Reference. Implementing a modal dependent type theory [gratzer-sterling-birkedal-2019]

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.

@article{gratzer-sterling-birkedal-2019,
  author = {Gratzer, Daniel and Sterling, Jonathan and Birkedal, Lars},
  location = {New York, NY, USA},
  publisher = {ACM},
  date = {2019-07},
  doi = {10.1145/3341711},
  issn = {2475-1421},
  journaltitle = {Proceedings of the ACM on Programming Languages},
  keywords = {Modal types,dependent types,normalization by evaluation,type-checking},
  number = {ICFP},
  pages = {107:1--107:29},
  title = {Implementing a Modal Dependent Type Theory},
  volume = {3},
}