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In weighted automata theory, many classical results on formal languages havebeen extended into a quantitative setting. Here, we investigate weightedcontext-free languages of infinite words, a generalization of$\omega$-context-free languages (Cohen, Gold 1977) and an extension of weightedcontext-free languages of finite words (Chomsky, Sch\"utzenberger 1963). As inthe theory of formal grammars, these weighted context-free languages, or$\omega$-algebraic series, can be represented as solutions of mixed$\omega$-algebraic systems of equations and by weighted $\omega$-pushdownautomata.  In our first main result, we show that (mixed) $\omega$-algebraic systems canbe transformed into Greibach normal form. We use the Greibach normal form inour second main result to prove that simple $\omega$-reset pushdown automatarecognize all $\omega$-algebraic series. Simple $\omega$-reset automata do notuse $\epsilon$-transitions and can change the stack only by at most one symbol.These results generalize fundamental properties of context-free languages toweighted context-free languages.

Greibach Normal Form for $\omega$-Algebraic Systems and Weighted Simple $\omega$-Pushdown Automata