Abstract
In the past few years, significant progress has been made on spatio-temporal cyber-physical systems in achieving spatio-temporal properties on several long-standing tasks. With the broader specification of spatio-temporal properties on various applications, the concerns over their spatio-temporal logics have been raised in public, especially after the widely reported safety-critical systems involving self-driving cars, intelligent transportation system, image processing. In this paper, we present a spatio-temporal specification language, STSL PC, by combining Signal Temporal Logic (STL) with a spatial logic S4 u, to characterize spatio-temporal dynamic behaviors of cyber-physical systems. This language is highly expressive: it allows the description of quantitative signals, by expressing spatio-temporal traces over real valued signals in dense time, and Boolean signals, by constraining values of spatial objects across threshold predicates. STSL PC combines the power of temporal modalities and spatial operators, and enjoys important properties such as finite model property. We provide a Hilbert-style axiomatization for the proposed STSL PC and prove the soundness and completeness by the spatio-temporal extension of maximal consistent set and canonical model. Further, we demonstrate the decidability of STSL PC and analyze the complexity of STSL PC. Besides, we generalize STSL to the evolution of spatial objects over time, called STSL OC, and provide the proof of its axiomatization system and decidability.
Highlights
It is a challenging work to model cyber-physical systems, because cyber-physical systems integrate cyber systems, physical environment and the interactive part of them, and because cyber-physical systems combine temporal and spatial aspects, discrete and continuous behaviors, and nondeterministic models [1]
3 We extend the expressiveness of STSLPC, called STSLOC, to the changes or evolution of spatial objects over time, and prove the incompleteness and undecidability
We propose the encoding of metric interval temporal logic (MITL) to Constraint Linear temporal logic (LTL) over clock (CLTLoc) [51], which is an extension of Constraint LTL [52] with clocks
Summary
It is a challenging work to model cyber-physical systems, because cyber-physical systems integrate cyber systems, physical environment and the interactive part of them, and because cyber-physical systems combine temporal and spatial aspects, discrete and continuous behaviors, and nondeterministic models [1]. Many works have been done with concurrent [2], hybrid [3,4,5] and stochastic [6, 7] behaviors of motion-based spatially distributed systems [8], but fewer researchers concentrate on spatio-temporal aspects. This work aims at building a spatio-temporal specification language by solving spatio-temporal constraints concerning dense time and real-valued variables, because an intelligent object in physical environment is equipped with changes in specified space and continuous time. While Boolean semantics evaluates Boolean signals from a trace which can be booleanized through a set of threshold predicates
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