$\text{TT}^{\Box}_{\mathcal C}$: a Family of Extensional Type Theories with Effectful Realizers of Continuity

TL;DR

This paper introduces a family of effectful, extensional type theories with a forcing interpretation that can internally realize continuity principles through stateful computations, bridging semantic and computational justifications.

Contribution

It identifies a subclass of these theories that internally compute moduli of continuity using stateful effects like reference cells.

Findings

Effectful computations can directly compute moduli of continuity.

A subclass of $ ext{TT}^{oxempty}_ ext{C}$ theories internally realizes continuity.

Stateful effects enable computational interpretation of classical continuity principles.

Abstract

$\text{TT}^{\Box}_{{\mathcal C}}$ 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.