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(LTL∇) 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. (shrink)

We consider a new fragment of first-order logic with two variables. This logic is defined over interval structures. It constitutes unary predicates, a binary predicate and a function symbol. Considering such a fragment of first-order logic is motivated by defining a general framework for event-based interval temporal logics. In this paper, we present a sound, complete and terminating decision procedure for this logic. We show that the logic is decidable, and provide a NEXPTIME complexity bound for satisfiability. This result shows (...) that even a simple decidable fragment of first-order logic has NEXPTIME complexity. (shrink)

Interval temporal logics provide both an insight into a nature of time and a framework for temporal reasoning in various areas of computer science. In this paper we present sound and complete relational proof systems in the style of dual tableaux for relational logics associated with modal logics of temporal intervals and we prove that the systems enable us to verify validity and entailment of these temporal logics. We show how to incorporate in the systems various relations between intervals and/or (...) various time orderings. (shrink)

In this paper, we investigate the expressiveness of the variety of propositional interval neighborhood logics , we establish their decidability on linearly ordered domains and some important subclasses, and we prove the undecidability of a number of extensions of PNL with additional modalities over interval relations. All together, we show that PNL form a quite expressive and nearly maximal decidable fragment of Halpern–Shoham’s interval logic HS.