More Than 1700 Years of Word Equations

TL;DR

This paper reviews recent advances in the study of word equations, highlighting a simple decision procedure for their solvability and describing the solution set as an EDT0L language, connecting to Diophantine equations and Hilbert's 10th problem.

Contribution

It presents a simplified decision procedure for word equations and characterizes their solutions as an EDT0L language, advancing understanding in formal language theory.

Findings

Decidable whether a word equation is solvable.

Solution sets can be described as EDT0L languages.

Connections established between word equations and Diophantine problems.

Abstract

Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a linear Diophantine equation can be viewed as a special case of a system of word equations over a unary alphabet, and, more importantly, a word equation can be viewed as a special case of a Diophantine equation. Hence, the problem WordEquations: "Is a given word equation solvable?" is intimately related to Hilbert's 10th problem on the solvability of Diophantine equations. This became clear to the Russian school of mathematics at the latest in the mid 1960s, after which a systematic study of that relation began. Here, we review some recent developments which led to an amazingly simple decision procedure for WordEquations, and to the description of the set of all solutions as an EDT0L language.