Call-by-Name and Call-by-Value in Normal Modal Logic

Kakutani, Yoshihiko

doi:10.1007/978-3-540-76637-7_27

Cited by 13 publications

(19 citation statements)

References 22 publications

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“…(commcontr) is a 'contraction' rule. This is also unfamiliar in dual-context calculiessentially for the same reasons as (commweak)-but is also well-known in Bierman-de Paiva style calculi as a 'garbage collection' rule: see (Goubault-Larrecq, 1996), (Bierman and de Paiva, 2000) and (Kakutani, 2007).…”

Section: Categorical Semanticsmentioning

confidence: 96%

Dual-context calculi for modal logic

Kavvos

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We show how to derive natural deduction systems and modal lambda calculi for constructive versions of the necessity fragments of the modal logics K, T, K4, GL by exploiting a pattern found in sequent calculus. We investigate the metatheory of the resulting calculi, and study a notion of reduction for them, for which confluence and strong normalisation hold. We also consider the subformula property, and present sound and complete categorical semantics.

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Section: Categorical Semanticsmentioning

confidence: 96%

Dual-context calculi for modal logic

Kavvos

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…This calculus is investigated in some papers [4,28,27]. Logical provability of the Gentzen-style modal type theory is equivalent to intuitionistic K. The desired syntactic properties such as strong normalization and categorical semantics are provided in the papers.…”

Section: Other Systemsmentioning

confidence: 99%

“…For the type theoretic aspects, on the other hand, the approaches are diverse. As of this writing, there are mainly three types of natural deduction systems that have gained popularity, called Gentzen-style [4,28], dual-context [18,30], and Fitch-style [37,13] systems. For the first two systems, their computational and categorical aspects are intensively investigated in a number of papers [17,29], and applied in multi-staged computation [15,39].…”

Section: Introductionmentioning

confidence: 99%

Modality via Iterated Enrichment

Nishiwaki

Kakutani

Murase

2018

Electronic Notes in Theoretical Computer Science

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This paper investigates modal type theories by using a new categorical semantics called change-of-base semantics. Change-of-base semantics is novel in that it is based on (possibly infinitely) iterated enrichment and interpretation of modality as hom objects. In our semantics, the relationship between meta and object levels in multi-staged computation exactly corresponds to the relationship between enriching and enriched categories. As a result, we obtain a categorical explanation of situations where meta and object logics may be completely different. Our categorical models include conventional models of modal type theory (e.g., cartesian closed categories with a monoidal endofunctor) as special cases and hence can be seen as a natural refinement of former results. On the type theoretical side, it is shown that Fitch-style modal type theory can be directly interpreted in iterated enrichment of categories. Interestingly, this interpretation suggests the fact that Fitch-style modal type theory is the right adjoint of dualcontext calculus. In addition, we present how linear temporal, S4, and linear exponential modalities are described in terms of change-of-base semantics. Finally, we show that the change-of-base semantics can be naturally extended to multi-staged effectful computation and generalized contextual modality a la Nanevski et al. We emphasize that this paper answers the question raised in the survey paper by de Paiva and Ritter in 2011, what a categorical model for Fitch-style type theory is like.

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“…A Gentzen-style calculus was first proposed by Bellin et al [2], and was later refined by one of the authors [11], [12]. Dualcontext calculi [5], [8] have been developed mainly for intuitionistic S4 (IS4), because the first dual-context calculus [1] was dedicated to intuitionistic linear logic with the exponential modality.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, a model for Gentzen-style and dual-context is believed to be a ccc with a monoidal endofunctor [2], [5], [10], [12], [13]. If we write F for a model of the dual-context calculus, semantics of the dual-context is as follows.…”

Section: Introductionmentioning

confidence: 99%

Dual-context Modal Logic as Left Adjoint of Fitch-style Modal Logic

Kakutani

Murase

Nishiwaki

2019

Journal of Information Processing

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In the stream of studies on intuitionistic modal logic, we can find mainly three kinds of natural deduction systems. For logical aspects, adding axiom schemata is a simple and popular way to construct a system. The Curry-Howard correspondence, however, gives us a connection between logic and computer science. From the viewpoint of programming languages, two more important systems, called a dual-context system and a Fitch-style system, have been proposed. While dual-context systems for S4 are heavily used in the field of staged computation, a dual-context system for K is also studied more recently. In our previous studies, categorical semantics for Fitch-style modal logic is proposed and usefulness of levels is noticed. This paper observes an interesting fact that the box modality of the dualcontext system is in fact a left adjoint of that of the Fitch-style system. In order to show the statement, we embed both the two systems, which are refined with levels, into the adjoint calculus that equips an adjunction a priori. Moreover, the adjunction is refined with polarity and the adjoint calculus is extended to polarized logic.

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Call-by-Name and Call-by-Value in Normal Modal Logic

Cited by 13 publications

References 22 publications

Dual-context calculi for modal logic

Dual-context calculi for modal logic

Modality via Iterated Enrichment

Dual-context Modal Logic as Left Adjoint of Fitch-style Modal Logic

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