Realising Intensional S4 and GL Modalities

Chen, Liang-Ting; Ko, Hsiang-Shang

doi:10.4230/lipics.csl.2022.14

Cited by 3 publications

References 28 publications

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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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Layered Modal Type Theory

Hu,

Pientka

2024

Lecture Notes in Computer Science

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A totally predictable outcome: an investigation of traversals of infinite structures

Bazerman

2022

Proceedings of the 15th ACM SIGPLAN International Haskell Symposium

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A Layered Approach to Intensional Analysis in Type Theory

Hu,

Pientka

2024

ACM Trans. Program. Lang. Syst.

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We introduce layering to modal type theory to support meta-programming and intensional analysis coherently. 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 λ -calculus with no modality. Layer 1 is obtained by extending the core language with one layer of contextual ◻ types to support pattern matching on potentially open code from layer 0 while retaining normalization. Although both layers fundamentally share a uniform syntax 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. Moreover, the uniform syntax at both layers enables quotation and code running. 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.

show abstract

Realising Intensional S4 and GL Modalities

Cited by 3 publications

References 28 publications

Layered Modal Type Theory

Layered Modal Type Theory

A totally predictable outcome: an investigation of traversals of infinite structures

A Layered Approach to Intensional Analysis in Type Theory

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