Proceedings Article•

# The structural lambda-calculus

23 Aug 2010-Vol. 6247, pp 381-395

TL;DR: In this article, an untyped structural λ-calculus, called λj, was introduced, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction.

Abstract: Inspired by a recent graphical formalism for λ-calculus based on Linear Logic technology, we introduce an untyped structural λ-calculus, called λj, which combines action at a distance with exponential rules decomposing the substitution by means of weakening, contraction and dereliction. Firstly, we prove fundamental properties such as confluence and preservation of β-strong normalisation. Secondly, we use λj to describe known notions of developments and superdevelopments, and introduce a more general one called XL-development. Then we show how to reformulate Regnier's s-equivalence in λj so that it becomes a strong bisimulation. Finally, we prove that explicit composition or de-composition of substitutions can be added to λj while still preserving β-strong normalisation.

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01 Jan 2017

TL;DR: Two resource aware typing systems for the λμ-calculus based on non-idempotent intersection and union types are defined and typability provides upper bounds for the length of head-reduction sequences and maximal reduction sequences.

Abstract: We define two resource aware typing systems for the λμ-calculus based on non-idempotent intersection and union types. The non-idempotent approach provides very simple combinatorial arguments –based on decreasing measures of type derivations– to characterize head and strongly normalizing terms. Moreover, typability provides upper bounds for the length of head-reduction sequences and maximal reduction sequences. 1998 ACM Subject Classification F.4.1 Mathematical Logic

27 citations

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01 Jan 1999

TL;DR: In this paper, a cyclic sharing theory for higher order extension of higher order extensions of cyclic graphs is presented. But it does not consider the relation between cyclic graph graphs and action calculi.

Abstract: Introduction.- Sharing Graphs and Equational Presentation.- Models of Acyclic Sharing Theroy.- Higher Order Extension.- Relating Models.- Models of Cyclic Sharing Theory.- Recursion from Cyclic Sharing.- Action Calculi.- Conclusion.- Proofs.- Bibliography.- Index.

26 citations

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29 Aug 2017

TL;DR: A call-by-need strategy for computing strong normal forms of open terms (reduction is admitted inside the body of abstractions and substitutions, and the terms may contain free variables), which guarantees that arguments are only evaluated when needed and at most once.

Abstract: We present a call-by-need strategy for computing strong normal forms of open terms (reduction is admitted inside the body of abstractions and substitutions, and the terms may contain free variables), which guarantees that arguments are only evaluated when needed and at most once. The strategy is shown to be complete with respect to β-reduction to strong normal form. The proof of completeness relies on two key tools: (1) the definition of a strong call-by-need calculus where reduction may be performed inside any context, and (2) the use of non-idempotent intersection types. More precisely, terms admitting a β-normal form in pure lambda calculus are typable, typability implies (weak) normalisation in the strong call-by-need calculus, and weak normalisation in the strong call-by-need calculus implies normalisation in the strong call-by-need strategy. Our (strong) call-by-need strategy is also shown to be conservative over the standard (weak) call-by-need.

25 citations

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11 Mar 2012TL;DR: The permutative λ-calculus is introduced and it is proved that confluence modulo the equations and preservation of beta-strong normalisation (PSN) by means of an auxiliary substitution calculus.

Abstract: We introduce the permutative λ-calculus, an extension of λ-calculus with three equations and one reduction rule for permuting constructors, generalising many calculi in the literature, in particular Regnier's sigma-equivalence and Moggi's assoc-equivalence. We prove confluence modulo the equations and preservation of beta-strong normalisation (PSN) by means of an auxiliary substitution calculus. The proof of confluence relies on M-developments, a new notion of development for λ-terms.

22 citations

##### References

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TL;DR: This column presents an intuitive overview of linear logic, some recent theoretical results, and summarizes several applications oflinear logic to computer science.

Abstract: Linear logic was introduced by Girard in 1987 [11] . Since then many results have supported Girard' s statement, \"Linear logic is a resource conscious logic,\" and related slogans . Increasingly, computer scientists have come to recognize linear logic as an expressive and powerful logic with connection s to a variety of topics in computer science . This column presents a.n intuitive overview of linear logic, some recent theoretical results, an d summarizes several applications of linear logic to computer science . Other introductions to linear logic may be found in [12, 361 .

2,304 citations

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24 Aug 1998TL;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

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01 Jan 1980

TL;DR: This proof is a direct generalization of Aczel's original proof, which is close to the well-known confluence proof for λ-calculus by Tait and Martin-Lof and gives an outline of a short proof of confluence.

Abstract: Combinatory reduction systems, or CRSs for short, were designed to combine the usual first-order format of term rewriting with the presence of bound variables as in pure λ-calculus and various typed λ-calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and left-linear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the well-known confluence proof for λ-calculus by Tait and Martin-Lof. There is a well-known connection between the parallel reduction featuring in the latter proof and the concept of “developments”, and a classical lemma in the theory of λ-calculus is that of “finite developments”, a strong normalization result. It turns out that the notion of “parallel reduction” used in Aczel's proof gives rise to a generalized form of developments which we call “superdevelopments” and on which we will briefly comment.

662 citations

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26 Jun 2007TL;DR: Decidability is obtained for the extensional equational theory of simply-typed λ-calculus extended with sum types for normalising and confluent extensional rewriting theory.

Abstract: Inspired by recent work on normalisation by evaluation for sums, we propose a normalising and confluent extensional rewriting theory for the simply-typed λ-calculus extended with sum types. As a corollary of confluence we obtain decidability for the extensional equational theory of simply-typed λ-calculus extended with sum types. Unlike previous decidability results, which rely on advanced rewriting techniques or advanced category theory, we only use standard techniques.

364 citations

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TL;DR: The chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism and subjectivism and that a representative class of algorithms can be modelized by means of standard mathematics.

Abstract: Publisher Summary This chapter describes the development of a semantics of computation free from the twin drawbacks of reductionism (that leads to static modification) and subjectivism (that leads to syntactical abuses, in other terms, bureaucracy). The new approach initiated in this chapter rests on the use of a specific C*-algebra Λ* that has the distinguished property of bearing a (non associative) inner tensor product. The chapter describes that a representative class of algorithms can be modelized by means of standard mathematics.

321 citations