Recognizing Numbers

TL;DR

This paper explores the concept of recognizability by finite monoids across various algebraic structures like integers, rationals, reals, and complex numbers, extending the classical automata theory.

Contribution

It characterizes recognizable subsets in different additive and multiplicative monoids over various number systems, highlighting their properties and limitations.

Findings

Recognizable sets are closed under Boolean operations.

Recognizable sets do not share all properties of word languages.

The paper extends automata theory to algebraic structures beyond words.

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.