Permutative Conversions in Intuitionistic Multiary Sequent Calculi with Cuts

Santo, José Espírito; Pinto, Luís

doi:10.1007/3-540-44904-3_20

Cited by 15 publications

(5 citation statements)

References 8 publications

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“…In this paper we study the λ Gtz -calculus, introduced in [8], and corresponding under the Curry-Howard correspondence to the intuitionistic sequent calculus. The interest of λ Gtz , and simultaneously the difference relative to λ is that: (i) at the logical level, no restriction is placed on left inferences; (ii) at the term-calculus level, λ Gtz has a single cut-construction that subsumes both explicit substitution and an enlarged concept of application, exhibiting the features of "multiarity" and "generality" [13]. The main result of this paper is the design of an intersection type assignment system λ Gtz ∩ which, we prove, characterises the strongly normalising λ Gtz -terms (i.e.…”

Section: Introductionmentioning

confidence: 99%

Characterising Strongly Normalising Intuitionistic Terms

Santo

Ivetić

Likavec

2012

Fundamenta Informaticae

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This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary λ-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary λ-terms, ΛJ-terms (λ-terms with generalised application), and λx-terms (λ-terms with explicit substitution). We explain via our system why the type systems in the natural deduction format for ΛJ and λx known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the β-strongly normalising λ-terms, as a corollary to a PSN-result, relating the λ-calculus and the intuitionistic sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known characterisation of the β-strongly normalising λ-terms in terms of assignment of strict types follows as an easy corollary of our results.

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Section: Introductionmentioning

confidence: 99%

Characterising Strongly Normalising Intuitionistic Terms

Santo

Ivetić

Likavec

2012

Fundamenta Informaticae

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“…However, this bijection failed to extend to sequent calculus with cuts. After that, intuitionistic sequent λ-calculi were proposed by Barendregt and Ghilezan [1], Dyckhoff and Pinto [4], Espirito-Santo and Pinto [5], among others. One of the most recently proposed systems is λ Gtzcalculus, developed by Espirito-Santo [6], whose simply typed version corresponds to the sequent calculus for intuitionistic implicational logic.…”

Section: Introductionmentioning

confidence: 99%

Intersection types for λGtz-calculus

Ghilezan

Ivetić

2007

Publ Inst Math (Belgr)

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“…Rule p2 is used in[6, 3] along with two other permutation rules p1 and p3 to reduce TJ -terms to a fragment isomorphic to natural deduction.…”

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A Faithful and Quantitative Notion of Distant Reduction for Generalized Applications (Long Version)

Santo¹,

Kesner²,

Peyrot³

2022

Preprint

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We introduce a call-by-name lambda-calculus λJ with generalized applications which integrates a notion of distant reduction that allows to unblock β-redexes without resorting to the permutative conversions of generalized applications. We show strong normalization of simply typed terms, and we then fully characterize strong normalization by means of a quantitative typing system. This characterization uses a non-trivial inductive definition of strong normalization -that we relate to others in the literature-, which is based on a weak-head normalizing strategy. Our calculus relates to explicit substitution calculi by means of a translation between the two formalisms which is faithful, in the sense that it preserves strong normalization. We show that our calculus λJ and the well-know calculus ΛJ determine equivalent notions of strong normalization. As a consequence, ΛJ inherits a faithful translation into explicit substitutions, and its strong normalization can be characterized by the quantitative typing system designed for λJ, despite the fact that quantitative subject reduction fails for permutative conversions.

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Permutative Conversions in Intuitionistic Multiary Sequent Calculi with Cuts

Cited by 15 publications

References 8 publications

Characterising Strongly Normalising Intuitionistic Terms

Characterising Strongly Normalising Intuitionistic Terms

Intersection types for λGtz-calculus

A Faithful and Quantitative Notion of Distant Reduction for Generalized Applications (Long Version)

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