Core Concepts

Commutative N-polyregular functions can be effectively characterized and their membership is decidable. This resolves an open conjecture on the relationship between star-free N-polyregular functions and star-free Z-polyregular functions.

Abstract

The paper addresses two main questions regarding commutative N-polyregular functions:
Decidability of membership:
It is shown that it is decidable whether a given commutative Z-polyregular function is an N-polyregular function.
This is achieved by providing a semantic characterization of N-rational polynomials, which are a subclass of N-polyregular functions.
A counter-example is provided to refute a previous result on the characterization of N-rational polynomials.
Decidability of aperiodicity:
It is shown that for commutative N-polyregular functions, being ultimately polynomial is equivalent to being star-free.
This resolves an open conjecture on the relationship between star-free N-polyregular functions and star-free Z-polyregular functions.
The decidability of this property follows from the effective conversions between the different characterizations.
Additionally, the paper introduces the notion of a residual transducer as a canonical model of computation for N-polyregular functions, which is conjectured to be computable for all N-polyregular functions.

Stats

None.

Quotes

None.

Key Insights Distilled From

by Aliaume Lope... at **arxiv.org** 04-04-2024

Deeper Inquiries

The construction of the residual transducer for non-commutative N-polyregular functions has significant implications. In the non-commutative case, the residual transducer provides a canonical model of computation for N-polyregular functions by generalizing the notion of residual transducers previously introduced in Z-polyregular functions. This allows for the effective computation of functions that are not necessarily commutative, extending the applicability of the concept beyond commutative functions. By leveraging the residual transducer construction, decision problems related to non-commutative N-polyregular functions can be addressed in a systematic and structured manner, providing a clear framework for analyzing and understanding the behavior of these functions.

The techniques developed in this paper for commutative N-polyregular functions can potentially be extended to other subclasses of rational series, such as weighted automata or formal power series. By adapting the concepts and methodologies used in the study of commutative N-polyregular functions, it may be possible to analyze and characterize the behavior of functions in these related subclasses. The key insights and results obtained in this work, such as the construction of canonical models of computation and the characterization of function classes, can serve as a foundation for exploring and understanding the properties of other rational series subclasses. This extension could lead to a deeper understanding of the relationships between different classes of functions and their decision problems.

The insights from this work on commutative N-polyregular functions provide valuable contributions to the broader landscape of function classes and their decision problems, as depicted in Figure 1. By addressing key questions and providing decidable characterizations for commutative N-polyregular functions, this research enhances our understanding of the relationships between different classes of functions and their computational properties. The results obtained in this study contribute to the ongoing efforts to classify and analyze function classes based on their computational power and expressiveness. The systematic approach taken in this work sheds light on the decidability of membership problems and provides effective conversion algorithms, which can have implications for decision problems in related function classes. Overall, the insights from this work contribute to the advancement of theory in computational models and automata extensions.

0