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Journal ArticleDOI

Linearity, Non-determinism and Solvability

01 Jan 2010-Fundamenta Informaticae (IOS Press)-Vol. 103, Iss: 1, pp 173-202
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.

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Citations
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Book ChapterDOI
23 May 2012
TL;DR: The value-substitution lambda-calculus is introduced, a simple calculus borrowing ideas from Herbelin and Zimmerman's call-by-value λ CBV calculus and from Accattoli and Kesner's substitution calculus λ sub .
Abstract: In the call-by-value lambda-calculus solvable terms have been characterised by means of call-by-name reductions, which is disappointing and requires complex reasonings. We introduce the value-substitution lambda-calculus, a simple calculus borrowing ideas from Herbelin and Zimmerman's call-by-value λ CBV calculus and from Accattoli and Kesner's substitution calculus λ sub . In this new setting, we characterise solvable terms as those terms having normal form with respect to a suitable restriction of the rewriting relation.

50 citations


Cites background from "Linearity, Non-determinism and Solv..."

  • ...Solvability has also been recently studied for some extensions of λ-calculus in [18,25], but both works consider a call-by-name calculus....

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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 "Linearity, Non-determinism and Solv..."

  • ...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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Book ChapterDOI
01 Sep 2014
TL;DR: The inhabitation problem for intersection types is known to be undecidable, but it is known that it is decidable in the case of non-idempotent intersection types.
Abstract: The inhabitation problem for intersection types is known to be undecidable. We study the problem in the case of non-idempotent intersection, and we prove decidability through a sound and complete algorithm. We then consider the inhabitation problem for an extended system typing the λ-calculus with pairs, and we prove the decidability in this case too. The extended system is interesting in its own, since it allows to characterize solvable terms in the λ-calculus with pairs.

31 citations

Journal ArticleDOI
TL;DR: This work defines essential models (a new class of logical models) through a parametric type assignment system using non-idempotent intersection types that provides a logical description of a family of λ-models arising from a category of sets and relations.
Abstract: Intersection type assignment systems can be used as a general framework for building logical models of �-calculus that allow to reason about the denotation of terms in a finitary way. We define essential models (a new class of logical models) through a parametric type assignment system using non idempotent intersection types. Under an interpretation of terms based on typings instead than the usual one based on types, every suitable instance of the parameters induces a �-model, whose theory is sensible. We prove that this type assignment system provides a logical description of a family of �-models arising from a category of sets and relations. In the general framework of denotational semantics of λ-calculus, logical models are a particular class of λ-models supplying a finitary description of the interpretation of terms, through type assignment systems. Types are built from a set of constants, via two type-constructors: the arrow (→) and the intersection (∧). Terms are interpreted as sets of types, so reasoning about the interpretation of a term in these models can be done via type inference; in fact, in order to prove the equivalence between two terms, it is sufficient to show that they can be assigned the same set oftypes. Although the type inference is undecidable, logical models are concrete tools for reasoning in finitary way on the interpretation of terms, since a (finite) derivation grasps a finite piece of the semantic-interpretations. The relationship between logical models and domain-theoretical models has been widely studied, and it has been proved that some interesting classes of such models can be described in logical form. For instance, filter models supply a logical description of a class of Scott models based on continuous functions, since they can be seen as a restriction of the domain theory in logical form, which goes back to Stone duality (see Abramsky (1991)). A first characterization of a logical model where types represent continuous functions is in Coppo et al. (1984), a sketch of the proof of the correspondence between

26 citations


Cites methods from "Linearity, Non-determinism and Solv..."

  • ...In order to prove this relationship between essential and relational models, we adapt the pattern followed in Paolini et al. (2009), where a logical description of models based on coherence spaces is given....

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Journal ArticleDOI
TL;DR: It is proved that the resulting typed λ-calculus is strongly normalising and features weak subject reduction and it is shown how to naturally encode matrices and vectors in this typed calculus.
Abstract: We describe a type system for the linear-algebraic λ-calculus. The type system accounts for the linear-algebraic aspects of this extension of λ-calculus: it is able to statically describe the linear combinations of terms that will be obtained when reducing the programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We prove that the resulting typed λ-calculus is strongly normalising and features weak subject reduction. Finally, we show how to naturally encode matrices and vectors in this typed calculus.

26 citations


Cites background from "Linearity, Non-determinism and Solv..."

  • ...It may also be viewed as being part of a series of works on probabilistic extensions of calculi, e.g. [13, 31] and [19, 21, 35] for λ-calculus more specifically....

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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
Dana Scott1
01 Sep 1976
TL;DR: In this article, the meaning of many kinds of expressions in programming languages can be taken as elements of certain spaces of partial objects, and these spaces are modeled in one universal domain.
Abstract: The meaning of many kinds of expressions in programming languages can be taken as elements of certain spaces of “partial” objects. In this report these spaces are modeled in one universal domain ${...

787 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

Book
02 Jan 1993
TL;DR: Cours de lambda-calcul: Definition, beta-reduction et confluence, Representation des fonctions recursives.
Abstract: Cours de lambda-calcul. Definition, beta-reduction et confluence. Representation des fonctions recursives. Modeles du lambda-calcul Logique combinatoire Types, systeme F

226 citations