Bounded-oscillation Pushdown Automata

TL;DR

This paper introduces a new method for approximating context-free languages by restricting pushdown automata to runs with bounded stack oscillation, enabling more efficient analysis of language properties.

Contribution

It defines k-oscillating runs in pushdown automata, relates them to context-free grammars, and analyzes the complexity of the associated emptiness problem.

Findings

k-emptiness problem is NLOGSPACE-complete when k is fixed.

Provides a filtering method for runs based on oscillation annotations.

Studies closure properties of k-oscillating language classes.

Abstract

We present an underapproximation for context-free languages by filtering out runs of the underlying pushdown automaton depending on how the stack height evolves over time. In particular, we assign to each run a number quantifying the oscillating behavior of the stack along the run. We study languages accepted by pushdown automata restricted to k-oscillating runs. We relate oscillation on pushdown automata with a counterpart restriction on context-free grammars. We also provide a way to filter all but the k-oscillating runs from a given PDA by annotating stack symbols with information about the oscillation. Finally, we study closure properties of the defined class of languages and the complexity of the k-emptiness problem asking, given a pushdown automaton P and k >= 0, whether P has a k-oscillating run. We show that, when k is not part of the input, the k-emptiness problem is NLOGSPACE-complete.