On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics

TL;DR

This paper explores two proof calculus styles for tense logics using nested sequents, establishing their equivalence and linking display postulates with deep inference, which enhances proof search capabilities.

Contribution

It introduces a correspondence between shallow calculi with display postulates and deep inference calculi, improving proof search in tense logics.

Findings

Deep inference calculi have the subformula property.

Shallow calculi are less suitable for proof search due to structural rules.

The two calculus styles are shown to be equivalent for various tense logic extensions.

Abstract

We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search.