The subpower membership problem SMP(𝐀) of a finite algebraic structure 𝐀 asks whether a given partial function from Aⁿ to A can be interpolated by a term operation of 𝐀, or not. While this problem can be EXPTIME-complete in general, Willard asked whether it is always solvable in polynomial time if 𝐀 is a Mal'tsev algebra. In particular, this includes many important structures studied in abstract algebra, such as groups, quasigroups, rings, Boolean algebras. In this paper we give an affirmative answer to Willard’s question for a big class of 2-nilpotent Mal'tsev algebras. We furthermore develop tools that might be essential in answering the question for general nilpotent Mal'tsev algebras in the future.
@InProceedings{kompatscher:LIPIcs.STACS.2024.46, author = {Kompatscher, Michael}, title = {{The Subpower Membership Problem of 2-Nilpotent Algebras}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {46:1--46:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.46}, URN = {urn:nbn:de:0030-drops-197562}, doi = {10.4230/LIPIcs.STACS.2024.46}, annote = {Keywords: subpower membership problem, Mal'tsev algebra, compact representation, nilpotence, clonoids} }
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