# On the Magnitude of Completeness Thresholds in Bounded Model Checking

Daniel Bundala ( University of Oxford )

Bounded model checking (BMC) is a highly successful bug-finding method that examines paths of bounded length for violations of a given regular or $\omega$-regular specification. A "completeness threshold" for a given model $M$ and specification $\varphi$ is a bound $k$ such that, if no counterexample to $\varphi$ of length $k$ or less can be found in $M$, then $M$ in fact satisfies $\varphi$. The quest for `small' completeness thresholds in BMC goes back to the very inception of the technique, over a decade ago, and remains a topic of active research.

For a fixed specification, completeness thresholds are typically expressed in terms of key attributes of the models under consideration, such as their diameter (length of the longest shortest path) and especially their "recurrence diameter" (length of the longest loop-free path). A recent research paper (Kroening et al.) identified a large class of LTL specifications having completeness thresholds "linear" in the models' recurrence diameter. However, the authors left open the question of whether linearity is in general even decidable.

In this talk, we settle the problem in the affirmative, by showing that the linearity problem for both regular and $\omega$-regular specifications (provided as automata and B\"uchi automata respectively) is PSPACE-complete. Moreover, we establish the following dichotomies: for regular specifications, completeness thresholds are either linear or exponential, whereas for $\omega$-regular specifications, completeness thresholds are either linear or at least quadratic.

This is joint work with Joel Ouaknine and James Worrell.