Satisfiability for MTL and TPTL over Non-monotonic Data Words

In the context of real-time systems, Metric Temporal Logic (MTL) and Timed Propositional Temporal Logic (TPTL) are prominent and widely used extensions of Linear Temporal Logic. In this paper, we examine the possibility of using MTL and TPTL to specify properties about classes of non-monotonic data languages over the natural numbers. Words in this class may model the behaviour of, e.g., one-counter machines. We proved, however, that the satisfiability problem for many reasonably expressive fragments of MTL and TPTL is undecidable, and thus the use of these logics is rather limited. On the positive side we prove that satisfiability for the existential fragment of TPTL is NP-complete.