Dimension-Minimality and Primality of Counter Nets

A k-Counter Net (k-CN) is a finite-state automaton equipped with k integer counters that are not allowed to become negative, but do not have explicit zero tests. This language-recognition model can be thought of as labelled vector addition systems with states, some of which are accepting. Certain decision problems for k-CNs become easier, or indeed decidable, when the dimension k is small. Yet, little is known about the effect that the dimension k has on the class of languages recognised by k-CNs. Specifically, it would be useful if we could simplify algorithmic reasoning by reducing the dimension of a given CN. To this end, we introduce the notion of dimension-primality for k-CN, whereby a k-CN is prime if it recognises a language that cannot be decomposed into a finite intersection of languages recognised by d-CNs, for some d<k. We show that primality is undecidable. We also study two related notions: dimension-minimality (where we seek a single language-equivalent d-CN of lower dimension) and language regularity. Additionally, we explore the trade-offs in expressiveness between dimension and non-determinism for CN.