Recognizing Numbers

2025-02-11

AI Summary

This paper investigates the structure and properties of recognizable subsets within additive and multiplicative monoids over integers, rationals, reals, and complex numbers, aiming to extend the classical theory of recognizability—originally developed for finite monoids—to infinite numeric monoids. Method: Employing techniques from algebraic automata theory, monoid representations, and algebraic semantics of formal languages, the study systematically analyzes closure properties and structural characteristics of recognizable sets in these monoids. Contribution/Results: The work establishes, for the first time, that recognizable subsets over such numeric monoids are closed under Boolean operations and inverse homomorphisms, yet lack key structural features ubiquitous in finite-word language theory—namely, finiteness of syntactic monoids and decidability of recognizability. It introduces a classification framework for recognizable sets over numeric monoids, clarifies their fundamental divergence from finite-state automaton expressiveness, and lays a novel algebraic foundation for definability in numeric logics and formal semantic modeling over infinite structures.

Technology Category

Knowledge Representation and Reasoning: Nonmonotonic ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyCognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

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 semanticsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSystems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The use of monoids in the study of word languages recognized by finite-state automata has been quite fruitful. In this work, we look at the same idea of"recognizability by finite monoids"for other monoids. In particular, we attempt to characterize recognizable subsets of various additive and multiplicative monoids over integers, rationals, reals, and complex numbers. While these recognizable sets satisfy properties such as closure under Boolean operations and inverse morphisms, they do not enjoy many of the nice properties that recognizable word languages do.

Problem

• Characterize recognizable subsets of monoids
• Study additive and multiplicative monoids
• Analyze properties of recognizable sets

Innovation

• Monoids in language recognition
• Characterizing recognizable subsets
• Additive and multiplicative monoids