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Book ChapterDOI

Solvability in resource lambda-calculus

20 Mar 2010-pp 358-373
TL;DR: This work defines a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity, and gives a syntactical, operational and logical characterization of this kind of solvability.
Abstract: The resource calculus is an extension of the λ-calculus allowing to model resource consumption. Namely, the argument of a function comes as a finite multiset of resources, which in turn can be either linear or reusable, giving rise to non-deterministic choices, expressed by a formal sum. Using the λ-calculus terminology, we call solvable a term that can interact with the environment: solvable terms represent meaningful programs. Because of the non-determinism, different definitions of solvability are possible in the resource calculus. Here we study the optimistic (angelical, or may) notion, and so we define a term solvable whenever there is a simple head context reducing the term into a sum where at least one addend is the identity. We give a syntactical, operational and logical characterization of this kind of solvability.

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Citations
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Journal ArticleDOI
TL;DR: This article explores the use of non-idempotent intersection types in the framework of the λ-calculus by replacing the reducibility technique with trivial combinatorial arguments.
Abstract: This article explores the use of non-idempotent intersection types in the framework of the λ-calculus. Different topics are presented in a uniform framework: head normalization, weak normalization, weak head normalization, strong normalization, inhabitation, exact bounds and principal typings. The reducibility technique, traditionally used when working with idempotent types, is replaced in this framework by trivial combinatorial arguments.

53 citations


Cites background from "Solvability in resource lambda-calc..."

  • ...Solvability is characterized by means of non-idempotent IT in a lambda-calculus with resources [51]....

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Proceedings ArticleDOI
01 Jan 2012
TL;DR: This work construction of a new model, which features a new duality, is presented, and how to use it for reducing normalization results in idempotent intersection types to purely combinatorial methods is explained.
Abstract: We proved recently that the extensional collapse of the relational model of linear logic coincides with its Scott model, whose objects are preorders and morphisms are downwards closed relations. This result is obtained by the construction of a new model whose objects can be understood as preorders equipped with a realizability predicate. We present this model, which features a new duality, and explain how to use it for reducing normalization results in idempotent intersection types (usually proved by reducibility) to purely combinatorial methods. We illustrate this approach in the case of the call-by-value lambda-calculus, for which we introduce a new resource calculus, but it can be applied in the same way to many different calculi.

49 citations

Journal ArticleDOI
Giulio Manzonetto1
TL;DR: An abstract ‘model theory’ for the untyped differential λ-calculus is developed and it is shown that the resource calculus can be interpreted by translation into every linear reflexive object living in a Cartesian closed differential category.
Abstract: The differential λ-calculus is a paradigmatic functional programming language endowed with a syntactical differentiation operator that allows the application of a program to an argument in a linear way. One of the main features of this language is that it is resource conscious and gives the programmer suitable primitives to handle explicitly the resources used by a program during its execution. The differential operator also allows us to write the full Taylor expansion of a program. Through this expansion, every program can be decomposed into an infinite sum (representing non-deterministic choice) of 'simpler' programs that are strictly linear. The aim of this paper is to develop an abstract 'model theory' for the untyped differential λ-calculus. In particular, we investigate what form a general categorical definition of a denotational model for this calculus should take. Starting from the work of Blute, Cockett and Seely on differential categories, we develop the notion of a Cartesian closed differential category and prove that linear reflexive objects living in such categories constitute sound and complete models of the untyped differential λ-calculus. We also give sufficient conditions for Cartesian closed differential categories to model the Taylor expansion. This requires that every model living in such categories equates all programs having the same full Taylor expansion. We then provide a concrete example of a Cartesian closed differential category modelling the Taylor expansion, namely the category MRel of sets and relations from finite multisets to sets. We prove that the extensional model of λ-calculus we have recently built in MRel is linear, and is thus also an extensional model of the untyped differential λ-calculus. In the same category, we build a non-extensional model and prove that it is, nevertheless, extensional on its differential part. Finally, we study the relationship between the differential λ-calculus and the resource calculus, which is a functional programming language combining the ideas behind the differential λ-calculus with those behind Boudol's λ-calculus with multiplicities. We define two translation maps between these two calculi and study the properties of these translations. In particular, this analysis shows that the two calculi share the same notion of a model, and thus that the resource calculus can be interpreted by translation into every linear reflexive object living in a Cartesian closed differential category.

41 citations

Journal ArticleDOI
TL;DR: This work studies the notion of solvability in the resource calculus, an extension of the λ-calculus modelling resource consumption, and gives a syntactical, operational and logical characterization for the may-solvability and only a partial characterization of the must-solvable.
Abstract: We study the notion of solvability in the resource calculus, an extension of the λ-calculus modelling resource consumption. Since this calculus is non-deterministic, two different notions of solvability arise, one optimistic (angelical, may) and one pessimistic (demoniac, must). We give a syntactical, operational and logical characterization for the may-solvability and only a partial characterization of the must-solvability. Finally, we discuss the open problem of a complete characterization of the must-solvability.

34 citations

Book ChapterDOI
04 Apr 2016
TL;DR: It is shown that call-by-need and call- by-name are observationally equivalent, so that in particular, the former turns out to be a correct implementation of the latter.
Abstract: We first develop a (semantical) characterization of call-by-need normalization by means of typability, i.e. we show that a term is normalizing in call-by-need if and only if it is typable in a suitable system with non-idempotent intersection types. This first result is used to derive a new completeness proof of call-by-need w.r.t. call-by-name. Concretely, we show that call-by-need and call-by-name are observationally equivalent, so that in particular, the former turns out to be a correct implementation of the latter.

34 citations


Cites background from "Solvability in resource lambda-calc..."

  • ...Different assignment systems with nonidempotent intersection types have been studied in the literature for different purposes [8, 10, 18, 19, 24, 26, 27, 30, 35, 34]....

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References
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Book
30 Apr 2012
TL;DR: In this article, the Lambda-Calculus has been studied as a theory of composition and reduction, and the theory of reduction has been used to construct models of Lambda Theories.
Abstract: Towards the Theory. Introduction. Conversion. Reduction. Theories. Models. Conversion. Classical Lambda Calculus. The Theory of Combinators. Classical Lambda Calculus (Continued). The Lambda-Calculus. Bohm Trees. Reduction. Fundamental Theorems. Strongly Equivalent Reductions. Reduction Strategies. Labelled Reduction. Other Notions of Reduction. Theories. Sensible Theories. Other Lambda Theories. Models. Construction of Models. Local Structure of Models. Global Structure of Models. Combinatory Groups. Appendices: Typed Lambda Calculus. Illative Combinatory Logic. Variables. References.

2,632 citations

Book ChapterDOI
21 Aug 2000
TL;DR: In the ambient logic of classical second order propositional calculus, the specification problem for a family of excluded middle like tautologies is solved and these are shown to be realized by sequential simulations of specific communication schemes for which they provide a safe typing mechanism.
Abstract: In the ambient logic of classical second order propositional calculus, we solve the specification problem for a family of excluded middle like tautologies. These are shown to be realized by sequential simulations of specific communication schemes for which they provide a safe typing mechanism.

1,119 citations

Book ChapterDOI
24 Aug 1998
TL;DR: It is proved that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true Σ\(^{\rm 0}_{\rm 1}\)-sentences).
Abstract: We place simple axioms on an elementary topos which suffice for it to provide a denotational model of call-by-value PCF with sum and product types. The model is synthetic in the sense that types are interpreted by their set-theoretic counterparts within the topos. The main result characterises when the model is computationally adequate with respect to the operational semantics of the programming language. We prove that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true Σ\(^{\rm 0}_{\rm 1}\)-sentences).

1,000 citations

Journal ArticleDOI
TL;DR: This work presents an extension of the lambda-calculus with differential constructions, and state and prove some basic results (confluence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-Calculus.

307 citations


"Solvability in resource lambda-calc..." refers background or methods in this paper

  • ...Definition 5....

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  • ...Ehrhard and Regnier designed the differential λ-calculus [2], drawing on insights gained from an analysis of some denotational models of linear logic....

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  • ...This paper deals extensively with the weakest notion of solvability, which asks that a term is solvable whenever a suitable context filled with it reduces to a sum, where at least one addend is the identity....

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Journal ArticleDOI

213 citations


"Solvability in resource lambda-calc..." refers background in this paper

  • ...…form [8], operationally if and only if the head reduction strategy applied to it eventually stops [8], logically if and only if it can be typed in a suitable intersection type assignment system [9], denotationally if and only if its denotation is not minimal in a suitable sensible model [10, 11]....

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