Branching within Time: an Expressively Complete and Elementarily Decidable Temporal Logic for Time Granularity

Suitable extensions of monadic second-order theories of k successors have been proposed in the literature to specify in a concise way reactive systems whose behaviour can be naturally modeled with respect to a (possibly infinite) set of differently-grained temporal domains. This is the case, for instance, of the wide-ranging class of real-time reactive systems whose components have dynamic behaviours regulated by very different time constants, e.g., days, hours, and seconds. In this paper, we focus on the theory of k-refinable downward unbounded layered structures MSO[<_{tot},(\downarrow_i)_{i=0}^{k-1}], that is, the theory of infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, whose satisfiability problem is nonelementarily decidable. We define a propositional temporal logic counterpart of MSO[<_{tot},(\downarrow_i)_{i=0}^{k-1}], with set quantification restricted to infinite paths, called CTSL, which features an original mix of linear and branching temporal operators. We prove the expressive completeness of CTSL with respect to such a path fragment of MSO[<_{tot}, (\downarrow_i)_{i=0}^{k-1}], and show that its satisfiability problem is 2EXPTIME-complete.