Display to Labeled Proofs and Back Again for Tense Logics

TL;DR

This paper establishes effective translations between display calculus proofs and labeled calculus proofs for tense logics, enhancing understanding of proof structures and their interrelations.

Contribution

It introduces a method to translate proofs between display and labeled calculi for tense logics, using labeled polytrees for canonical representation.

Findings

Effective translation from display to labeled calculus for Kt with path axioms.

Canonical representation of display sequents as labeled polytrees.

Insights into proof correspondence for tense logics.

Abstract

We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special form that is translatable to a derivation in the associated display calculus. A key insight in this converse translation is a canonical representation of display sequents as labeled polytrees. Labeled polytrees, which represent equivalence classes of display sequents modulo display postulates, also shed light on related correspondence results for tense logics.