Analysis of spatio-temporal properties of stochastic systems using TSTL

In this article,we present Three-Valued spatio-temporal Logic (TSTL), which enriches the available spatiotemporal analysis of properties expressed in Signal spatio-temporal Logic (SSTL), to give further insight into the dynamic behavior of systems. Our novel analysis starts from the estimation of satisfaction probabilities of given SSTL properties and allows the analysis of their temporal and spatial evolution. Moreover, in our verification procedure, we use a three-valued approach to include the intrinsic and unavoidable uncertainty related to the simulation-based statistical evaluation of the estimates; this can be also used to assess the appropriate number of simulations to use depending on the analysis needs. We present the syntax and three-valued semantics of TSTL and specific extended monitoring algorithms to check the validity of TSTL formulas. We introduce a reliability requirement for TSTL monitoring and an automatic procedure to verify it. Two case studies demonstrate how TSTL broadens the application of spatio-temporal logics in realistic scenarios, enabling analysis of threat monitoring and privacy preservation based on spatial stochastic population models.