I have the following statement in a paper:
Let $\Psi$ be the formal power series defined over the alphabet $\Omega$ and the log semiring by: $(\Psi, (a, b)) = -log(c((a,b)))$ for $(a,b) \in \Omega$, and let $S$ be the formal pwer series $S$ over the log semiring defined by: $S=\Omega^*+\Psi+\Omega^*$ (an alphabet is a finite set of symbols and $\Omega$ contains pairs of such symbols.)
It continues: $S$ is a rational power series as a +-product and closure of the polynomial power series $\Omega$ and $\Psi$.
What exactly is meant here? I know about the automata theoretical aspects, but I haven't heard of the notion "formal power series" over an alphabet and a semiring. How can I think of this?
The paper is at http://www.cs.nyu.edu/~mohri/pub/ ,  and  (virtually identical), page 15 and page 17 in the 22-page paper. It is a verbatim copy.