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

TL;DR

This paper extends classical formal language results to weighted infinite words, showing that weighted $ ext{omega}$-algebraic systems can be transformed into Greibach normal form and that simple $ ext{omega}$-reset pushdown automata recognize all such series.

Contribution

It introduces a normal form for weighted $ ext{omega}$-algebraic systems and proves the equivalence with simple $ ext{omega}$-reset pushdown automata, generalizing context-free language properties.

Findings

Weighted $ ext{omega}$-algebraic systems can be transformed into Greibach normal form.

Simple $ ext{omega}$-reset pushdown automata recognize all $ ext{omega}$-algebraic series.

Generalization of classical context-free language properties to weighted infinite words.

Abstract

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 $\omega$-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Sch\"utzenberger 1963). As in the 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 $\omega$-pushdown automata. In our first main result, we show that (mixed) $\omega$-algebraic systems can be transformed into Greibach normal form. We use the Greibach normal form in our second main result to prove that simple $\omega$-reset pushdown automata recognize all $\omega$-algebraic series. Simple $\omega$-reset automata do not use $\epsilon$-transitions and can change the stack only by at most one symbol. These results generalize fundamental properties of context-free languages to weighted context-free languages.