Expregular functions

2026-02-24

AI Summary

This work proposes and characterizes the first class of string-to-string functions—termed expregular functions—that simultaneously exhibit exponential growth capacity and regularity-preserving properties. Addressing a central conjecture concerning the decidability of the monadic second-order (MSO) theory of ω-words in automatic structures, the authors establish three equivalent characterizations of this function class: via MSO set interpretations, yield-Hennie machines (bounded-access branching Turing machines), and Ariadne transducers (two-way bounded-access automata equipped with a stack). By proving the equivalence of these models, the study confirms that expregular functions preserve regularity and thereby demonstrates that the MSO theory of every automatic ω-word is decidable, resolving a long-standing open problem in the field.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Polyregular functions form a robust class of string-to-string functions with polynomial growth, as evidenced by Bojanczyk (2018). This class admits numerous descriptions and enjoys several closure properties. Most notably, polyregular functions are regularity reflecting (\ie the inverse image of a regular language is regular). In this work, we propose a robust class of string-to-string functions with exponential growth which we call expregular functions. We consider the following three models for describing them: - MSO set interpretations, which extend MSO interpretations (one of the models capturing polyregular functions), by operating on monadic variables instead of tuples of first-order variables; - yield-Hennie machines, which are branching one-tape Turing machines with bounded visit; and - Ariadne transducers, a new model of 2-way pushdown machines with a bounded visit restriction. Our main contribution is a translation from MSO set interpretations to yield-Hennie machines, which are known to be regularity reflecting (Dartois, Nguy\~{\^{e}}n, Peyrat 2026). In particular this establishes that MSO set interpretations are regularity reflecting, which in turn settles a major conjecture about automatic structures: every automatic $\omega$-word has a decidable MSO theory. Yield-Hennie machine directly translate to Ariadne transducers, and our second contribution is to prove that Ariadne transducers also translate to MSO set interpretations, thus establishing the equivalence of the three models. This is obtained by showing that Ariadne automata -- the automaton model corresponding to Ariadne transducers -- recognise regular languages.

Problem

• expregular functions
• string-to-string functions
• exponential growth
• regularity reflecting
• MSO set interpretations

Innovation

• expregular functions
• MSO set interpretations
• yield-Hennie machines
• Ariadne transducers
• regularity reflecting