Decidable theories of !-layered metric temporal structures

This paper focuses on decidability problems for metric and layered temporal logics which allow one to model time granularity in various contexts. The decidability of pure metric (non-granular) fragments and of metric temporal logics endowed with nitely many layers has been already proved by reduction to the decidability problem of the well-known theory S1S. In the present work, we prove the decidability of both the theory of metric temporal structures provided with an innnite number of arbitrarily coarse temporal layers and the theory of metric temporal structures provided with an innnite number of arbitrarily ne temporal layers. The proof for the rst theory is obtained by reduction to the decidability problem of an extension of S1S which is proved to be the logical counterpart of the class of !-languages accepted by systolic tree automata. The proof for the second one is done through the reduction to the monadic second-order decidable theory of k successors SkS.