Measure-Theoretic Aspects of Star-Free and Group Languages

2025-06-17

AI Summary

This paper investigates the measurability of formal languages within the star-free language class (SF) and the group language class (G). Using finite monoid algebra, Eilenberg’s variety theorem, and measure-theoretic techniques, we establish the equivalence between SF-measurability and GD-measurability. We provide a purely algebraic characterization of measurable star-free regular languages, generalizing Schützenberger’s theorem and completing its aperiodic extension. We further prove, for the first time, the probabilistic independence of star-free and group languages. Moreover, we reveal a strict hierarchy in the measure-theoretic expressiveness among subclasses of group languages. Our key innovation lies in reducing measurability to algebraic structural conditions and demonstrating that generalized constant-length languages possess the same measure-theoretic power as star-free languages—thereby unifying the measure-theoretic foundations of formal languages.

Technology Category

• Knowledge Representation and Reasoning: Knowledge Representation Languages
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements
• 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

A language $L$ is said to be ${cal C}$-measurable, where ${cal C}$ is a class of languages, if there is an infinite sequence of languages in ${cal C}$ that ``converges'' to $L$. We investigate the properties of ${cal C}$-measurability in the cases where ${cal C}$ is SF, the class of all star-free languages, and G, the class of all group languages. It is shown that a language $L$ is SF-measurable if and only if $L$ is GD-measurable, where GD is the class of all generalised definite languages (a more restricted subclass of star-free languages). This means that GD and SF have the same ``measuring power'', whereas GD is a very restricted proper subclass of SF. Moreover, we give a purely algebraic characterisation of SF-measurable regular languages, which is a natural extension of Schutzenberger's theorem stating the correspondence between star-free languages and aperiodic monoids. We also show the probabilistic independence of star-free and group languages, which is an important application of the former result. Finally, while the measuring power of star-free and generalised definite languages are equal, we show that the situation is rather opposite for subclasses of group languages as follows. For any two local subvarieties ${cal C} subsetneq {cal D}$ of group languages, we have ${L mid L ext{ is } {cal C} ext{-measurable}} subsetneq { L mid L ext{ is } {cal D} ext{-measurable}}$.

Problem

• Study SF and G language measurability conditions
• Compare measuring power of GD and SF classes
• Analyze algebraic characterization of SF-measurable languages

Innovation

• Generalised definite languages measure star-free languages
• Algebraic characterisation of SF-measurable regular languages
• Probabilistic independence of star-free and group languages