Graph easy sets of mute lambda terms

Bucciarelli, Antonio; Carraro, Amedeo; Favro, Giordano; Salibra, Antonino

doi:10.1016/j.tcs.2015.12.024

Cited by 5 publications

(4 citation statements)

References 25 publications

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“…Grellois-Melliès' infinitary model of Linear Logic in [12], [13]). The semantical implications of the main theorem (every term is R-typable) remain to be understood and the proof techniques presented here can certainly be used to study infinitary models or coinductive/recursive type systems before they are endowed with some validity or guard condition, or maybe to build other models of pure λ-calculus, for instance, to get some semantical proof of the easiness [14] of sets of mute terms, as in [4].…”

Section: Discussionmentioning

confidence: 99%

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Every λ-Term is Meaningful for the Infinitary Relational Model

Vial

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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Infinite types and formulas are known to have really curious and unsound behaviors. For instance, they allow to type Ω, the auto-autoapplication and they thus do not ensure any form of normalization/productivity. Moreover, in most infinitary frameworks, it is not difficult to define a type R that can be assigned to every λ-term. However, these observations do not say much about what coinductive (i.e. infinitary) type grammars are able to provide: it is for instance very difficult to know what types (besides R) can be assigned to a term in this setting. We begin with a discussion on the expressivity of different forms of infinite types. Using the resource-awareness of sequential intersection types (system S) and tracking, we then prove that infinite types are able to characterize the order (arity) of every λ-terms and that, in the infinitary extension of the relational model, every term has a "meaning" i.e. a non-empty denotation. From the technical point of view, we must deal with the total lack of productivity guarantee for typable terms: we do so by importing methods inspired by first order model theory.

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Section: Discussionmentioning

confidence: 99%

“…As it turns out, C 1 is the support of a type e.g., (4 figure below). By contrast, no type T may satisfy supp(T ) = C 2 , because a non-terminal node of a type (necessarily an arrow) should have a child on track 1 (by convention, its right-hand side), but 4 ∈ C 2 and 4 · 1 / ∈ C 2 .…”

Section: A a Toy Example: Support Candidates For Typesmentioning

confidence: 99%

Every λ-Term is Meaningful for the Infinitary Relational Model

Vial

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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“…x n .u, where u is not an abstraction. The integer n is sometimes called the order (as in [7]) of t| a . We say then that a is an order n position.…”

Section: Typing Normal Forms and Subject Expansionmentioning

confidence: 99%

Sequence Types and Infinitary Semantics

Vial¹

2021

Preprint

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We introduce a new representation of non-idempotent intersection types, using sequences (families indexed with natural numbers) instead of lists or multisets. This allows scaling up intersection type theory to the infinitary λ-calculus. We thus characterize hereditary head normalization (Klop's Problem) and we give a unique type to all hereditary permutators (TLCA Problem #20), which is not possible in a finite system. On our way, we use non-idempotent intersection to retrieve some well-known results on infinitary terms. This paper begins with a gentle, high-level introduction to intersection type theory and to the infinitary calculus.

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“…Terms of order 0 are, by definition, terms that cannot be converted to a lambda abstraction. Historically, these terms have sometimes been called zero terms [21,5]. At other times, the expression "zero terms" has been used, even by the same authors, to refer to the class of unsolvable terms of order zero.…”

Section: 2mentioning

confidence: 99%

The fixed point property and a technique to harness double fixed point combinators

Manzonetto

Polonsky

Saurin

et al. 2019

Journal of Logic and Computation

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The ${\lambda }$-calculus enjoys the property that each ${\lambda }$-term has at least one fixed point, which is due to the existence of a fixed point combinator. It is unknown whether it enjoys the ‘fixed point property’ stating that each ${\lambda }$-term has either one or infinitely many pairwise distinct fixed points. We show that the fixed point property holds when considering possibly open fixed points. The problem of counting fixed points in the closed setting remains open, but we provide sufficient conditions for a ${\lambda }$-term to have either one or infinitely many fixed points. In the main result of this paper we prove that in every sensible ${\lambda }$-theory there exists a ${\lambda }$-term that violates the fixed point property. We then study the open problem concerning the existence of a double fixed point combinator and propose a proof technique that could lead towards a negative solution. We consider interpretations of the ${\lambda } {\mathtt{Y}}$-calculus into the ${\lambda }$-calculus together with two reduction extension properties, whose validity would entail the non-existence of any double fixed point combinators. We conjecture that both properties hold when typed ${\lambda } {\mathtt{Y}}$-terms are interpreted by arbitrary fixed point combinators. We prove reduction extension property I for a large class of fixed point combinators. Finally, we prove that the ${\lambda }{\mathtt{Y}}$-theory generated by the equation characterizing double fixed point combinators is a conservative extension of the ${\lambda }$-calculus.

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Graph easy sets of mute lambda terms

Cited by 5 publications

References 25 publications

Every λ-Term is Meaningful for the Infinitary Relational Model

Every λ-Term is Meaningful for the Infinitary Relational Model

Sequence Types and Infinitary Semantics

The fixed point property and a technique to harness double fixed point combinators

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