Right Propositional Neighborhood Logic over Natural Numbers with Integer Constraints for Interval Lengths

Abstract

Interval temporal logics are based on interval structures over linearly (or partially) ordered domains, where time intervals, rather than time instants, are the primitive ontological entities. In this paper we introduce and study Right Propositional Neighborhood Logic over natural numbers with integer constraints for interval lengths, which is a propositional interval temporal logic featuring a modality for the 'right neighborhood' relation between intervals and explicit integer constraints for interval lengths. We prove that it has the bounded model property with respect to ultimately periodic models and is therefore decidable. In addition, we provide an EXPSPACE procedure for satisfiability checking and we prove EXPSPACE-hardness by a reduction from the exponential corridor tiling problem.

A Decidable Spatial Generalization of Metric Interval Temporal Logic

“…For any given horizontal k-sequence σ in L, we will denote by h σ q the first point of the q-th occurrence of σ. Hereafter, whenever σ will be evident from the context, we will write h q for h σ q . The next Lemma is analogous to Lemma 5.12 in [1]. However, in the spatial setting, to be able to reduce the size of the model we must also guarantee the existence of a certain number of occurrences of the sequence before a given point h q .…”

“…In the following, we will show how the techniques developed in [1] for the metric temporal logic RPNL+INT can be exploited to prove the decidability of DAC. We first give a bound on the size of finite fulfilling LSSs and then we show that, in the infinite case, we can safely restrict ourselves to infinite fulfilling LSSs with a finite bounded representation.…”

“…However, it (almost) never comes for free: it involves a blow up in complexity, that can possibly yield undecidability. In this paper, we study a spatial generalization of the decidable metric interval temporal logic RPNL+INT [1]. The main goal of spatial formal systems is to capture common-sense knowledge about space and to provide a calculus of spatial information.…”

“…In this paper, we present the Directional Area Calculus (DAC), that can be viewed as a two-dimensional variant of RPNL+INT [1]. DAC allows one to reason with basic shapes, such as lines, points, and rectangles, directional relations, and (a weak form of) areas.…”

Maximal Decidable Fragments of Halpern and Shoham’s Modal Logic of Intervals

“…The decidability of AĀ, over various classes of linear orders, has been proved by Bresolin et al [3] by reducing its satisfiability problem to that of the two-variable fragment of first-order logic over the same classes of linear orders [17]. An optimal (NEXPTIME) tableau-based decision procedure for AĀ over the integers has been given in [5] and later extended to the classes of all (resp., dense, discrete) linear orders [6], while a decidable metric extension of the future fragment of AĀ over the natural numbers has been proposed in [7] and later extended to the full logic [4]. Finally, a number of undecidable extensions of AĀ have been given in [2,3].…”