We investigate the solution set of the pseudoperiodic extension of the classical Lyndon and Sch\"utzenberger word equations. Consider u_1 ... u_l = v_1 ... v_m w_1 ... w_n, where u_i is in {u, theta(u)} for all 1 <= i <= l, v_j is in {v, theta(v)} for all 1 <= j <= m, w_k is in {w, theta(w)} for all 1 <= k <= n and u, v and w are variables, and theta is an antimorphic involution. A solution is called pseudoperiodic, if u,v,w are in {t, theta(t)}^+ for a word t. [Czeizler et al./I&C/2011] established that for small values of l, m, and n non-periodic solutions exist, and that for large enough values all solutions are pseudoperiodic. However, they leave a gap between those bounds which we close for a number of cases. Namely, we show that for l = 3 and either m,n >= 12 or m,n >= 5 and either m and n are not both even or not all u_i's are equal, all solutions are pseudoperiodic.
@InProceedings{manea_et_al:LIPIcs.FSTTCS.2013.475, author = {Manea, Florin and M\"{u}ller, Mike and Nowotka, Dirk}, title = {{On the Pseudoperiodic Extension of u^l = v^m w^n}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2013)}, pages = {475--486}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-64-4}, ISSN = {1868-8969}, year = {2013}, volume = {24}, editor = {Seth, Anil and Vishnoi, Nisheeth K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2013.475}, URN = {urn:nbn:de:0030-drops-43948}, doi = {10.4230/LIPIcs.FSTTCS.2013.475}, annote = {Keywords: Word equations, Pseudoperiodicity, Lyndon-Sch\"{u}tzenberger equation} }
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