Branching-time model checking of one-counter processes

One-counter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from the modal mu-calculus for this problem. First, we analyze the periodic behaviour of CTL over OCPs 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. Thus, model checking fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This generalizes a 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 OCP, CTL model checking is PSPACE-hard. Third, we show that there already exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. For the latter, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable. We demonstrate that our approach can be used to answer further open questions.

💡 Research Summary

The paper investigates the computational complexity of model checking Computation Tree Logic (CTL) over one‑counter processes (OCPs), a subclass of push‑down systems that use a single unary stack symbol. An OCP consists of a finite set of control locations together with a single non‑negative integer counter; transitions may increment, decrement, or test the counter for zero. Because the counter can grow arbitrarily, the state space is infinite, yet its structure is highly regular.

The authors first observe that a PSPACE upper bound for CTL model checking on OCPs follows immediately from the known PSPACE bound for the modal μ‑calculus, which subsumes CTL. However, this bound is coarse and does not reflect the influence of the formula’s syntactic shape or the size of the underlying process. To obtain a finer analysis, they introduce a new syntactic measure called leftward‑until depth (LUD). LUD is defined as the maximal nesting depth of “until” operators that appear on the left‑hand side of a formula (i.e., the number of times a sub‑formula of the form E U ψ χ is itself nested inside another such sub‑formula). Intuitively, LUD captures how far a CTL formula may force the verification procedure to walk “backwards” along the counter, thereby limiting the amount of counter decrement that must be examined.

Using LUD, the authors prove a periodicity theorem for OCPs: for any fixed control location q and any two counter values c and c + L, where L is a polynomial function of the number of control locations |Q| and the LUD of the formula, the truth value of the formula at (q,c) and (q,c + L) coincides. This result follows from the observation that the set of reachable counter values from a given configuration eventually repeats with a period bounded by the product of the maximal counter change per transition and the number of control locations. Consequently, to decide whether an OCP satisfies a CTL formula φ it suffices to explore only a finite “representative window” of counter values, namely the interval