On the characterization of models of H*: The semantical aspect

Breuvart, Flavien; Miller, Dale

doi:10.2168/lmcs-12(2:4)2016

Cited by 5 publications

(19 citation statements)

References 23 publications

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“…We will prove separately the two inclusions. 10 Notice that in a relational model [17] this issue would not hold (even if other problems would come later) since in any elements of the interpretation (a, α) ∈ λx.M the a is a finite multiset which can only "see" a finite number occurences of z.…”

Section: Technical Lemmamentioning

confidence: 99%

“…To overcome this we propose two different techniques leading to two different proofs of the main result: one purely semantical and the other purely syntactical. In this article we only present the former, the latter being the object of a companion paper [8].…”

Section: Introductionmentioning

confidence: 99%

“…

We give a characterization, with respect to a large class of models of untyped λ-calculus, of those models that are fully abstract for head-normalization, i.e., whose equational theory is H * (observations for head normalization). An extensional K-model D is fully abstract if and only if it is hyperimmune, i.e., not well founded chains of elements of D cannot be captured by any recursive function.This article, together with its companion paper [8] form the long version of [10]. It is a standalone paper that presents a purely semantical proof of the result as opposed to its companion paper that presents an independent and purely syntactical proof of the same result.

CCCreative Commons 2 F. BREUVART or K-models [24,6].

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confidence: 99%

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(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Accattoli¹,

Lago²

2016

Logical Methods in Computer Science

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Abstract. Slot and van Emde Boas' weak invariance thesis states that reasonable machines can simulate each other within a polynomial overhead in time. Is λ-calculus a reasonable machine? Is there a way to measure the computational complexity of a λ-term? This paper presents the first complete positive answer to this long-standing problem. Moreover, our answer is completely machine-independent and based on a standard notion in the theory of λ-calculus: the length of a leftmost-outermost derivation to normal form is an invariant, i.e. reasonable, cost model. Such a theorem cannot be proved by directly relating λ-calculus with Turing machines or random access machines, because of the size-explosion problem: there are terms that in a linear number of steps produce an exponentially large output. The first step towards the solution is to shift to a notion of evaluation for which the length and the size of the output are linearly related. This is done by adopting the linear substitution calculus (LSC), a calculus of explicit substitutions modeled after linear logic proof nets and admitting a decomposition of leftmost-outermost derivations with the desired property. Thus, the LSC is invariant with respect to, say, random access machines. The second step is to show that the LSC is invariant with respect to the λ-calculus. The size explosion problem seems to imply that this is not possible: having the same notions of normal form, evaluation in the LSC is exponentially longer than in the λ-calculus. We solve such an impasse by introducing a new form of shared normal form and shared reduction, called useful. Useful evaluation produces a compact, shared representation of the normal form, by avoiding those steps that only unshare the output without contributing to β-redexes, i.e. the steps that cause the blow-up in size. The main technical contribution of the paper is indeed the definition of useful reductions and the thorough analysis of their properties. ACM CCS: [Theory of computation]:Models of computation-Computability-Lambda calculus; Models of computation-Abstact machines; Logic-Proof theory / Linear logic / Equational logic and rewriting; Computational complexity and cryptography-Complexity theory and logic.

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Section: Technical Lemmamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

CCCreative Commons 2 F. BREUVART or K-models [24,6].

…”

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confidence: 99%

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(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Accattoli¹,

Lago²

2016

Logical Methods in Computer Science

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“…The proof techniques used in [16] are however more standard as those categories are not necessarily models of differential linear logic. A breakthrough in this subject has been recently achieved by Breuvart in [5], where he was able to provide a precise characterisation of those Krivine's models (K-models, for short) having theory H * . Indeed he proved that an extensional K-model has theory H * if and only if the unfolding of equivalent arrow types is governed by a hyperimmune function, a notion widely used in recursion theory.…”

Section: Related and Further Workmentioning

confidence: 99%

Relational Graph Models, Taylor Expansion and Extensionality

Manzonetto¹,

Ruoppolo²

2014

Electronic Notes in Theoretical Computer Science

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International audienceWe define the class of relational graph models and study the induced order-and equational-theories. Using the Taylor expansion, we show that all λ-terms with the same Böhm tree are equated in any relational graph model. If the model is moreover extensional and satisfies a technical condition, then its order-theory coincides with Morris's observational pre-order. Finally, we introduce an extensional version of the Taylor expansion, then prove that two λ-terms have the same extensional Taylor expansion exactly when they are equivalent in Morris's sense

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“…The calculi with tests played a central role in this paper. The idea of test mechanisms as syntactic extensions of the λ-calculus was first used by Bucciarelli et al [9] and developed further by the author [4,5,7] for Krivine-models. The one presented in this paper is yet an other generalization to the broader (extensional and distributive) filter models.…”

Section: Introductionmentioning

confidence: 99%

Refining Properties of Filter Models: Sensibility, Approximability and Reducibility

Breuvart¹

2018

Preprint

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In this paper, we study the tedious link between the properties of sensibility and approximability of models of untyped λ-calculus. Approximability is known to be a slightly, but strictly stronger property that sensibility. However, we will see that so far, each and every (filter) model that have been proven sensible are in fact approximable. We explain this result as a weakness of the sole known approach of sensibility: the Tait reducibility candidates and its realizability variants.In fact, we will reduce the approximability of a filter model D for the λ-calculus to the sensibility of D but for an extension of the λ-calculus that we call λ-calculus with D-tests. Then we show that traditional proofs of sensibility of D for the λ-calculus are smoothly extendable for this λ-calculus with D-tests.

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On the characterization of models of H*: The semantical aspect

Cited by 5 publications

References 23 publications

(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

(Leftmost-Outermost) Beta Reduction is Invariant, Indeed

Relational Graph Models, Taylor Expansion and Extensionality

Refining Properties of Filter Models: Sensibility, Approximability and Reducibility

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