Temporalized logics and automata for time granularity

TL;DR

This paper develops monadic second-order theories and automata for layered structures representing different levels of time granularity, enabling expressive and decidable temporal logics for complex temporal models.

Contribution

It introduces temporalized automata as automata-theoretic counterparts to temporalized logics, facilitating analysis of layered time structures.

Findings

Automata are closed under Boolean operations.

Decidability of the proposed automata and logics.

Expressive equivalence between automata and temporal logics.

Abstract

Suitable extensions of the monadic second-order theory of k successors have been proposed in the literature to capture the notion of time granularity. In this paper, we provide the monadic second-order theories of downward unbounded layered structures, which are infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, and of upward unbounded layered structures, which consist of a finest domain and an infinite number of coarser and coarser domains, with expressively complete and elementarily decidable temporal logic counterparts. We obtain such a result in two steps. First, we define a new class of combined automata, called temporalized automata, which can be proved to be the automata-theoretic counterpart of temporalized logics, and show that relevant properties, such as closure under Boolean operations, decidability, and expressive equivalence with respect to temporal logics, transfer from component automata to temporalized ones. Then, we exploit the correspondence between temporalized logics and automata to reduce the task of finding the temporal logic counterparts of the given theories of time granularity to the easier one of finding temporalized automata counterparts of them.