On the relationships between theories of time granularity and the monadic second-order theory of one successor

In this paper we explore the connections between the monadic second-order theory of one successor (MSO[<] for short) and the theories of ω-layered structures for time granularity. We first prove that the decision problem for MSO[<] and that for a suitable first-order theory of the upward unbounded layered structure are inter-reducible. Then, we show that a similar result holds for suitable chain variants of the MSO theory of the totally unbounded layered structure (this allows us to solve some decision problems about theories of time granularity left open by Franceschet et al. [FRA 06]).