Metric propositional neighborhood logics on natural numbers

Abstract

Interval logics formalize temporal reasoning on interval structures over linearly (or partially) ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities 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 alternative to well-established logical frameworks such as Duration 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 FO2[N,=,<,s] of first-order logic for linear orders with successor function, interpreted over natural numbers. Moreover, we show that MPNL can be extended in a natural way to cover full FO2[N,=,<,s], but, unexpectedly, the latter (and hence the former) turns out to be undecidable

Finite satisfiability of propositional interval logic formulas with multi-objective evolutionary algorithms

“…Expressiveness and (potential) applications of these languages have been extensively discussed in [4,5]: we refer the reader to such a publication for details. Here, we provide a short summary of its outcomes.…”

“…The fragment A, seen as a sub-logic of AA (PNL) and therefore also of its metric version (MPNL [5]), is interesting from the practical point of view. Expressiveness and (potential) applications of these languages have been extensively discussed in [4,5]: we refer the reader to such a publication for details.…”

Prompt Interval Temporal Logic

“…Last but not least, we are interested in comparing the expressiveness of the logics PROMPT-PNL and PROMPT d -PNL with that of metric PNL, that is, the metric extension of PNL introduced and systematically studied in [7].…”

Metric propositional neighborhood logic with an equivalence relation

“…It is worth pointing out that, as in [2], the successor is assumed to be in relational form (it can be easily checked that the relational and functional characterizations of the successor are not equivalent in FO 2 răs). As a matter of fact, the addition of a successor function to PNL immediately yields undecidability, as shown in [3].…”