A spatio-temporal specification language and its completeness & decidability

Abstract

AbstractIn 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, STSLPC, by combining Signal Temporal Logic (STL) with a spatial logic S4u, 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. STSLPCcombines 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 STSLPCand prove the soundness and completeness by the spatio-temporal extension of maximal consistent set and canonical model. Further, we demonstrate the decidability of STSLPCand analyze the complexity of STSLPC. Besides, we generalize STSL to the evolution of spatial objects over time, called STSLOC, and provide the proof of its axiomatization system and decidability.