On Explicit Substitution with Names

Rose, Kristoffer H.; Bloo, Roel; Lang, Frédéric

doi:10.1007/s10817-011-9222-5

Cited by 8 publications

(5 citation statements)

References 37 publications

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“…In other words, these calculi shed light on the execution models of higher-order languages. In fact, the development of a calculus with explicit substitutions faithful to the λ -calculus, in the sense of the preservation of some desired properties were the main motivation for such a long list of calculi with explicit substitutions invented in the last decades [1,23,6,10,19,15,8,11,17].…”

Section: An Explicit Substitution Operatormentioning

confidence: 99%

A Formalized Extension of the Substitution Lemma in Coq

Lima,

de Moura

2023

Electron. Proc. Theor. Comput. Sci.

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Section: An Explicit Substitution Operatormentioning

confidence: 99%

A Formalized Extension of the Substitution Lemma in Coq

Lima,

de Moura

2023

Electron. Proc. Theor. Comput. Sci.

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“…[DK97,Rit99]) has focussed on Melliès' observation that, contrary to what one might expect from the lambda calculus, such calculi may not be strongly normalising [Mel95] (see e.g. [RBL11] for an overview). Two-dimensional type theories, on the other hand, first arose from Seely's observation [See87] that η-expansion and β-reduction form the unit and counit of a lax (directed) cartesian closed structure, a perspective advocated further by Jay & Ghani [Gha95,JG95] and put to use by Hilken [Hil96] for a proof-theoretic account of rewriting.…”

Section: A Type Theory For Biclonesmentioning

confidence: 99%

Cartesian closed bicategories: type theory and coherence

Saville

2020

Preprint

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In this thesis I lift the Curry-Howard-Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories-2-categories 'up to isomorphism' equipped with similarly weak products and exponentials-arise in logic, categorical algebra, and game semantics. However, calculations in such bicategories quickly fall into a quagmire of coherence data. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence-in terms of the difficulty of calculating-bring the data of cartesian closed bicategories down to the familiar level of cartesian closed categories. viii corresponding categories. Broadly speaking, this is the what I do in this thesis: I construct a type theory, show it corresponds to a special class of categories, and then-by proving something about the type theory-solve a problem about the class of categories.The problem is called coherence. The special categories I work with-the 'cartesian closed bicategories' of the title-have uses in other areas of category theory, as well as in algebra and in the study of programming languages, but they are intricate. As well as the ways of getting from A to B, they include the routes between these routes. Imagine A and B are Cambridge and Oxford. Then the routes between them might be walking directions for the various routes, and the routes-between-routes might be the ways you can change one set of directions into the other: change 'left' for 'right' at this junction, replace '100 yards' with '2 miles', and so on. Or you can imagine studying programs, and the ways of transforming them stage-by-stage into something that you can run in 0s and 1s on your hardware. In this example, you might have two programs with the same input type and the same output type-such as those in (1) above-and think about the ways of transforming one into another: replacing yˆ6 3 by y ˆ2, and x 2 ˆ2 by just x, and so on. Precisely describing these two levels, and the ways they must interact, requires many axioms and many checks at every stage of a calculation. This quickly becomes tedious, and leads to proofs that are so long it is hard to check they are correct, let alone fit them onto a page so that they can be verified by the community. In this thesis I show that cartesian closed bicategories have the property that any equation you can write down for any cartesian closed bicategory (not relying on any special properties of a specific one) must hold. This means that those long tedious calculations are dramatically simplified: all those things that you had to check before are now guaranteed to hold by the theorem.In Part I, then, I construct a type theory for describing cartesian closed bicategories. If a type theory is a logic for programs, this is a logic for programs and ways of transforming programs into one another. I show that expressions i...

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“…Van Oostrom [54] and later Kesner and Lengrand [30], applying ideas from linear logic [29], proposed to extend λ-calculus with explicit substitution [30] with operators to control the use of variables (resources). Their linear λlxr-calculus is an extension of the λx-calculus [9,47] with operators for linear substitution, erasure and duplication, preserving at the same time confluence and full composition of explicit substitutions. The simply typed version of this calculus corresponds to the intuitionistic fragment of linear logic proof-nets, according to Curry-Howard correspondence, and it enjoys strong normalisation and subject reduction.…”

Section: Typeability ⇒ Sn In λ ∩mentioning

confidence: 99%

Resource control and intersection types: an intrinsic connection

Ghilezan¹,

Ivetić²,

Lescanne³

et al. 2014

Preprint

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In this paper we investigate the λ -calculus, a λ-calculus enriched with resource control. Explicit control of resources is enabled by the presence of erasure and duplication operators, which correspond to thinning and contraction rules in the type assignment system. We introduce directly the class of λ -terms and we provide a new treatment of substitution by its decomposition into atomic steps. We propose an intersection type assignment system for λ -calculus which makes a clear correspondence between three roles of variables and three kinds of intersection types. Finally, we provide the characterisation of strong normalisation in λ -calculus by means of an intersection type assignment system. This process uses typeability of normal forms, redex subject expansion and reducibility method.

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On Explicit Substitution with Names

Cited by 8 publications

References 37 publications

A Formalized Extension of the Substitution Lemma in Coq

A Formalized Extension of the Substitution Lemma in Coq

Cartesian closed bicategories: type theory and coherence

Resource control and intersection types: an intrinsic connection

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