Dimension-Minimality and Primality of Counter Nets

TL;DR

This paper investigates the properties of $k$-Counter Nets, focusing on their dimension-primality, minimality, and the impact of dimension on language recognition, revealing undecidability results and expressiveness trade-offs.

Contribution

It introduces the concept of dimension-primality for $k$-Counter Nets and studies related notions like minimality and regularity, providing new insights into their computational properties.

Findings

Primality of $k$-CN is undecidable.

Dimension-minimality can be characterized and analyzed.

Trade-offs exist between dimension and non-determinism in expressiveness.

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.