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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.)

$S$ 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/, [99], page 15 and page 17.

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    $\begingroup$ Is that a verbatim copy from the paper? $\endgroup$ – Mariano Suárez-Álvarez Jan 24 '11 at 5:02
  • $\begingroup$ Could you provide a link or a precise reference to the paper, please? $\endgroup$ – t.b. Jan 24 '11 at 5:04
  • $\begingroup$ yes. updated. thanks. $\endgroup$ – Felix Dombek Jan 24 '11 at 5:17
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A forum power series is a "Symbolic series" and not necessarily one where the object the series is over is a "number".

The idea comes from group there where you have the operations of multiplication and addition in a ring. you can form the expressions $ax + bx^2 + cx^3$, etc... e.g., the polynomials or P[x] but note x may not necessarily be in the ring. We can still write such expressions and take x as a sort of place holder or symbol

Polynomial ring

Formal power series

The idea is an extension from the idea of a "normal" power series but applied to a "formal object". Those two links should clue you in on the exact meaning.

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  • $\begingroup$ Sorry, the Wikipedia article didn't help me much :( $\endgroup$ – Felix Dombek Jan 24 '11 at 5:26

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