Sliding window property testing for regular languages

TL;DR

This paper introduces ultra-efficient sliding window property testing algorithms for regular languages, achieving logarithmic and constant space complexities for deterministic and randomized testers, respectively, in streaming models.

Contribution

It combines sliding window models with property testing, providing the first ultra-efficient algorithms for recognizing regular languages in streaming settings.

Findings

Deterministic testers use logarithmic space.

Randomized testers with two-sided error use constant space.

The approach extends previous space complexity characterizations to property testing.

Abstract

We study the problem of recognizing regular languages in a variant of the streaming model of computation, called the sliding window model. In this model, we are given a size of the sliding window $n$ and a stream of symbols. At each time instant, we must decide whether the suffix of length $n$ of the current stream ("the active window") belongs to a given regular language. Recent works showed that the space complexity of an optimal deterministic sliding window algorithm for this problem is either constant, logarithmic or linear in the window size $n$ and provided natural language theoretic characterizations of the space complexity classes. Subsequently, those results were extended to randomized algorithms to show that any such algorithm admits either constant, double logarithmic, logarithmic or linear space complexity. In this work, we make an important step forward and combine the sliding window model with the property testing setting, which results in ultra-efficient algorithms for all regular languages. Informally, a sliding window property tester must accept the active window if it belongs to the language and reject it if it is far from the language. We consider deterministic and randomized sliding window property testers with one-sided and two-sided errors. In particular, we show that for any regular language, there is a deterministic sliding window property tester that uses logarithmic space and a randomized sliding window property tester with two-sided error that uses constant space.