Coming to terms with modal logic : on the interpretation of modalities in typed lambda-calculus

Borghuis, Tijn

doi:10.6100/ir427575

Cited by 3 publications

(2 citation statements)

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“…Nanevski et al [17] formulated their original contextual modal type theory in dual-context style [19,6,11], which has judgments with two-level contexts. In contrast, we formulate λ [] in so-called Fitch-or Kripke-style [4,1,15,6,31]. We choose this design because the Fitch-style formulation provides Lisp-like quote/unquote syntax, which is akin to that in linear-temporal type theories [5,30], and hence it is easier to compare these two type theories.…”

Section: Simple Fitch-style Contextual Modal Type Theorymentioning

confidence: 99%

“…This section provides a proof of strong normalization of β-reduction in λ ∀[] . A common approach to proving strong normalization of a modal calculus is to provide a reduction-preserving translation to another strongly normalizing calculus such as simply typed lambda calculi [15,1]. We tried this approach, reducing strong normalization of λ ∀[] to that of System F [8].…”

Section: Parametric Reducibility and Strong Normalizationmentioning

confidence: 99%

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Contextual Modal Type Theory with Polymorphic Contexts

Murase

Nishiwaki²,

Igarashi

2023

Lecture Notes in Computer Science

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Modal types—types that are derived from proof systems of modal logic—have been studied as theoretical foundations of metaprogramming, where program code is manipulated as first-class values. In modal type systems, modality corresponds to a type constructor for code types and controls free variables and their types in code values. Nanevski et al. have proposed contextual modal type theory, which has modal types with fine-grained information on free variables: modal types are explicitly indexed by contexts—the types of all free variables in code values.This paper presents $$\lambda _{\forall []}$$ λ ∀ [ ] , a novel extension of contextual modal type theory with parametric polymorphism over contexts. Such an extension has been studied in the literature but, unlike earlier proposals, $$\lambda _{\forall []}$$ λ ∀ [ ] is more general in that it allows multiple occurrence of context variables in a single context. We formalize $$\lambda _{\forall []}$$ λ ∀ [ ] with its type system and operational semantics given by $$\beta $$ β -reduction and prove its basic properties including subject reduction, strong normalization, and confluence. Moreover, to demonstrate the expressive power of polymorphic contexts, we show a type-preserving embedding from a two-level fragment of Davies’ $$\lambda _{\bigcirc }$$ λ ◯ , which is based on linear-time temporal logic, to $$\lambda _{\forall []}$$ λ ∀ [ ] .

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Section: Simple Fitch-style Contextual Modal Type Theorymentioning

confidence: 99%

Section: Parametric Reducibility and Strong Normalizationmentioning

confidence: 99%

Contextual Modal Type Theory with Polymorphic Contexts

Murase

Nishiwaki²,

Igarashi

2023

Lecture Notes in Computer Science

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Layered Modal Type Theory

Hu,

Pientka

2024

Lecture Notes in Computer Science

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We introduce layering to modal type theory to combine type theory with intensional analysis. In particular, we demonstrate this idea by developing a 2-layered modal type theory. At the core of this type theory (layer 0) is a simply typed $$\lambda $$ λ -calculus with no modality. Layer 1 is obtained by extending the core language with one layer of contextual $$\square $$ □ types to support pattern matching on potentially open code from layer 0 while retaining normalization. Although both layers fundamentally share the same language and the same typing judgment, we only allow computation at layer 1. As a consequence, layer 0 accurately captures the syntactic representation of code in contrast to the computational behaviors at layer 1. The system is justified by normalization by evaluation (NbE) using a presheaf model. The normalization algorithm extracted from the model is sound and complete and is implemented in Agda.Layered modal type theory provides a uniform foundation for meta-programming with intensional analysis. We see this work as an important step towards a foundational way to support meta-programming in proof assistants.

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Borghuis

1998

Journal of Logic, Language and Information

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Coming to terms with modal logic : on the interpretation of modalities in typed lambda-calculus

Cited by 3 publications

References 0 publications

Contextual Modal Type Theory with Polymorphic Contexts

Contextual Modal Type Theory with Polymorphic Contexts

Layered Modal Type Theory

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