A History of Until

Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: AUB 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 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 history 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.