Back from the future


Abstract
Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: A∪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 ẃ. 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 ẃ. 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 N for a linear-time logic LTL∇ endowed with the new history operator. We show that LTL∇ is equivalent to the linear temporal logic LTL with until, which follows by formalizing back and forth translations between the two logics. We also define an indirect translation from LTL∇ into LTL via temporal logics with past operators; such a result provides an upper bound to the problem of satisfiability for LTL∇ formulas.
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DOI 10.3166/jancl.20.241-277
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References found in this work BETA

Past, present and future.Arthur Prior - 1967 - Revue Philosophique de la France Et de l'Etranger 157:476-476.
Basic Proof Theory.Roy Dyckhoff - 2001 - Bulletin of Symbolic Logic 7 (2):280-280.
Natural Deduction: A Proof-Theoretical Study.Richmond Thomason - 1965 - Journal of Symbolic Logic 32 (2):255-256.
Labelled Deductive Systems.Dov M. Gabbay - 1996 - Oxford University Press.
Logics of Time and Computation.Robert Goldblatt - 1990 - Studia Logica 49 (2):284-286.

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