Robustness of Temporal Logic Specifications

In this paper, we consider the robust interpretation of Metric Temporal Logic (MTL)formulas over signals that take values in metric spaces. For such signals, whichare generated by systems whose states are equipped with nontrivial metrics, forexample continuous or hybrid, robustness is not only natural, but also a criticalmeasure of system performance. Thus, we propose multi-valued semantics for MTLformulas, which capture not only the usual Boolean satisfiability of the formula,but also topological information regarding the distance, ε, from unsatisfiability. Weprove that any other signal that remains ε-close to the initial one also satisfies thesame MTL specification under the usual Boolean semantics. Finally, our frameworkis applied to the problem of testing formulas of two fragments of MTL, namelyMetric Interval Temporal Logic (MITL) and closed Metric Temporal Logic (clMTL),over continuous-time signals using only discrete-time analysis. The motivating ideabehind our approach is that if the continuous-time signal fulfills certain conditionsand the discrete time signal robustly satisfies the temporal logic specification, thenthe corresponding continuous-time signal should also satisfy the same temporal logicspecification.