### Abstract

We describe a dependent type theory, and a denotational model for it, that incorporates both intensional and extensional semantic universes. In the former, terms and types are interpreted as strategies on certain graph games, which are concrete data structures of a generalized form, and in the latter as stable functions on event domains. The concrete data structures themselves form an event domain, with which we may interpret an (extensional) universe type of (intensional) types. A dependent game corresponds to a stable function into this domain; we use its trace to define dependent product and sum constructions as it captures precisely how unfolding moves combine with the dependency to shape the possible interaction in the game. Since each strategy computes a stable function on CDS states, we can lift typing judgements from the intensional to the extensional setting, giving an expressive type theory with recursively defined type families and type operators. We define an operational semantics for intensional terms, giving a functional programming language based on our type theory, and prove that our semantics for it is computationally adequate. By extending it with a simple non-local control operator on intensional terms, we can precisely characterize behaviour in the intensional model. We demonstrate this by proving full abstraction and full completeness results.

Original language | English |
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Title of host publication | Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018 |

Publisher | IEEE |

Pages | 95-104 |

Number of pages | 10 |

Volume | Part F138033 |

ISBN (Electronic) | 9781450355834 |

DOIs | |

Publication status | Published - 9 Jul 2018 |

Event | 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018 - Oxford, UK United Kingdom Duration: 9 Jul 2018 → 12 Jul 2018 |

### Publication series

Name | Proceedings - Symposium on Logic in Computer Science |
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Publisher | IEEE |

ISSN (Electronic) | 2575-5528 |

### Conference

Conference | 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018 |
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Country | UK United Kingdom |

City | Oxford |

Period | 9/07/18 → 12/07/18 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018*(Vol. Part F138033, pp. 95-104). (Proceedings - Symposium on Logic in Computer Science). IEEE. https://doi.org/10.1145/3209108.3209206

**Extensional and intensional semantic universes : A denotational model of dependent types.** / Blot, Valentin; Laird, Jim.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018.*vol. Part F138033, Proceedings - Symposium on Logic in Computer Science, IEEE, pp. 95-104, 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK United Kingdom, 9/07/18. https://doi.org/10.1145/3209108.3209206

}

TY - GEN

T1 - Extensional and intensional semantic universes

T2 - A denotational model of dependent types

AU - Blot, Valentin

AU - Laird, Jim

PY - 2018/7/9

Y1 - 2018/7/9

N2 - We describe a dependent type theory, and a denotational model for it, that incorporates both intensional and extensional semantic universes. In the former, terms and types are interpreted as strategies on certain graph games, which are concrete data structures of a generalized form, and in the latter as stable functions on event domains. The concrete data structures themselves form an event domain, with which we may interpret an (extensional) universe type of (intensional) types. A dependent game corresponds to a stable function into this domain; we use its trace to define dependent product and sum constructions as it captures precisely how unfolding moves combine with the dependency to shape the possible interaction in the game. Since each strategy computes a stable function on CDS states, we can lift typing judgements from the intensional to the extensional setting, giving an expressive type theory with recursively defined type families and type operators. We define an operational semantics for intensional terms, giving a functional programming language based on our type theory, and prove that our semantics for it is computationally adequate. By extending it with a simple non-local control operator on intensional terms, we can precisely characterize behaviour in the intensional model. We demonstrate this by proving full abstraction and full completeness results.

AB - We describe a dependent type theory, and a denotational model for it, that incorporates both intensional and extensional semantic universes. In the former, terms and types are interpreted as strategies on certain graph games, which are concrete data structures of a generalized form, and in the latter as stable functions on event domains. The concrete data structures themselves form an event domain, with which we may interpret an (extensional) universe type of (intensional) types. A dependent game corresponds to a stable function into this domain; we use its trace to define dependent product and sum constructions as it captures precisely how unfolding moves combine with the dependency to shape the possible interaction in the game. Since each strategy computes a stable function on CDS states, we can lift typing judgements from the intensional to the extensional setting, giving an expressive type theory with recursively defined type families and type operators. We define an operational semantics for intensional terms, giving a functional programming language based on our type theory, and prove that our semantics for it is computationally adequate. By extending it with a simple non-local control operator on intensional terms, we can precisely characterize behaviour in the intensional model. We demonstrate this by proving full abstraction and full completeness results.

UR - http://www.scopus.com/inward/record.url?scp=85051120455&partnerID=8YFLogxK

U2 - 10.1145/3209108.3209206

DO - 10.1145/3209108.3209206

M3 - Conference contribution

VL - Part F138033

T3 - Proceedings - Symposium on Logic in Computer Science

SP - 95

EP - 104

BT - Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018

PB - IEEE

ER -