Model checking restricted sets of timed paths

In this paper, we study the complexity of model-checking formulas of four important real-time logics (TPTL,MTL,MITL, and TCTL) over restricted sets of timed paths. The classes of restricted sets of timed paths that we consider are (i) a single finite (or ultimately periodic) timed path, (ii) an infinite set of finite (or infinite) timed paths defined by a finite (or ultimately periodic) path in a region graph, (iii) an infinite set of finite (or infinite) timed paths defined by a finite (or ultimately periodic) path in a zone graph.Several results are quite negative: TPTL and MTL remain undecidable along region- and zone-paths. On the other hand, we obtained PTIME algorithms for model-checking TCTL along a region path, and for MTL along a single timed path.