On the total variation distance of semi-markov chains

Semi-Markov chains (SMCs) are continuous-time probabilistic transition systems where the residence time on states is governed by generic distributions on the positive real line. This paper shows the tight relation between the total variation distance on SMCs and their model checking problem over linear real-time specifications. Specifically, we prove that the total variation between two SMCs coincides with the maximal difference w.r.t. the likelihood of satisfying arbitrary MTL formulas or ω-languages recognized by timed automata. Computing this distance (i.e., solving its threshold problem) is NPhard and its decidability is an open problem. Nevertheless, we propose an algorithm for approximating it with arbitrary precision