Dimension-Minimality and Primality of Counter Nets

Abstract: 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.