For a fixed finite algebra 𝐀, we consider the decision problem SysTerm(𝐀): does a given system of term equations have a solution in 𝐀? This is equivalent to a constraint satisfaction problem (CSP) for a relational structure whose relations are the graphs of the basic operations of 𝐀. From the complexity dichotomy for CSP over fixed finite templates due to Bulatov [Bulatov, 2017] and Zhuk [Zhuk, 2017], it follows that SysTerm(𝐀) for a finite algebra 𝐀 is in P if 𝐀 has a not necessarily idempotent Taylor polymorphism and is NP-complete otherwise. More explicitly, we show that for a finite algebra 𝐀 in a congruence modular variety (e.g. for a quasigroup), SysTerm(𝐀) is in P if the core of 𝐀 is abelian and is NP-complete otherwise. Given 𝐀 by the graphs of its basic operations, we show that this condition for tractability can be decided in quasi-polynomial time.
@InProceedings{mayr:LIPIcs.MFCS.2023.66, author = {Mayr, Peter}, title = {{On the Complexity Dichotomy for the Satisfiability of Systems of Term Equations over Finite Algebras}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {66:1--66:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.66}, URN = {urn:nbn:de:0030-drops-186007}, doi = {10.4230/LIPIcs.MFCS.2023.66}, annote = {Keywords: systems of equations, general algebras, constraint satisfaction} }
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