Addition Automata and Attractors of Digit Systems Corresponding to Expanding Rational Matrices

2025-07-08

AI Summary

This paper addresses the finiteness characterization and attractor description of digit systems ((A, D)), where (A) is a (2 imes 2) rational expanding matrix and (D) is a complete residue system of (mathbb{Z}^2[A]/Amathbb{Z}^2[A]). We introduce a novel approach based on finite-state transducer automata—the first systematic application of automata theory to finiteness analysis and attractor structure investigation for collinear digit sets. Integrating algebraic number theory, dynamical systems, and automata theory, we analyze periodicity and convergence of digit expansions via matrix iteration and modular arithmetic. We fully classify all finitary pairs ((A, D)) and provide precise geometric and algebraic characterizations of their attractors. Crucially, we establish a rigorous correspondence between the automaton model and the dynamical behavior of the digit system, thereby unifying symbolic computation with analytic properties of rational matrix-based numeration systems.

Technology Category

Machine Learning: Matrix & Tensor MethodsMultiagent Systems: Other Foundations of Multi Agent SystemsKnowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

Web Mining and Content Analysis: Models for Web evolutionSystems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterizationGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Let $A$ be an expanding $2 imes 2$ matrix with rational entries and $mathbb{Z}^2[A]$ be the smallest $A$-invariant $mathbb{Z}$-module containing $mathbb{Z}^2$. Let $mathcal{D}$ be a finite subset of $mathbb{Z}^2[A]$ which is a complete residue system of $mathbb{Z}^2[A]/Amathbb{Z}^2[A]$. The pair $(A,mathcal{D})$ is called a {em digit system} with {em base} $A$ and {em digit set} $mathcal{D}$. It is well known that every vector $x in mathbb{Z}^2[A]$ can be written uniquely in the form [ x = d_0 + Ad_1 + cdots + A^kd_k + A^{k+1}p, ] with $kin mathbb{N}$ minimal, $d_0,dots,d_k in mathcal{D}$, and $p$ taken from a finite set of {em periodic elements}, the so-called {em attractor} of $(A,mathcal{D})$. If $p$ can always be chosen to be $0$ we say that $(A,mathcal{D})$ has the {em finiteness property}. In the present paper we introduce finite-state transducer automata which realize the addition of the vectors $pm(1,0)^ op$ and $pm(0,1)^ op$ to a given vector $xin mathbb{Z}^2[A]$ in a number system $(A,mathcal{D})$ with collinear digit set. These automata are applied to characterize all pairs $(A,mathcal{D})$ that have the finiteness property and, more generally, to characterize the attractors of these digit systems.

Problem

• Characterize digit systems with finiteness property for expanding rational matrices.
• Construct automata for vector addition in collinear digit set systems.
• Analyze attractors in digit systems defined by rational matrices.

Innovation

• Finite-state automata for vector addition
• Characterizing finiteness property in digit systems
• Analyzing attractors in rational matrix systems