Modular temporal logic

D. Therien and T. Wilke (1996) characterized the Until hierarchy of linear temporal logic in terms of aperiodic monoids. Here, a temporal operator able to count modulo q is introduced. Temporal logic augmented with such operators is found decidable as it is shown to express precisely the solvable regular languages. Natural hierarchies are shown to arise when modular and conventional operators are interleaved. Modular operators are then cast as special cases of more general “group” temporal operators which, added to temporal logic, allow capturing any regular language L in much the same way that the syntactic monoid of L is constructed from groups and aperiodic monoids in the sense of Krohn-Rhodes