MTL Satisfiability over the Integers

We investigate the satisfiability problem for metric temporal logic (MTL) with both past and future operators over linear discrete bi-infinite time models — where time is unbounded both in the future and in the past — isomorphic to the integer numbers. We provide a technique to reduce satisfiability over the integers to satisfiability over the well-known mono-infinite time model of natural numbers, and we show how to implement the technique through an automata-theoretic approach. We also prove that MTL satisfiability over the integers is EXPSPACE-complete, hence the given algorithm is optimal in the worst case.