A logical characterization of timed regular languages

CLTLoc (Constraint LTL over clocks) is a quantifier-free extension of LTL allowing variables behaving like clocks over real numbers. CLTLoc is in PSPACE [1] and its satisfiability can polynomially be reduced to a Satisfiability Modulo Theories (SMT) problem, allowing a feasible implementation of a decision procedure. We used CLTLoc to capture the semantics of metric temporal logics over continuous time, such as Metric Interval Temporal Logic (MITL), resulting in the first successful implementation of a tool for checking MITL satisfiability [2], [3]. In this paper, we assess the expressive power of CLTLoc, by comparing it with various temporal formalisms over dense time. We restrict the analysis to well initialized models of formulae where the value of all clocks in the origin is either 0 or equal to a positive real constant. Under this assumption, when interpreted over timed words, the class of timed languages defined by CLTLoc formulae coincides with the class defined by Timed Automata. We also define a timed monadic first order logic of order, extending the one introduced by Kamp, which is expressively equivalent to CLTLoc for the class of timed languages that are defined by well initialized models.