Turn Complexity and Bounded Languages

Abstract

A turn or reversal 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 automaton, we first consider, for each string in its language, the minimum number of turns made in accepting computations (weak measure). We prove that it is decidable whether a pushdown automaton accepts a bounded language in a constant number of turns and whether it accepts a bounded language in k turns, for any given k>=0. This is in contrast to the general case, in which these problems are known to be undecidable, with the exception of acceptance in 0 turns, which is decidable. Furthermore, we prove that when the number of turns sufficient to accept a bounded language is not limited by any constants, it linearly grows with respect to the input length. Also this is in contrast with the general case where, for each nonnegative k, there exists a language for which the number of turns necessary and sufficient is of the order of log^(k), the k times composition of the logarithm with itself. We also prove that, when the costs of all accepting computations are taken into account (accept measure), a linear lower bound for the number of turns, if not limited by any constants, holds even removing the restriction to bounded languages.

In Proceedings AFL 2026, arXiv:2608.23071