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The Trichotomy of Regular Property Testing

Gabriel Bathie, Nathanaël Fijalkow, Corto Mascle

TL;DR

This work resolves the query-complexity landscape for property testing of regular languages under the Hamming distance by establishing a trichotomy: trivial, easy, and hard classes determined by the structure of minimal blocking sequences. It introduces blocking sequences (and their strong variant) and portals/SCC-paths to capture how words force non-membership across automata components, enabling tight upper and lower bounds. A near-optimal tester with O(log(1/ε)/ε) queries is shown for strongly connected NFAs, with a distribution-based lower bound proving hardness when infinitely many minimal blocking factors exist; these ideas extend to all NFAs, culminating in a PSPACE-complete decision procedure to classify automata into the three classes. The results consolidate and sharpen prior work, provide a polynomial-space characterization, and deliver practical testers and structural insights for regular-language property testing, with implications for streaming and sublinear-time decision procedures.

Abstract

Property testing is concerned with the design of algorithms making a sublinear number of queries to distinguish whether the input satisfies a given property or is far from having this property. A seminal paper of Alon, Krivelevich, Newman, and Szegedy in 2001 introduced property testing of formal languages: the goal is to determine whether an input word belongs to a given language, or is far from any word in that language. They constructed the first property testing algorithm for the class of all regular languages. This opened a line of work with improved complexity results and applications to streaming algorithms. In this work, we show a trichotomy result: the class of regular languages can be divided into three classes, each associated with an optimal query complexity. Our analysis yields effective characterizations for all three classes using so-called minimal blocking sequences, reasoning directly and combinatorially on automata.

The Trichotomy of Regular Property Testing

TL;DR

This work resolves the query-complexity landscape for property testing of regular languages under the Hamming distance by establishing a trichotomy: trivial, easy, and hard classes determined by the structure of minimal blocking sequences. It introduces blocking sequences (and their strong variant) and portals/SCC-paths to capture how words force non-membership across automata components, enabling tight upper and lower bounds. A near-optimal tester with O(log(1/ε)/ε) queries is shown for strongly connected NFAs, with a distribution-based lower bound proving hardness when infinitely many minimal blocking factors exist; these ideas extend to all NFAs, culminating in a PSPACE-complete decision procedure to classify automata into the three classes. The results consolidate and sharpen prior work, provide a polynomial-space characterization, and deliver practical testers and structural insights for regular-language property testing, with implications for streaming and sublinear-time decision procedures.

Abstract

Property testing is concerned with the design of algorithms making a sublinear number of queries to distinguish whether the input satisfies a given property or is far from having this property. A seminal paper of Alon, Krivelevich, Newman, and Szegedy in 2001 introduced property testing of formal languages: the goal is to determine whether an input word belongs to a given language, or is far from any word in that language. They constructed the first property testing algorithm for the class of all regular languages. This opened a line of work with improved complexity results and applications to streaming algorithms. In this work, we show a trichotomy result: the class of regular languages can be divided into three classes, each associated with an optimal query complexity. Our analysis yields effective characterizations for all three classes using so-called minimal blocking sequences, reasoning directly and combinatorially on automata.
Paper Structure (20 sections, 26 theorems, 8 equations, 3 figures, 2 algorithms)

This paper contains 20 sections, 26 theorems, 8 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Main result and overview of the paper
  4. Preliminaries
  5. Hard Languages for Strongly Connected NFAs
  6. Positional words, blocking factors and strongly connected NFAs
  7. An efficient property tester for strongly connected NFAs.
  8. An efficient sampling algorithm
  9. The tester
  10. Lower bound from infinitely many minimal blocking factors
  11. The structure of $\textsf{MBF}\xspace(\mathcal{A})$
  12. Constructing a Hard Distribution $\mathcal{D}$
  13. Properties of the distribution $\mathcal{D}$
  14. Characterisation of Hard Languages for All NFAs
  15. Blocking sequences
  16. ...and 5 more sections

Key Result

Theorem 5

Let $L$ be an infinite regular language recognized by an NFA $\mathcal{A}$ and let $\textsf{MBS}\xspace(\mathcal{A})$ denote the set of minimal blocking sequences of $\mathcal{A}$. The complexity of testing $L$ is characterized by $\textsf{MBS}\xspace(\mathcal{A})$ as follows:

Figures (3)

  • Figure 2: An automaton $\mathcal{A}_1$ that recognizes the language $L_1 = (a+c)^* (a+b)^*$.
  • Figure 3: An automaton $\mathcal{A}_2$ that recognizes the language $L_2 = [((c+d+e)^* b (b+e)^* d)^* a] (b+c+d+e)^*$.
  • Figure 4: Automaton used for \ref{['ex:SCCpath']}.

Theorems & Definitions (49)

  • Definition 3: Hard, easy and trivial languages
  • Remark 4
  • Theorem 5
  • Definition 6
  • Definition 7
  • Definition 9: Positional words
  • Definition 11: Blocking factors
  • Theorem 12
  • Corollary 13
  • Corollary 14
  • ...and 39 more