A History of Until

TL;DR

This paper introduces a new temporal operator to explicitly formalize the dual nature of the 'until' operator in linear-time logics, enabling well-behaved deduction rules and a sound, complete natural deduction system.

Contribution

It proposes a novel operator that captures the 'history' of 'until', facilitating the development of formal deduction systems for temporal logics with 'until'.

Findings

Developed a labeled natural deduction system for the new operator.

Proved the system's soundness and completeness relative to LTL with 'until'.

Provided a translation method linking the new system to existing temporal logic.

Abstract

Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: A until B holds at the current time instant w iff either B holds at w or there exists a time instant w' in the future at which B holds and such that A holds in all the time instants between the current one and w'. This "ambivalent" nature poses a significant challenge when attempting to give deduction rules for until. In this paper, in contrast, we make explicit this duality of until to provide well-behaved natural deduction rules for linear-time logics by introducing a new temporal operator that allows us to formalize the "history" of until, i.e., the "internal" universal quantification over the time instants between the current one and w'. This approach provides the basis for formalizing deduction systems for temporal logics endowed with the until operator. For concreteness, we give here a labeled natural deduction system for a linear-time logic endowed with the new operator and show that, via a proper translation, such a system is also sound and complete with respect to the linear temporal logic LTL with until.