In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of ω-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Schützenberger 1963). As in the theory of formal grammars, these weighted languages, or ω-algebraic series, can be represented as solutions of mixed ω-algebraic systems of equations and by weighted ω-pushdown automata.

In our first main result, we show that mixed ω-algebraic systems can be transformed into Greibach normal form. Our second main result proves that simple ω-reset pushdown automata recognize all ω-algebraic series that are a solution of an ω-algebraic system in Greibach normal form. Simple reset automata do not use ε-transitions and can change the stack only by at most one symbol. These results generalize fundamental properties of context-free languages to weighted languages.