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The Subpower Membership Problem of 2-Nilpotent Algebras

Author Michael Kompatscher

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LIPIcs.STACS.2024.46.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • subpower membership problem
  • Mal'tsev algebra
  • compact representation
  • nilpotence
  • clonoids

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Abstract

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.

Cite As Get BibTex

Michael Kompatscher. The Subpower Membership Problem of 2-Nilpotent Algebras. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.46

Author Details

Michael Kompatscher
  • Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

Funding

This paper was supported by projects PRIMUS/24/SCI/008 and UNCE/SCI/022 of Charles University.

Acknowledgements

I would like to thank Peter Mayr for introducing me to difference clonoids and giving several helpful comments on earlier versions of this paper. Furthermore I would like to thank the anonymous referees for their many helpful remarks.

References

  1. Erhard Aichinger and Peter Mayr. Finitely generated equational classes. Journal of Pure and Applied Algebra, 220(8):2816-2827, 2016. URL: https://doi.org/10.1016/j.jpaa.2016.01.001.
  2. Erhard Aichinger and Nebojša Mudrinski. Some applications of higher commutators in Mal’cev algebras. Algebra universalis, 63(4):367-403, 2010. URL: https://doi.org/10.1007/s00012-010-0084-1.
  3. Libor Barto, Andrei Krokhin, and Ross Willard. Polymorphisms, and how to use them. In Dagstuhl Follow-Ups, volume 7. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017. URL: https://doi.org/10.4230/DFU.Vol7.15301.1.
  4. Clifford Bergman. Universal algebra: Fundamentals and selected topics. CRC Press, 2011. Google Scholar
  5. Zarathustra Brady. Notes on CSPs and Polymorphisms. arXiv preprint arXiv:2210.07383v1, 2022. Google Scholar
  6. Bruno Buchberger. Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Journal of symbolic computation, 41(3-4):475-511, 2006. URL: https://doi.org/10.1016/j.jsc.2005.09.007.
  7. Andrei Bulatov and Víctor Dalmau. A simple algorithm for Mal'tsev constraints. SIAM Journal on Computing, 36(1):16-27, 2006. URL: https://doi.org/10.1137/050628957.
  8. Andrei Bulatov, Marcin Kozik, Peter Mayr, and Markus Steindl. The subpower membership problem for semigroups. International Journal of Algebra and Computation, 26(07):1435-1451, 2016. URL: https://doi.org/10.1142/S0218196716500612.
  9. Andrei Bulatov, Peter Mayr, and Ágnes Szendrei. The subpower membership problem for finite algebras with cube terms. Logical Methods in Computer Science, 15(1), 2019. URL: https://doi.org/10.23638/LMCS-15(1:11)2019.
  10. Jakub Bulín and Michael Kompatscher. Short definitions in constraint languages. In Proceedings of the 48th International Symposium on Mathematical Foundations of Computer Science, (MFCS 2023), volume 272 of LIPIcs, pages 28:1-28:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.MFCS.2023.28.
  11. Stanley Burris and Hanamantagouda Sankappanavar. A course in universal algebra, volume 78. Springer, 1981. Google Scholar
  12. Victor Dalmau and Peter Jeavons. Learnability of quantified formulas. Theoretical Computer Science, 306(1-3):485-511, 2003. URL: https://doi.org/10.1016/S0304-3975(03)00342-6.
  13. Stefano Fioravanti. Closed sets of finitary functions between finite fields of coprime order. Algebra universalis, 81(4):Art. No 52, 2020. URL: https://doi.org/10.1007/s00012-020-00683-5.
  14. Ralph Freese and Ralph McKenzie. Commutator theory for congruence modular varieties, volume 125. CUP Archive, 1987. Google Scholar
  15. Merrick Furst, John Hopcroft, and Eugene Luks. Polynomial-time algorithms for permutation groups. In 21st Annual Symposium on Foundations of Computer Science (sfcs 1980), pages 36-41. IEEE, 1980. Google Scholar
  16. Christian Herrmann. Affine algebras in congruence modular varieties. Acta Universitatis Szegediensis, 41:119-11, 1979. Google Scholar
  17. Paweł Idziak, Petar Marković, Ralph McKenzie, Matthew Valeriote, and Ross Willard. Tractability and learnability arising from algebras with few subpowers. SIAM Journal on Computing, 39(7):3023-3037, 2010. Google Scholar
  18. Piotr Kawałek, Michael Kompatscher, and Jacek Krzaczkowski. Circuit equivalence in 2-nilpotent algebras. arXiv preprint arXiv:1909.12256, accepted for publication in STACS 2024. Google Scholar
  19. Donald E Knuth. Efficient representation of perm groups. Combinatorica, 11(1):33-43, 1991. URL: https://doi.org/10.1007/BF01375471.
  20. Michael Kompatscher, Peter Mayr, and Patrick Wynne. Personal communication. presented at AAA102, Szeged, 2022. URL: https://www.math.u-szeged.hu/aaa102/Slides/Mayr,%20Peter2.pdf.
  21. Dexter Kozen. Complexity of finitely presented algebras. In Proceedings of the 9th annual ACM Symposium on Theory of Computing (STOC), pages 164-177, 1977. URL: https://doi.org/10.1145/800105.803406.
  22. Dexter Kozen. Lower bounds for natural proof systems. In 18th Annual Symposium on Foundations of Computer Science (sfcs 1977), pages 254-266, 1977. URL: https://doi.org/10.1109/SFCS.1977.16.
  23. Marcin Kozik. A finite set of functions with an EXPTIME-complete composition problem. Theoretical Computer Science, 407(1-3):330-341, 2008. Google Scholar
  24. Ernst W Mayr and Albert R Meyer. The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in mathematics, 46(3):305-329, 1982. URL: https://doi.org/10.1016/0001-8708(82)90048-2.
  25. Peter Mayr. The subpower membership problem for Mal'cev algebras. International Journal of Algebra and Computation, 22(07):1250075, 2012. URL: https://doi.org/10.1142/S0218196712500750.
  26. Peter Mayr. Vaughan-Lee’s nilpotent loop of size 12 is finitely based. Algebra universalis, 85(1):1-12, 2024. URL: https://doi.org/10.1007/s00012-023-00832-6.
  27. Peter Mayr and Patrick Wynne. Clonoids between modules. arXiv preprint arXiv:2307.00076, 2023. Google Scholar
  28. Vladimir Shpilrain and Alexander Ushakov. Thompson’s group and public key cryptography. In Applied Cryptography and Network Security: Third International Conference, ACNS 2005, New York, NY, USA, June 7-10, 2005. Proceedings 3, pages 151-163. Springer, 2005. URL: https://doi.org/10.1007/11496137_11.
  29. Vladimir Shpilrain and Gabriel Zapata. Using the subgroup membership search problem in public key cryptography. Contemporary Mathematics, 418:169, 2006. URL: https://doi.org/10.1090/conm/418/07955.
  30. Charles C Sims. Computational methods in the study of permutation groups. In Computational problems in abstract algebra, pages 169-183. Elsevier, 1970. URL: https://doi.org/10.1016/B978-0-08-012975-4.50020-5.
  31. Joel VanderWerf. Wreath products of algebras: generalizing the Krohn-Rhodes theorem to arbitrary algebras. Semigroup Forum, 52(1):93-100, 1996. URL: https://doi.org/10.1007/BF02574084.
  32. Ross Willard. Four unsolved problems in congruence permutable varieties, 2007. Talk at the Conference on Order, Algebra, and Logics, Nashville. URL: https://www.math.uwaterloo.ca/~rdwillar/documents/Slides/willard_nashville_2007_slides.pdf.
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