# Automata theory on sliding windows

## TL;DR

This paper extends the analysis of space complexity for sliding window algorithms on regular languages, providing characterizations, bounds, and decision procedures for constant and logarithmic space classes.

## Contribution

It offers natural characterizations for constant and logarithmic space classes and relates them to language growth, advancing understanding of automata-based streaming algorithms.

## Key findings

- Characterizations for constant and logarithmic space classes
- Tight bounds on space complexity relative to automata size
- Decidability results for language classes with specific space bounds

## Abstract

In a recent paper we analyzed the space complexity of streaming algorithms whose goal is to decide membership of a sliding window to a fixed language. For the class of regular languages we proved a space trichotomy theorem: for every regular language the optimal space bound is either constant, logarithmic or linear. In this paper we continue this line of research: We present natural characterizations for the constant and logarithmic space classes and establish tight relationships to the concept of language growth. We also analyze the space complexity with respect to automata size and prove almost matching lower and upper bounds. Finally, we consider the decision problem whether a language given by a DFA/NFA admits a sliding window algorithm using logarithmic/constant space.