The Triple-Pair Construction for Weighted $\omega$-Pushdown Automata

TL;DR

This paper introduces the triple-pair construction, a generalization of the triple construction, for weighted omega-pushdown automata, establishing a connection with mixed algebraic systems and context-free grammars over complete semirings.

Contribution

It presents the triple-pair construction method, linking weighted omega-pushdown automata to algebraic systems and context-free grammars, extending classical techniques to the omega setting.

Findings

Existence of a mixed algebraic system representing automaton behavior

Construction of a mixed context-free grammar for specific semirings

Generalization of the triple construction to omega automata

Abstract

Let S be a complete star-omega semiring and Sigma be an alphabet. For a weighted omega-pushdown automaton P with stateset 1...n, n greater or equal to 1, we show that there exists a mixed algebraic system over a complete semiring-semimodule pair ((S<<Sigma*>>)^nxn, (S<<Sigma^omega>>)^n) such that the behavior ||P|| of P is a component of a solution of this system. In case the basic semiring is the Boolean semiring or the semiring of natural numbers (augmented with infinity), we show that there exists a mixed context-free grammar that generates ||P||. The construction of the mixed context-free grammar from P is a generalization of the well known triple construction and is called now triple-pair construction for omega-pushdown automata.