LTL over integer periodicity constraints

Periodicity constraints are present in many logical formalisms, in fragments of Presburger LTL, in calendar logics, and in logics for access control, to quote a few examples. We introduce the logic PLTL, an extension of Linear-Time Temporal Logic LTL with past-time operators whose atomic formulae are defined from a first-order constraint language dealing with periodicity. The underlying constraint language is a fragment of Presburger arithmetic shown to admit a pspace-complete satisfiability problem and we establish that PLTL model-checking and satisfiability problems are in pspace as plain LTL. The logic PLTL is a quite rich and concise language to express periodicity constraints. We show that adding logical quantification to PLTL provides expspace-hard problems. As another application, we establish that the equivalence problem for extended single-string automata, known to express the equality of time granularities, is pspace-complete. The paper concludes by presenting a bunch of open problems related to fragments of Presburger LTL.