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Square-free words over partially commutative alphabets

Lookup NU author(s): Dr Lukasz Mikulski



There exist many constructions of infinite words over three-letter alphabet avoiding squares. However, the characterization of the lexicographically minimal square-free word is an open problem. Efficient construction of this word is not known. We show that the situation changes when some letters commute with each other. We give two characterizations (morphic and recursive) of the lexicographically minimal square-free word v in the case of a partially commutative alphabet of size three. We consider the only non-trivial relation of partial commutativity, for which v exists: there are two commuting letters, while the third one is blocking (does not commute at all). We also show that the n-th letter of v can be computed in time logarithmic with respect to n.

Publication metadata

Author(s): Mikulski L, Piatkowski M, Rytter W

Publication type: Report

Publication status: Published

Series Title: School of Computing Science Technical Report Series

Year: 2014

Pages: 21

Print publication date: 01/11/2014

Report Number: 1439

Institution: School of Computing Science, University of Newcastle upon Tyne

Place Published: Newcastle upon Tyne