Branching-Time Model Checking of One-Counter Processes and Timed Automata

One-counter automata (OCA) are pushdown automata which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) on transition systems induced by one-counter automata. A PSPACE upper bound is inherited from the modal μ-calculus for this problem proven by Serre. First, we analyze the periodic behaviour of CTL over OCA and derive a model checking algorithm whose running time is exponential only in the number of control locations and a syntactic notion of the formula that we call leftward until depth. In particular, model checking fixed OCA against CTL formulas with a fixed leftward until depth is in P. This generalizes a corresponding recent result of the first author, Mayr and To for the expression complexity of CTL’s fragment EF. Second, we prove that already over some fixed OCA, CTL model checking is PSPACE-hard, i.e., expression complexity is PSPACE-hard. Third, we show that there already exists a fixed CTL formula for which model checking of OCA is PSPACE-hard, i.e., data complexity is PSPACE-hard as well. To obtain the latter result, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspace-uniform NC and (ii) PSPACE is AC-serializable.We demonstrate that our approach can be used to obtain further results. We show that model-checking CTL’s fragment EF over OCA is hard for P , thus establishing a matching lower bound. We moreover show that the following problem is hard for PSPACE: Given a one-counter Markov decision process, a set of target states with counter value zero each, and an initial state, to decide whether the probability that the initial state will eventually reach one of the target states is arbitrarily close to 1. This improves a recently proven lower bound for every level of the boolean hierarchy shown by Brazdil et al. Finally, we prove that there is a fixed CTL formula for which model checking 2-clock timed automata is PSPACE-hard, generalizing a PSPACEhardness result for the combined complexity by Laroussinie et al.