Constructing Unprejudiced Extensional Type Theories with Choices via Modalities.

Liron Cohen, Vincent Rahli

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Time-progressing expressions, i.e., expressions that compute to different values over time such as Brouwerian choice sequences or reference cells, are a common feature in many frameworks. For type theories to support such elements, they usually employ sheaf models. In this paper, we provide a general framework in the form of an extensional type theory incorporating various time-progressing elements along with a general possible-worlds forcing interpretation parameterized by modalities. The modalities can, in turn, be instantiated with topological spaces of bars, leading to a general sheaf model. This parameterized construction allows us to capture a distinction between theories that are “agnostic”, i.e., compatible with classical reasoning in the sense that classical axioms can be validated, and those that are “intuitionistic”, i.e., incompatible with classical reasoning in the sense that classical axioms can be proven false. This distinction is made via properties of the modalities selected to model the theory and consequently via the space of bars instantiating the modalities. We further identify a class of time-progressing elements that allows deriving “intuitionistic” theories that include not only choice sequences but also simpler operators, namely reference cells.

Original languageEnglish
Title of host publication7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022
EditorsAmy P. Felty
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages10:1-10:23
Number of pages23
ISBN (Electronic)9783959772334
DOIs
StatePublished - 2022
Event7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022 - Haifa, Israel
Duration: 2 Aug 20225 Aug 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume228
ISSN (Print)1868-8969

Conference

Conference7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022
Country/TerritoryIsrael
CityHaifa
Period2/08/225/08/22

Keywords

  • Agda
  • Choice sequences
  • Classical Logic
  • Constructive Type Theory
  • Extensional Type Theory
  • Intuitionism
  • Realizability
  • References
  • Theorem proving

ASJC Scopus subject areas

  • Software

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