Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types

Bernadet, Alexis; Lengrand, Stéphane

doi:10.1007/978-3-642-19805-2_7

Cited by 22 publications

(24 citation statements)

References 25 publications

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“…This improved on previous results, e.g. [Bernadet and Graham-Lengrand 2013b;Bernadet and Lengrand 2011] where multi types provided the exact measure of longest evaluation paths plus the size of the normal forms which, as discussed above, can be exponentially bigger. Finally, they enrich the structure of base types so that, for those typing derivations providing the exact lengths, the type of a term gives the structure (and hence the size) of its normal form.…”

Section: Contributionssupporting

confidence: 70%

Tight typings and split bounds

Accattoli

Graham-Lengrand

Kesner

2018

Proc. ACM Program. Lang.

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Multi types-aka non-idempotent intersection types-have been used to obtain quantitative bounds on higherorder programs, as pioneered by de Carvalho. Notably, they bound at the same time the number of evaluation steps and the size of the result. Recent results show that the number of steps can be taken as a reasonable time complexity measure. At the same time, however, these results suggest that multi types provide quite lax complexity bounds, because the size of the result can be exponentially bigger than the number of steps.Starting from this observation, we refine and generalise a technique introduced by Bernadet & Graham-Lengrand to provide exact bounds for the maximal strategy. Our typing judgements carry two counters, one measuring evaluation lengths and the other measuring result sizes. In order to emphasise the modularity of the approach, we provide exact bounds for four evaluation strategies, both in the λ-calculus (head, leftmostoutermost, and maximal evaluation) and in the linear substitution calculus (linear head evaluation).Our work aims at both capturing the results in the literature and extending them with new outcomes. Concerning the literature, it unifies de Carvalho and Bernadet & Graham-Lengrand via a uniform technique and a complexity-based perspective. The two main novelties are exact split bounds for the leftmost strategythe only known strategy that evaluates terms to full normal forms and provides a reasonable complexity measure-and the observation that the computing device hidden behind multi types is the notion of substitution at a distance, as implemented by the linear substitution calculus.

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Section: Contributionssupporting

confidence: 70%

Tight typings and split bounds

Accattoli

Graham-Lengrand

Kesner

2018

Proc. ACM Program. Lang.

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“…Non-idempotent intersection types have been used as technical tool for a characterisation of strong normalisation in Bernadet and Lengrand (2011b). Nonidempotency has a quantitative flavour, in fact they have been used for proving interesting quantitative properties about the complexity of the β-reduction in de Carvalho (2009), Bernadet and Lengrand (2011a) and De Benedetti and Ronchi Della Rocca (2013). Some observations about the use of non-idempotent intersection types in the setting of implicit computational complexity have been made in Terui (2006).…”

Section: Related Workmentioning

confidence: 99%

Essential and relational models

Paolini

Piccolo

Rocca

2015

Math. Struct. Comp. Sci.

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Essential and relational models † Intersection type assignment systems can be used as a general framework for building logical models of λ-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 λ-model, whose theory is sensible. We prove that this type assignment system provides a logical description of a family of λ-models arising from a category of sets and relations.

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“…2. The intersection types that we use here differ from those of [BL11a], in that the associativity and commutativity (AC) of the intersection ∩ are only featured "on the surface" of types, and not underneath functional arrows →. This will make the typing rules much more syntax-directed, simplifying the proofs of soundness and completeness of typing with respect to the strong normalisation property.…”

Section: Definition 23 (Types)mentioning

confidence: 99%

“…While the resource calculi along the lines of [BEM10] are well-suited to de Carvalho's study of head reductions, our interest in longest reduction sequences (no matter where the redexes are) lead us to explicit substitution calculi along the lines of [KL05,KL07,Kes07,Ren11]. Hence the extension of our complexity results (already presented in [BL11a] for λ) to λS and λlxr. 2 Filter models and strong normalisation.…”

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confidence: 94%

Non-idempotent intersection types and strong normalisation

Bernadet¹,

Lengrand²

2013

Logical Methods in Computer Science

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Abstract. We present a typing system with non-idempotent intersection types, typing a term syntax covering three different calculi: the pure λ-calculus, the calculus with explicit substitutions λS, and the calculus with explicit substitutions, contractions and weakenings λlxr.In each of the three calculi, a term is typable if and only if it is strongly normalising, as it is the case in (many) systems with idempotent intersections.Non-idempotency brings extra information into typing trees, such as simple bounds on the longest reduction sequence reducing a term to its normal form. Strong normalisation follows, without requiring reducibility techniques.Using this, we revisit models of the λ-calculus based on filters of intersection types, and extend them to λS and λlxr. Non-idempotency simplifies a methodology, based on such filter models, that produces modular proofs of strong normalisation for well-known typing systems (e.g. System F ). We also present a filter model by means of orthogonality techniques, i.e. as an instance of an abstract notion of orthogonality model formalised in this paper and inspired by classical realisability. Compared to other instances based on terms (one of which rephrases a now standard proof of strong normalisation for the λ-calculus), the instance based on filters is shown to be better at proving strong normalisation results for λS and λlxr.Finally, the bounds on the longest reduction sequence, read off our typing trees, are refined into an exact measure, read off a specific typing tree (called principal ); in each of the three calculi, a specific reduction sequence of such length is identified. In the case of the λ-calculus, this complexity result is, for longest reduction sequences, the counterpart of de Carvalho's result for linear head-reduction sequences.

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Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types

Cited by 22 publications

References 25 publications

Tight typings and split bounds

Tight typings and split bounds

Essential and relational models

Non-idempotent intersection types and strong normalisation

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