# Chomsky Schützenberger enumeration theorem

In many textbooks the Chomsky-Schützenberger enumeration theorem is stated as that the characteristic formal power series of a language is $\mathbb N$-algebraic, if the grammar is unambigious. In some other books the formulation is given, that the structure describing function $$f_L(x)=\sum_{n \in \mathbb N}l_nx^n$$ is algebraic over $\mathbb Q$ iff the grammar is unambigious, where $l_n = \vert \Sigma^n \cap L \vert$. I understand that by replacing every terminal by the same terminal in the system of equations associated with a grammar $G$ it is possible, to obtain the structure describing function, but I couldn't find a proof anywhere,(and don't know how I could construct one) that then the (analytical) power series $f_L(x)$ is not transcendental.

• So what would I need to read in the Kuich-Salomaa book? I'm a logician, so this is not really my topic, but I've always wanted to learn about this proof; I remember reading that the transcendence of $\pi$ was equivalent to some language being inherently ambiguous and found that quite neat, although I'm not sure I remember the details correctly. – Jori Jul 28 '20 at 19:39