Sweep Complexity Revisited

TL;DR

This paper investigates the sweep complexity of one-way jumping DFA, revealing that higher complexity does not necessarily imply acceptance of non-regular languages and establishing new separation results.

Contribution

It provides a detailed analysis of sweep complexity, showing its limitations in distinguishing regular from non-regular languages and establishing new complexity class separations.

Findings

Higher than constant sweep complexity can still accept only regular languages

Sweep complexity does not differentiate between regular and non-regular languages

Exponential/logarithmic relations between input factors are verified by these automata

Abstract

We study the sweep complexity of DFA in one-way jumping mode answering several questions posed earlier. This measure is the number of times in the worst case that such machines have to return to the beginning of their input after having skipped some of the symbols. The class of languages accepted by these machines strictly includes the regular class and constant sweep complexity allows exactly the acceptance of regular languages. However, we show that there exist machines with higher than constant complexity still only accepting regular languages and that in general the sweep complexity of an automaton does not distinguish between accepting regular and non-regular languages. We establish separation results for asymptotic classes defined by this complexity measure and give a surprising exponential/logarithmic relation between factors of certain inputs which can be verified by such machines.