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On the Counting Complexity of the Skolem Problem (2403.00098v1)

Published 29 Feb 2024 in cs.CC and cs.LO

Abstract: The Skolem Problem asks, given an integer linear recurrence sequence (LRS), to determine whether the sequence contains a zero term or not. Its decidability is a longstanding open problem in theoretical computer science and automata theory. Currently, decidability is only known for LRS of order at most 4. On the other hand, the sole known complexity result is NP-hardness, due to Blondel and Portier. A fundamental result in this area is the celebrated Skolem-Mahler-Lech theorem, which asserts that the zero set of any LRS is the union of a finite set and finitely many arithmetic progressions. This paper focuses on a computational perspective of the Skolem-Mahler-Lech theorem: we show that the problem of counting the zeros of a given LRS is #P-hard, and in fact #P-complete for the instances generated in our reduction.

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References (24)
  1. PRIMES is in P. Ann. Math. (2), 160(2):781–793, 2004. doi:10.4007/annals.2004.160.781.
  2. There are infinitely many Carmichael numbers. Annals of Mathematics, 139(3):703–722, 1994. URL: http://www.jstor.org/stable/2118576.
  3. On the complexity of numerical analysis. SIAM Journal on Computing, 38(5):1987–2006, 2009.
  4. Deciding ω𝜔\omegaitalic_ω-regular properties on linear recurrence sequences. Proc. ACM Program. Lang., 5(POPL), 2021.
  5. On the zeros of linear recurrence sequences. Acta Arithmetica, 147(4):387–396, 2011.
  6. Computational Complexity: A Modern Approach. Cambridge University Press, USA, 1st edition, 2009.
  7. On the complexity of pattern matching for highly compressed two-dimensional texts. Journal of Computer and System Sciences, 65(2):332–350, 2002. URL: https://www.sciencedirect.com/science/article/pii/S0022000002918520, doi:https://doi.org/10.1006/jcss.2002.1852.
  8. Skolem Meets Schanuel. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of Leibniz International Proceedings in Informatics (LIPIcs), pages 20:1–20:15, Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. URL: https://drops.dagstuhl.de/opus/volltexte/2022/16818, doi:10.4230/LIPIcs.MFCS.2022.20.
  9. The presence of a zero in an integer linear recurrent sequence is NP-hard to decide. Linear Algebra and its Applications, 351-352:91–98, August 2002. doi:10.1016/s0024-3795(01)00466-9.
  10. Counting value sets: Algorithm and complexity, 2011. URL: https://arxiv.org/abs/1111.1224, doi:10.48550/ARXIV.1111.1224.
  11. On the complexity of the orbit problem. J. ACM, 63(3), jun 2016. doi:10.1145/2857050.
  12. Introduction to Algorithms. McGraw-Hill Higher Education, 2nd edition, 2001.
  13. Recurrence Sequences. American Mathematical Society, July 2003. doi:10.1090/surv/104.
  14. Skolem’s problem - on the border between decidability and undecidability. Technical Report 683, 2005.
  15. Michael Krüger. On the Complexity of Alternative Solutions. PhD thesis, Friedrich-Schiller-University Jena, Germany, 2008.
  16. Noam Livne. A note on #P-completeness of NP-witnessing relations. Inf. Process. Lett., 109(5):259–261, feb 2009. doi:10.1016/j.ipl.2008.10.009.
  17. The distance between terms of an algebraic recurrence sequence. Journal für die reine und angewandte Mathematik, 349, 1984.
  18. Positivity Problems for Low-Order Linear Recurrence Sequences, pages 366–379. 2014. URL: https://epubs.siam.org/doi/abs/10.1137/1.9781611973402.27, arXiv:https://epubs.siam.org/doi/pdf/10.1137/1.9781611973402.27, doi:10.1137/1.9781611973402.27.
  19. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6(1):64 – 94, 1962. doi:10.1215/ijm/1255631807.
  20. Complexity of Restricted Variants of Skolem and Related Problems. In Kim G. Larsen, Hans L. Bodlaender, and Jean-Francois Raskin, editors, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017), volume 83 of Leibniz International Proceedings in Informatics (LIPIcs), pages 78:1–78:14, Dagstuhl, Germany, 2017. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. URL: http://drops.dagstuhl.de/opus/volltexte/2017/8130, doi:10.4230/LIPIcs.MFCS.2017.78.
  21. Completeness in the polynomial-time hierarchy a compendium. Sigact News - SIGACT, 33, 01 2002.
  22. N. K. Vereshchagin. Occurrence of zero in a linear recursive sequence. Mathematical Notes of the Academy of Sciences of the USSR, 38(2):609–615, August 1985. doi:10.1007/bf01156238.
  23. J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press, 2013. URL: https://books.google.co.in/books?id=AE5PN5QGgvUC.
  24. Counting curves and their projections. Comput. Complex., 6(1):64–99, 1997. doi:10.1007/BF01202042.

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