TY - GEN
T1 - Constructing Unprejudiced Extensional Type Theories with Choices via Modalities.
AU - Cohen, Liron
AU - Rahli, Vincent
N1 - Funding Information: Funding This research was partially supported by Grant No. 2020145 from the United States-Israel Binational Science Foundation (BSF). Publisher Copyright: © Liron Cohen and Vincent Rahli
PY - 2022/6/28
Y1 - 2022/6/28
N2 - 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.
AB - 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.
KW - Agda
KW - Choice sequences
KW - Classical Logic
KW - Constructive Type Theory
KW - Extensional Type Theory
KW - Intuitionism
KW - Realizability
KW - References
KW - Theorem proving
UR - http://www.scopus.com/inward/record.url?scp=85133719958&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.FSCD.2022.10
DO - https://doi.org/10.4230/LIPIcs.FSCD.2022.10
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 10:1-10:23
BT - 7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022
A2 - Felty, Amy P.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022
Y2 - 2 August 2022 through 5 August 2022
ER -