# Is there any equation-based method for transforming Büchi-automata to omega-regular language?

I know there exists an equation-based method for transforming finite automata into regular language (or `regular expression'). The main idea is as follows. First we construct a set of equations named "(1)" based on an automaton $$A$$: \begin{aligned}&l_1 = \varepsilon + p_{11}l_1 + p_{12}l_2 + ... + p_{1n}l_n\\ &l_2 = \varepsilon + p_{21}l_1 + p_{22}l_2 + ... + p_{2n}l_n\\ &...\\ &l_n = \varepsilon + p_{n1}l_1 + p_{n2}l_2 + ... + p_{nn}l_n, \end{aligned} where are $$l_1,...,l_n$$ are the locations (or called "states") of $$A$$ with $$l_1$$ the initial location, $$p_{ij}$$ are regular expressions. Then we apply Arden's rule:

"$$X = p^*q$$ is the unique solution for equation $$X = q + pX$$"

to solve the equations. The result of $$l_1$$ is the regular expression we want.

Now my question is, is there a similar method for transforming Büchi automata into omega-regular language?

Sure. In fact, the translation from Büchi automata to $$\omega$$-regular expressions is only a small extension of the one for finite-word languages.
Recall that an $$\omega$$-regular expression is of the form $$s_1\cdot t_1^\omega+\ldots+s_k \cdot t_k^\omega$$, where all the $$s_i$$ and $$t_i$$ are regular expressions.
The translation of an NBW $$A$$ to such an expression is based on the following observation: let $$A_{p,q}$$ be the NFA obtained from $$A$$ by setting $$p$$ as the initial state, and $$q$$ as the single accepting state, then $$L(A)=\bigcup_{q\in \alpha} L(A_{q_0,q})\cdot L(A_{q,q})^\omega$$ where $$\alpha$$ is the set of accepting states of $$A$$.
Indeed, a word is accepted by $$A$$ iff there is a run on it that leads from $$q_0$$ to an accepting state $$q$$, that is then visited infinitely often.