Automata: Theoretic Aspects of Formal Power Series
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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)| . 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 |