Absolute Convergence of Rational Series Is Semi-decidable

Abstract

We study real-valued absolutely convergent rational series, i.e. functions $r: {\it\Sigma}^* \rightarrow {\mathbb R}$, defined over a free monoid ${\it\Sigma}^*$, that can be computed by a multiplicity automaton A and such that $\sum_{w\in {\it\Sigma}^*}|r(w)|<\infty$. We prove that any absolutely convergent rational series r can be computed by a multiplicity automaton A which has the property that r _|A| is simply convergent, where r _|A| is the series computed by the automaton |A| derived from A by taking the absolute values of all its parameters. Then, we prove that the set ${\cal A}^{rat}({\it\Sigma})$ composed of all absolutely convergent rational series is semi-decidable and we show that the sum $\sum_{w\in \Sigma^*}|r(w)|$ can be estimated to any accuracy rate for any $r\in {\cal A}^{rat}({\it\Sigma})$. We also introduce a spectral radius-like parameter ρ _|r| which satisfies the following property: r is absolutely convergent iff ρ _|r|< 1.