An optimal bound on the solution sets of one-variable word equations and its consequences
: Nowotka Dirk, Saarela Aleksi
: Ioannis Chatzigiannakis, Christos Kaklamanis, Daniel Marx, Donald Sannella
: International Colloquium on Automata, Languages and Programming
Publisher: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
: 2018
: LIPICS – Leibniz international proceedings in informatics
: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
: Leibniz International Proceedings in Informatics, LIPIcs
: LIPIcs: Leibniz International Proceedings in Informatics
: 107
: 136:1
: 136:13
: 978-3-95977-076-7
: 1868-8969
DOI: https://doi.org/10.4230/LIPIcs.ICALP.2018.136(external)
: http://drops.dagstuhl.de/opus/volltexte/2018/9140(external)
: https://research.utu.fi/converis/portal/detail/Publication/35729244(external)
We solve two long-standing open problems on word equations. Firstly, we prove that a onevariable word equation with constants has either at most three or an infinite number of solutions. The existence of such a bound had been conjectured, and the bound three is optimal. Secondly, we consider independent systems of three-variable word equations without constants. If such a system has a nonperiodic solution, then this system of equations is at most of size 17. Although probably not optimal, this is the first finite bound found. However, the conjecture of that bound being actually two still remains open.
