A Decidable Spatial Generalization of Metric Interval Temporal Logic

Abstract

Temporal reasoning plays an important role in\ud artificial intelligence. Temporal logics provide a natural framework\ud for its formalization and implementation. A standard\ud way of enhancing the expressive power of temporal logics is\ud to replace their unidimensional domain by a multidimensional\ud one. In particular, such a dimensional increase can be exploited\ud to obtain spatial counterparts of temporal logics. Unfortunately,\ud it often involves a blow up in complexity, possibly losing\ud decidability. In this paper, we propose a spatial generalization\ud of the decidable metric interval temporal logic RPNL+INT,\ud called Directional Area Calculus (DAC). DAC features two\ud modalities, that respectively capture (possibly empty) rectangles\ud to the north and to the east of the current one, and\ud metric operators, to constrain the size of the current rectangle.\ud We prove the decidability of the satisfiability problem for\ud DAC, when interpreted over frames built on natural numbers,\ud and we analyze its complexity. In addition, we consider a\ud weakened version of DAC, called WDAC, which is expressive\ud enough to capture meaningful qualitative and quantitative\ud spatial properties and computationally better

Analysis of Spatio-temporal Properties of Stochastic Systems Using TSTL

“…Other important logic-based formalisms to mention are those that have been proposed for reasoning about the topological [6] or directional [7] aspects of the interacting entities; for networks of processes [39], which can express spatio-temporal properties in discrete time; and for (graph) rewrite theories [2], [36], bigraphs [15], and data structure as graphs [9] and heaps [8]. Model checking of these logics usually has high computational complexity [7] or is even undecidable [35].…”

Monitoring mobile and spatially distributed cyber-physical systems

“…Several logic-based formalisms have been proposed to specify the behavior and the spatial structure of concurrent systems [7] and for reasoning about the topological [4] or directional [6] aspects of the interacting entities. In topological reasoning [4], the spatial objects are sets of points and the relation between them is preserved under translation, scaling and rotation.…”

“…In directional reasoning, the relation between objects depends on their relative position. These logics are usually highly computationally complex [6] or even undecidable [16].…”

SpaTeL

“…Several logics have been proposed for specifying the behavior and the spatial structure of concurrent systems [9] and for reasoning about the topological [5] or directional [7] aspects of the interacting entities. In topological reasoning [5], the spatial objects are sets of points and the relation between them is preserved under translation, scaling and rotation.…”

“…In directional reasoning, the relation between objects depends on their relative position. These logics are usually highly computationally complex [7] or even undecidable [22]. Even though there has been a lot work done in spatial logics and temporal logics with applications to several domains [8,13], spatiotemporal reasoning is scarcely explored.…”