A Non-Wellfounded and Labelled Sequent Calculus for Bimodal Provability Logic

TL;DR

This paper introduces a labelled, non-wellfounded sequent calculus for bimodal provability logic CS, leveraging cyclic proof techniques to model semantics and establish soundness, completeness, and decision procedures.

Contribution

It develops a novel non-wellfounded, labelled calculus for bimodal provability logic CS, incorporating cyclic proof methods for semantic modeling and proof-theoretic properties.

Findings

Proves soundness and completeness of the calculus

Provides a primitive decision procedure

Offers a method to extract countermodels

Abstract

We present a labelled and non-wellfounded calculus for the bimodal provability logic CS. The system is obtained by modelling the Kripke-like semantics of this logic. As in arXiv:2309.00532, we enforce the second-order property of converse wellfoundedness by using techniques from cyclic proof theory. We will prove soundness and completeness of this system with respect to the semantics and provide a primitive decision procedure together with a way to extract countermodels.