Logics of time and computation
Logics of time and computation
A propositional modal logic of time intervals
Journal of the ACM (JACM)
Temporal Logic with Forgettable Past
LICS '02 Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science
A Tableau System for Linear-TIME Temporal Logic
TACAS '97 Proceedings of the Third International Workshop on Tools and Algorithms for Construction and Analysis of Systems
A New One-Pass Tableau Calculus for PLTL
TABLEAUX '98 Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
Automated Natural Deduction for Propositional Linear-Time Temporal Logic
TIME '07 Proceedings of the 14th International Symposium on Temporal Representation and Reasoning
Combining Classical and Intuitionistic Implications
FroCoS '07 Proceedings of the 6th international symposium on Frontiers of Combining Systems
Labeled Natural Deduction Systems for a Family of Tense Logics
TIME '08 Proceedings of the 2008 15th International Symposium on Temporal Representation and Reasoning
A Labeled Natural Deduction System for a Fragment of CTL*
LFCS '09 Proceedings of the 2009 International Symposium on Logical Foundations of Computer Science
Labelled Tableaux for Distributed Temporal Logic
Journal of Logic and Computation
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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.