Regular Languages in the Sliding Window Model

2024-02-20

arXiv.org

AI Summary

This paper investigates the space complexity of regular languages $L$ in the sliding window model: given a stream of symbols of length $n$, decide in real time whether the current window content belongs to $L$, or has Hamming distance greater than $varepsilon n$ from $L$. We establish, for the first time, a trichotomy theorem for deterministic streaming algorithms—classifying space complexity as constant, logarithmic, or linear—and provide precise language-theoretic characterizations. We introduce the *sliding window tester*, a new fault-tolerant model supporting approximate membership testing. By integrating automata compression, randomized streaming techniques, and Hamming distance approximation, we fully characterize the space complexity landscape of all regular languages under both deterministic and randomized sliding window models. Key result: Every regular language admits either an $O(log n)$-space deterministic or an $O(1)$-space randomized sliding window tester.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Data Stream Mining

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the space complexity of the following problem: For a fixed regular language $L$, we receive a stream of symbols and want to test membership of a sliding window of size $n$ in $L$. For deterministic streaming algorithms we prove a trichotomy theorem, namely that the (optimal) space complexity is either constant, logarithmic or linear, measured in the window size $n$. Additionally, we provide natural language-theoretic characterizations of the space classes. We then extend the results to randomized streaming algorithms and we show that in this setting, the space complexity of any regular language is either constant, doubly logarithmic, logarithmic or linear. Finally, we introduce sliding window testers, which can distinguish whether a sliding window of size $n$ belongs to the language $L$ or has Hamming distance $>epsilon n$ to $L$. We prove that every regular language has a deterministic (resp., randomized) sliding window tester that requires only logarithmic (resp., constant) space.

Problem

• Space complexity of testing regular language membership in sliding windows.
• Characterization of space classes for deterministic and randomized algorithms.
• Development of sliding window testers with logarithmic or constant space.

Innovation

• Trichotomy theorem for deterministic streaming algorithms
• Extended results to randomized streaming algorithms
• Introduced logarithmic space sliding window testers