, the set of operators to which belongs ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. L'evy, "Explicit Substitutions", Functional Programming 1 (4) (1991) 375-416.

....using l would require higher kinds, complicating the semantic model. We avoid this complication by having quantifiers implicitly bind de Bruijn indices [6] represented as n instead of named variables in the type terms. The above term, for example, looks like # #F 0#. We use explicit substitution [1] rules given in Figure 11 to manipulate the terms. 4 For reasoning about recursive types, we need to know which types are contractive (as in the ideal model [7] or the indexed model [5] The type a # ######3#a# is not meaningful because the operator ##### is not contractive, but list # ### # ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit substitutions. In Seventeenth Annual ACM Symp. on Principles of Prog. Languages, pages 31--46. ACM Press, Jan 1990.

....kinds. Since our current goal in the FPCC project is to be able to compile core ML, it is enough for us to restrict ourselves to first order kinds. We have type functions arising out of the uses of type operators like rec and and we use de Bruijn indices [7] with explicit substitutions [1] to manage applications of arguments to these type functions. Our de Bruijn numbering starts at 0. We do not use an explicit # anywhere; in our calculus, the arity of a type expression (possibly with free variables) is known from the context in which it appears, so the # is unnecessary. ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit substitutions. In Seventeenth Annual ACM Symp. on Principles of Prog. Languages, pages 31--46. ACM Press, Jan 1990.

....in the call by value calculus that generalizes superdevelopments a la Aczel. 1 Introduction Implementing the calculus and, in general, functional languages has long been the subject of research. One line of research, exemplified notably by calculi of explicit substitutions such as [1], is to find first order rewrite systems, i.e. without binders such as that create opportunities for bugs in implementations. While implements reduction correctly, it is not terminating in the simply typed case [12] and is only confluent on so called semi closed terms [14] The question of ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. L evy. Explicit substitutions. In Proceedings of the 17th Annual ACM Symposium on Principles of Programming Languages, pages 31--46, San Francisco, California, January 1990.

....nicely captures the operational notion of sharing; it corresponds with the implementation technique of preventing work duplication by updating a closure with its result. Applications The Beta rule expresses the fundamental axiom of the # calculus, # reduction, by means of explicit substitution [1]: rather than having a substitution operation, substitutions are expressed in the language itself. We do this by adding the argument to the let environment, and then evaluating the body of the rule. All initial terms are assumed to be closed, i.e. contain no free variables. As a consequence there ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit substitutions. In 17th ACM Symp. on Principles of Programming Languages (POPL'90), pages 31--46, San Francisco, California, January 1990. 4.1

.... xy : A z : B:x. Type abstractor ( is written ( e.g. x; y : A; z : B)x denotes xy : A z : B:x. 3 calculus We only introduce here the notions and notations which are useful in this work. For a general presentation of explicit substitution calculi, the reader is referred for example, to [1,13]. Syntax For all x 2 V and f 2 F : terms t : x j f(t; t) j ft; tg j [t] t) j t t j thsi substitutions s : ID j j j t:s j s s In the term syntax, t u denotes a rewrite rule (a.k.a. abstraction) t] u) represents the application of t on u, the application of substitution ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit substitutions. In ACM, editor, Conf. Rec. 17th Symp. POPL, pages 31-46, 1990.

....nicely captures the operational notion of sharing; it corresponds with the implementation technique of preventing work duplication by updating a closure with its result. Applications The Beta rule expresses the fundamental axiom of the calculus, reduction, by means of explicit substitution [1]: rather than having a substitution operation, substitutions are expressed in the language itself. We do this by adding the argument to the let environment, and then evaluating the body of the rule. All initial terms are assumed to be closed, i.e. contain no free variables. As a consequence there ....

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit substitutions. In 17th ACM Symp. on Principles of Programming Languages (POPL'90), pages 31-46, San Francisco, California, January 1990.

....presented in this paper focuses on the basics of the calculus. It is easy to extend the ASF SDF specification to cover other reduction strategies, to extend to a typed calculus, to translate to De Bruijn sequences [Bru72] to experiment with the explicit substitutions in the oe calculus [ACCL90] and so on. Again, having specifications of these immediately provides one with tools to experiment with them. Our specification shows that from a purely formal definition, inter active environments can be generated. We realize that the calculus is not a very complex example, which is ....

M. Abadi, L. Cardelli, P.-L. Currien, and J.-J. L'evy. Explicit substitutions. In Proceedings of the 17th conference on the Principles of Programming Languages, 1990.

....steps t 0 t 1 : t n is also written t 0 t n (n 0) Subterms are characterized by occurrences (paths) which are either equal to [ for the entire term or to a sequence of integers (the branches) n 1 ; nm ] m 0) representing the access path to the subterm. The occurrence [1, 2] 1 ASF SDF is the name of the formalism used to specify programming languages; it originated from combining the Algebraic Specification Formalism ASF and the Syntax Definition Formalism SDF. 4 denotes the second son of the first son of the root, i.e. for term f(g(a; b) c) it denotes subterm b. ....

....indicate what remains of a term during rewriting. In the orthogonal case, the org function always yields a set consisting of at most one element. 1.4 Example As an example, Figure 3 shows a reduction step of a typical type checker. The redex tc(E 1 E 2 ) is contracted at occurrence [1] in the given context. Following the definition of the function org just given, origins for nodes within the context are mapped onto themselves. The context positions (on top of or next to the redex) are [ 2] and [2,1] denoting conc , undeclared var , and foo . For these, we have, org( ....

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M. Abadi, L. Cardelli, P.-L. Currien, and J.-J. L'evy. Explicit substitutions. In Proceedings of the 17th conference on Principles of Programming Languages, pages 31--46, 1990.

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M. Abadi, L. Cardelli, P.-L. Curien, and J.-J.L'evy. Explicit substitutions. In ACM Symp. on Principles of Programming Languages, 1990.

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Abadi, M., Cardelli, L., Curien, P.-L., and L evy, J.-J. 1990. Explicit substitutions. In Symposium on Principles of Programming Languages, POPL'90. ACM, San Francisco, California, 31--46.

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M. Abadi, L. Cardelli, P.-L. Curien, J.-J. L evy. Explicit substitutions. Proceedings of Principle Of Programming Languages, 1990.

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M. Abadi, L. Cardelli, P.-L. Curien, J.-J. L'evy. "Explicit substitutions". Journal of Functional Programming, 1(4):375--416, 1991.

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M. Abadi, L. Cardelli, P.-L. Curien, J.-J. L evy. Explicit substitutions. Proceedings of Principle Of Programming Languages, 1990.

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Abadi, M., Cardelli, L., Curien, P.-L., and Levy, J.-J. (1991). Explicit substitutions. Journal of Functional Programming, 1(4):375--416.

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M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Lvy. Explicit substitutions. J. Func. Progr., 1(4):375416, 1991.

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M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit Substitutions. J. of Func. Programming, 1(4):375{ 416, 1991.

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M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. L'evy, "Explicit Substitutions", Functional Programming 1 (4) (1991) 375-416.

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Abadi, M., Cardelli, L., Curien, P.-L. and L'evy, J.-J., (1991) Explicit substitutions, Functional Programming 1 (4), 375-416.

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M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. Levy. Explicit substitutions. In Proc. 17th POPL, pages 31--46, 1990.

No context found.

M. Abadi, L. Cardelli, P.-L. Curien, and J.-J. L'evy. Explicit substitutions. In Seventeenth Annual ACM Symposium on Principles of Programming Languages, pages 31--46, 1990.

No context found.

Abadi, M., L. Cardelli, P.-L. Curien, and J.-J. Levy: 1991, `Explicit Substitutions '. Journal of Functional Programming 1(4), 375-416.

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