Turn Complexity of Context-free Languages, Pushdown Automata and One-Counter Automata

TL;DR

This paper investigates the complexity of computation turns in pushdown and one-counter automata, revealing undecidability results, trade-offs, and hierarchies based on the number of turns in accepting computations.

Contribution

It introduces new undecidability results, establishes a hierarchy of complexity classes based on turn bounds, and explores the minimal turns needed for language acceptance.

Findings

Deciding if the number of turns is bounded by any constant is undecidable.

There is a non-recursive trade-off between pushdown and one-counter automata accepting in finite turns.

Languages exist that require sublinear but non-constant turns for acceptance.

Abstract

A turn in a computation of a pushdown automaton is a switch from a phase in which the height of the pushdown store increases to a phase in which it decreases. Given a pushdown or one-counter automaton, we consider, for each string in its language, the minimum number of turns made in accepting computations. We prove that it cannot be decided if this number is bounded by any constants. Furthermore, we obtain a non-recursive trade-off between pushdown and one-counter automata accepting in a finite number of turns and finite-turn pushdown automata, that are defined requiring that the constant bound is satisfied by each accepting computation. We prove that there are languages accepted in a sublinear but not constant number of turns, with respect to the input length. Furthermore, there exists an infinite proper hierarchy of complexity classes, with the number of turns bounded by different sublinear functions. In addition, there is a language requiring a number of turns which is not constant but grows slower than each of the functions defining the above hierarchy.