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Solving One Variable Word Equations in the Free Group in Cubic Time

Authors Robert Ferens , Artur Jeż



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Author Details

Robert Ferens
  • Institute of Computer Science, University of Wrocław, Poland
Artur Jeż
  • Institute of Computer Science, University of Wrocław, Poland

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Robert Ferens and Artur Jeż. Solving One Variable Word Equations in the Free Group in Cubic Time. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 30:1-30:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.30

Abstract

A word equation with one variable in a free group is given as U = V, where both U and V are words over the alphabet of generators of the free group and X, X⁻¹, for a fixed variable X. An element of the free group is a solution when substituting it for X yields a true equality (interpreted in the free group) of left- and right-hand sides. It is known that the set of all solutions of a given word equation with one variable is a finite union of sets of the form {α wⁱ β : i ∈ ℤ}, where α, w, β are reduced words over the alphabet of generators, and a polynomial-time algorithm (of a high degree) computing this set is known. We provide a cubic time algorithm for this problem, which also shows that the set of solutions consists of at most a quadratic number of the above-mentioned sets. The algorithm uses only simple tools of word combinatorics and group theory and is simple to state. Its analysis is involved and focuses on the combinatorics of occurrences of powers of a word within a larger word.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Theory of computation → Formalisms
  • Computing methodologies → Equation and inequality solving algorithms
Keywords
  • Word equations
  • free group
  • one-variable equations

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