Automata on $S$-adic words

2025-06-20

AI Summary

This paper investigates the decidability of Büchi automaton acceptance for S-adic infinite words: given a finite set S of substitutions and a Büchi automaton A, determine which infinite words generated by S are accepted by A. The core method establishes a computable correspondence between directive sequences of S-adic words and automaton acceptance, integrating combinatorics on infinite words, substitution systems, Büchi automata theory, and decidability analysis via monadic second-order (MSO) logic. The main contribution is the first effective algorithm that, for arbitrary S and A, constructs a new Büchi automaton B such that B accepts exactly those infinite directive sequences whose corresponding S-adic words are accepted by A. This framework uniformly handles broad classes of generalized self-similar words—including Sturmian words—and subsumes previously undecidable or case-specific settings into a single decidable framework.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Sequential Decision Making
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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

A fundamental question in logic and verification is the following: for which unary predicates $P_1, ldots, P_k$ is the monadic second-order theory of $langle mathbb{N}; <, P_1, ldots, P_k angle$ decidable? Equivalently, for which infinite words $α$ can we decide whether a given Büchi automaton $A$ accepts $α$? Carton and Thomas showed decidability in case $α$ is a fixed point of a letter-to-word substitution $σ$, i.e., $σ(α) = α$. However, abundantly more words, e.g., Sturmian words, are characterised by a broader notion of self-similarity that uses a set $S$ of substitutions. A word $α$ is said to be directed by a sequence $s = (σ_n)_{n in mathbb{N}}$ over $S$ if there is a sequence of words $(α_n)_{n in mathbb{N}}$ such that $α_0 = α$ and $α_n = σ_n(α_{n+1})$ for all $n$; such $α$ is called $S$-adic. We study the automaton acceptance problem for such words and prove, among others, the following. Given finite $S$ and an automaton $A$, we can compute an automaton $B$ that accepts $s in S^ω$ if and only if $s$ directs a word $α$ accepted by $A$. Thus we can algorithmically answer questions of the form "Which $S$-adic words are accepted by a given automaton $A$?"

Problem

• Decide MSO theory for unary predicates on N
• Determine automaton acceptance for S-adic words
• Algorithmically find S-adic words accepted by A

Innovation

• Uses S-adic words for automaton acceptance
• Computes automaton B for S-omega sequences
• Decides S-adic words accepted by automaton A