Abstract
Given a class K of partial Boolean functions and a partial Boolean function f of n variables, a subset U of its variables is called sufficient for the implementation of f in K if there exists an extension of f in K with arguments in U. We consider the problem of recognizing all subsets sufficient for the implementation of f in K. For some classes defined by relations, we propose the algorithms of solving this problem with complexity of O(2nn2) bit operations. In particular, we present some algorithms of this complexity for the class P*2 of all partial Boolean functions and the class M*2 of all monotone partial Boolean functions. The proposed algorithms use the Walsh-Hadamard and Möbius transforms.
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References
O. I. Golubeva, “Construction of Permissible Functions and their Application for Fault Tolerance,” in Proceedings of International Siberian Conference on Control and Communications (SIBCON), (Tomsk, Russia, Apr. 18–20, 2019) (TUSUR, Tomsk, 2019), pp. 1–5.
A. A. Voronenko, “On a Decomposition Method for Recognizing Membership in Invariant Classes,” Diskret. Mat. 14 (4), 110–116 (2002)
A. A. Voronenko, Discrete Math. Appl. 12 (6), 607–614 (2002)].
S. N. Selezneva, “Constructing Polynomials for Functions over Residue Rings Modulo a Composite Number in Linear Time,” in Lecture Notes in Computer Science, Vol. 7353: Computer Science—Theory and Applications (Springer, Heidelberg, 2012), pp. 303–312.
V. B. Alekseev and N. R. Emel’yanov, “A Method for Constructing Fast Algorithms in k-Valued Logic,” Mat. Zametki 38 (1), 148–156, 171 (1985)
V. B. Alekseev and N. R. Emel’yanov, Math. Notes 38 (1), 595–600 (1985)].
V. B. Alekseev, “Stepwise Bilinear Algorithms and Recognition of Completeness in k-Valued Logics,” Izv. Vyssh. Uchebn. Zaved. Mat. No. 7, 19–27 (1988)
V. B. Alekseev, Soviet Math. (Izv. VUZ) 32 (7), 31–42 (1988)].
V. B. Alekseev, “Logical Semirings and Their Use in Constructing Fast Algorithms,” Vestnik Moskov. Univ. Ser. I, Mat. Mekh. No. 1, 22–29, 78–79 (1997)
V. B. Alekseev, Moscow Univ. Math. Bull. 52 (1), 22–28 (1997)].
N. G. Parvatov, “Generating Sufficient Sets for Partial Boolean Functions,” Vestnik Tomsk. Gos. Univ., Appendex. No 23, 44–48 (2007).
V. N. Sachkov, Combinatorial Methods of Discrete Mathematics (Nauka, Moscow, 1977) [in Russian].
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977; Svyaz’, Moscow, 1979).
S. K. Rushforth, “Fast Fourier-Hadamard Decoding of Orthogonal Codes,” Inform. and Control 15 (1), 33–37 (1969).
A. A. Malyutin, “Fast Correlation Decoding of Some Subsets of Words of the First-Order Reed-Muller Code,” Diskret. Mat. 2 (2), 155–158 (1990)
A. A. Malyutin, Discrete Math. Appl. 2 (2), 155–158 (1992).
Yu. D. Karyakin, “Fast Correlation Decoding of Reed-Muller Codes,” Problemy Peredachi Informatsii 23 (2), 40–49 (1987).
N. N. Tokareva, “Generalizations of Bent Functions: A Survey of Publications,” Diskretn. Anal. Issled. Oper. 17 (1), 34–64, 99 (2010)
N. N. Tokareva, J. Appl. Indust. Math. 5 (1), 110–129 (2011)].
L. Weisner, “Abstract Theory of Inversion of Finite Series,” Trans. Amer. Math. Soc. 38, 474–484 (1935).
P. Hall, “The Eulerian Functions of a Group,” Quart. J. Math. 7, 134–151 (1936).
G. C. Rota, “On the Foundations of Combinatorial Theory. I. Theory of Möbius Functions,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete. 2, 340–368 (1964).
R. P. Stanley, Enumerative Combinatorics, Vol. 1 (Cambridge University Press, Cambridge, 1997).
N. G. Parvatov, “On Recognizing of Properties of Discrete Functions by Boolean Circuits,” Vestnik Tomsk. Gos Univ. Appendix 14, 233–236 (2005).
A. Schönhage and V. Straßen, “Schnelle Multiplikation großer Zahlen,” Computing 7, 281–292 (1971).
M. Fürer, “Faster Integer Multiplication,” In Proceedings of the Thirty-Ninth ACM Symposium on Theory of Computing, STOC 2007 (San Diego, CA, USA, June 11–13, 2007) (ACM Press., New York, NY, USA, 2007), pp. 57–66.
A. De, C. Saha, P. Kurur, and R. Saptharishi, “Fast Integer Multiplication Using Modular Arithmetic,” in Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC) (Victoria, Canada, May 17–20, 2008) (ACM, New York, NY, 2008), pp. 499–506
A. De, C. Saha, P. Kurur, and R. Saptharishi, SIAM J. Comput. 42 (2), 685–699 (2013)].
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Russian Text © The Author(s), 2020, published in Diskretnyi Analiz i Issledovanie Operatsii, 2020, Vol. 27, No. 1, pp. 110–126.
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Parvatov, N.G. Finding the Subsets of Variables of a Partial Boolean Function Which Are Sufficient for Its Implementation in the Classes Defined by Predicates. J. Appl. Ind. Math. 14, 186–195 (2020). https://doi.org/10.1134/S1990478920010172
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DOI: https://doi.org/10.1134/S1990478920010172