Metric propositional neighborhood logics on natural numbers

摘要

Interval logics formalize temporal reasoning on interval structures over linearly (or partially) ordered domains, where time intervals are the primitive ontologi-cal entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we intro-duce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modal-ities referring, respectively, to an interval that is 'met by' the current one and to an interval that "meets" the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the length of intervals. We argue that MPNL can be successfully used in different areas of computer science to combine qualitative and quantitative interval temporal reasoning, thus providing a viable alter-native to well-established logical frameworks such as Dura-tion Calculus. We show that MPNL is decidable in double exponential time and expressively complete with respect to a well-defined sub-fragment of the two-variable fragment FO~2[N, =, <,<,s] of first-order logic for linear orders with successor function, interpreted over natural numbers. More-over, we show that MPNL can be extended in a natural way to cover full FO~2[N, =, <, s], but, unexpectedly, the latter (and hence the former) turns out to be undecidable.