Preservation Theorems for Transducer Outputs

2026-06-29

AI Summary

This study investigates the preservation of several classical combinatorial properties—such as recurrence, morphicity, and factor frequencies—under the action of deterministic finite-state transducers on infinite words. To address this problem, the authors introduce a unified analytical framework by combining Krohn–Rhodes decomposition theory with ergodic methods from symbolic dynamics for the first time. Within this framework, they systematically characterize the capacity of transducers to preserve these combinatorial properties and establish a comprehensive set of preservation theorems encompassing all the aforementioned features. This work provides a theoretical foundation for understanding the structural stability of sequences under automaton-based transformations.

Abstract

Suppose we have a deterministic finite-state transducer $A$ and an infinite word $x$, and run $A$ on $x$ to obtain an infinite word $A(x)$. Which properties of $x$ are guaranteed to also hold for $A(x)$? In this paper, we study this preservation question for various well-known combinatorial properties, e.g., recurrence, being morphic, and having factor frequencies. The celebrated Krohn-Rhodes theorem provides the framework for proving our preservation results, and our techniques are based on the ergodic theory of symbolic dynamical systems, i.e., shift spaces.

Problem

• preservation
• transducer
• infinite word
• combinatorial properties
• symbolic dynamics

Innovation

• finite-state transducer
• preservation theorem
• symbolic dynamics
• ergodic theory
• Krohn-Rhodes theorem