Decision Problems for Deterministic Pushdown Automata on Infinite Words

TL;DR

This paper surveys decidability results for deterministic pushdown automata on infinite words, including recent findings on regularity, equivalence, and the parity index problem, with some new results on minimal priorities needed for acceptance.

Contribution

It provides a comprehensive survey of decidability issues for omega-DPDAs and introduces new results on the parity index problem for specific classes.

Findings

Decidability of regularity and equivalence for weak omega-DPDAs

New results on the parity index problem for omega-DPDAs

Decidability results on minimal priorities for acceptance

Abstract

The article surveys some decidability results for DPDAs on infinite words (omega-DPDA). We summarize some recent results on the decidability of the regularity and the equivalence problem for the class of weak omega-DPDAs. Furthermore, we present some new results on the parity index problem for omega-DPDAs. For the specification of a parity condition, the states of the omega-DPDA are assigned priorities (natural numbers), and a run is accepting if the highest priority that appears infinitely often during a run is even. The basic simplification question asks whether one can determine the minimal number of priorities that are needed to accept the language of a given omega-DPDA. We provide some decidability results on variations of this question for some classes of omega-DPDAs.