Affiliation:
1. Aarhus University, Denmark
2. Carnegie Mellon University, USA
Abstract
Modalities are everywhere in programming and mathematics! Despite this, however, there are still significant technical challenges in formulating a core dependent type theory with modalities. We present a dependent type theory
MLTT
🔒
supporting the connectives of standard Martin-Löf Type Theory as well as an
S4
-style necessity operator.
MLTT
🔒
supports a smooth interaction between modal and dependent types and provides a common basis for the use of modalities in programming and in synthetic mathematics. We design and prove the soundness and completeness of a type checking algorithm for
MLTT
🔒
, using a novel extension of normalization by evaluation. We have also implemented our algorithm in a prototype proof assistant for
MLTT
🔒
, demonstrating the ease of applying our techniques.
Funder
Villum Fonden
Air Force Office of Scientific Research
Natur og Univers, Det Frie Forskningsråd
Publisher
Association for Computing Machinery (ACM)
Subject
Safety, Risk, Reliability and Quality,Software
Cited by
12 articles.
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1. Normalization by evaluation for modal dependent type theory;Journal of Functional Programming;2023
2. A Category Theoretic View of Contextual Types: From Simple Types to Dependent Types;ACM Transactions on Computational Logic;2022-10-20
3. On Coevaluation Behavior and Equivalence;Mathematics;2022-10-14
4. Normalization for fitch-style modal calculi;Proceedings of the ACM on Programming Languages;2022-08-29
5. Normalization for Multimodal Type Theory;Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science;2022-08-02