On coarser interval temporal logics

Abstract

The primary characteristic of interval temporal logic is that intervals, rather than points, are taken as the primitive ontological entities. Given their generally bad computational behavior of interval temporal logics, several techniques exist to produce decidable and computationally affordable temporal logics based on intervals. In this paper we take inspiration from Golumbic and Shamir's coarser interval algebras, which generalize the classical Allen's Interval Algebra, in order to define two previously unknown variants of Halpern and Shoham's logic (HS) based on coarser relations. We prove that, perhaps surprisingly, the satisfiability problem for the coarsest of the two variants, namely HS 3 , not only is decidable, but PSpace-complete in the finite/discrete case, and PSpace-hard in any other case; besides proving its complexity bounds, we implement a tableau-based satisfiability checker for it and test it against a systematically generated benchmark. Our results are strengthened by showing that not all coarser-than-Allen's relations are a guarantee of decidability, as we prove that the second variant, namely HS 7 , remains undecidable in all interesting cases.

Decision Tree Learning with Spatial Modal Logics

“…Very briefly, it is worth recalling that satisfiability for H S , its one-dimensional version, is undecidable [21], and that various strategies have been considered in the literature to define fragments or variants of H S with better computational behaviour. These include constraining the underlying temporal structure [31], restricting the set of modal operators [1,9], softening the semantics to a reflexive one [27,29], restricting the nesting of modal operators [10], restricting the propositional power of the languages [11], and considering coarser interval temporal logics based on interval relations that describe a less precise relationship between intervals (similarly to what topological relations do) [33]. In the case of H S 2 , only the sub-languages for H S 2 RCC8 and H S 2 RCC5 have been studied, in [25], and their satisfiability problem is undecidable as well, even under very simple assumptions, or can be proven so by exploiting the results on the one-dimensional case.…”

Dynamic spatio‐temporal logic based on RCC‐8

“…Reynolds 41 has proved that the satisfiability for ‐formulas is a PSPACE‐ complete problem over all strict and finite strict linear orders. Muñoz‐Velasco et al 3 propose a Halpern‐Shoham interval temporal logic based on coarser relations , which is a variant of Halpern‐Shoham's logic , 29 and also prove that the satisfiability for is PSPACE‐ complete in the case of the natural numbers and the integers. The spatial logic 4 is one of the most influential formalism for topological relations, which extends modal logic with some universal modalities.…”

“…There are many formalisms used to reason temporal and spatial knowledge. Regarding qualitative temporal reasoning, many researchers have already proposed some temporal logics, such as Allen's Interval calculus, 1 propositional temporal logic , 2 and coarser Halpern‐Shoham temporal logic 3 . Regarding qualitative spatial reasoning, there have been a lot of works investigating spatial logics, such as, 4 (or ‐8), 5‐7 and ‐8 8…”

On neighbourhood singleton-style consistencies for qualitative spatial and temporal reasoning

“…More formally, QSTR restricts the vocabulary of rich mathematical theories that deal with spatial and temporal entities to simple qualitative constraint languages. Thus, QSTR provides a concise framework that allows for rather inexpensive reasoning about entities located in space and time and, hence, further boosts research and applications to a plethora of areas and domains that include, but are not limited to, dynamic GIS [3], cognitive robotics [4], deep learning [5], spatio-temporal design [6], qualitative model generation from video [7], ambient intelligence [8,9], visual explanation [10] and sensemaking [11], semantic question-answering [12], qualitative simulation [13], spatio-temporal data mining [14,15,16], and modal logic [17,18,19]. The interested reader may look into a more comprehensive review of the emerging applications, the trends, and the future directions of QSTR in [20,21].…”