More Than 1700 Years of Word Equations

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.Comment: The paper will appear as an invited address in the LNCS proceedings of CAI 2015, Stuttgart, Germany, September 1 - 4, 201

Solution sets for equations over free groups are EDT0L languages

© World Scientific Publishing Company. We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Jez, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to NSPACE(n log n) here