Equivalence Between Model-Checking Flat Counter Systems and Presburger Arithmetic

We show that model-checking flat counter systems over CTL* (with arithmetical constraints on counter values) has the same complexity as the satisfiability problem for Presburger arithmetic. The lower bound already holds with the temporal operator EF only, no arithmetical constraints in the logical language and with guards on transitions made of simple linear constraints. This complements our understanding of model-checking flat counter systems with linear-time temporal logics, such as LTL for which the problem is already known to be (only) NP-complete with guards restricted to the linear fragment.