Liron Cohen ; Vincent Rahli - $\text{TT}^{\Box}_{\mathcal C}$: a Family of Extensional Type Theories with Effectful Realizers of Continuity

lmcs:11666 - Logical Methods in Computer Science, June 26, 2024, Volume 20, Issue 2 - https://doi.org/10.46298/lmcs-20(2:18)2024
$\text{TT}^{\Box}_{\mathcal C}$: a Family of Extensional Type Theories with Effectful Realizers of ContinuityArticle

Authors: Liron Cohen ; Vincent Rahli

    $\text{TT}^{\Box}_{{\mathcal C}}$ is a generic family of effectful, extensional type theories with a forcing interpretation parameterized by modalities. This paper identifies a subclass of $\text{TT}^{\Box}_{{\mathcal C}}$ theories that internally realizes continuity principles through stateful computations, such as reference cells. The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. In this paper, the modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled internally in the theory.


    Volume: Volume 20, Issue 2
    Published on: June 26, 2024
    Accepted on: April 17, 2024
    Submitted on: July 28, 2023
    Keywords: Computer Science - Logic in Computer Science

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