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On the Complexity of Multivalued Logic Functions over Some Infinite Basis

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Abstract

Under study is the complexity of the realization of k-valued logic functions (k ≥ 3) by logic circuits in the infinite basis consisting of the Post negation (i.e., the function (x + 1) mod k) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. For an arbitrary function f, we find the lower and upper bounds of complexity, which differ from one another at most by 1 and have the form 3 log3(d(f)+ 1)+O(1), where d(f) is the maximal number of the decrease of the value of f taken over all increasing chains of tuples of values of the variables. We find the exact value of the corresponding Shannon function which characterizes the complexity of the most complex function of a given number of variables.

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References

  1. N. A. Karpova, “Some Properties of Shannon Functions,” Mat. Zametki 8 (5), 663–674 (1970) [Math. Notes 8 (5), 843–849 (1970)].

    MathSciNet  Google Scholar 

  2. O.M. Kasim-Zade, “Complexity of Circuits in an Infinite Basis,” Vestnik Moskov. Univ. Ser. 1, No. 6, 40–44 (1994).

    MathSciNet  MATH  Google Scholar 

  3. O. M. Kasim-Zade, “Orders of Growth of Shannon Functions for Circuit Complexity over Infinite Bases,” VestnikMoskov.Univ. Ser. 1,Mat. Mekh.No. 3, 55–57 (2013) [Moscow Univ.Math. Bull. 68 (3), 170–172 (2013)].

    MATH  Google Scholar 

  4. V. V. Kochergin, “Theory of Rectifier Circuits (the Current State),”in Discrete Mathematics and Its Applications, Vol. 7 (Inst. Prikl. Mat. Ross. Akad. Nauk, Moscow, 2013), pp. 23–40 [Available at http://www.keldysh.ru/dmschool/datastore/media/dmsc9_lect7.pdf (accessed Nov. 16, 2017)].

    Google Scholar 

  5. V. V. Kochergin and A. V. Mikhailovich, “On the Complexity of Circuits in Bases Containing Monotone Elements with Zero Weights,” Prikl. Diskretn.Mat. No. 4, 24–31 (2015).

    Article  Google Scholar 

  6. V. V. Kochergin and A. V. Mikhailovich, “The Minimum Number of Negations in Circuits for Systems of Multi-Valued Functions,” Diskretn. Mat. 28 (4), 80–90 (2016) [Discrete Math. Appl. 27 (5), 295–302 (2017)].

    Article  Google Scholar 

  7. O. B. Lupanov, “The Synthesis of Circuits from Threshold Elements,” in Problems of Cybernetics, Vol. 26 (Nauka, Moscow, 1973), pp. 109–140.

    Google Scholar 

  8. O. B. Lupanov, Asymptotic Estimations of Complexity of Control Systems (Moskov. Gos. Univ., Moscow, 1984) [in Russian].

    Google Scholar 

  9. A. A. Markov, “On the Inversion Complexity of a System of Functions,” Dokl. Akad. Nauk SSSR 116 (6), 917–919 (1957) [J. ACM 5 (4), 331–334 (1958)].

    MathSciNet  MATH  Google Scholar 

  10. A. A. Markov, “On the Inversion Complexity of Systems of Boolean Functions,” Dokl. Akad. Nauk SSSR 150 (3), 477–479 (1963) [Sov.Math. Dokl. 4, 694–696 (1963)].

    MathSciNet  Google Scholar 

  11. E. I. Nechiporuk, “On the Complexity of Circuits in Some Bases Containing Nontrivial Elements with Zero Weights,” in Problems of Cybernetics, Vol. 8 (Fizmatgiz, Moscow, 1962), pp. 123–160.

    MathSciNet  MATH  Google Scholar 

  12. E. I. Nechiporuk, “On a Synthesis of Schemes of Threshold Elements,” in Problems of Cybernetics, Vol. 11 (Nauka, Moscow, 1964), pp. 49–62.

    MathSciNet  MATH  Google Scholar 

  13. O. V. Podolskaya, “Circuit Complexity of Symmetric Boolean Functions in an Antichain Basis,” Diskretn. Mat. 27 (3), 95–107 (2015) [DiscreteMath. Appl. 26 (1), 31–40 (2016)].

    Article  Google Scholar 

  14. J. E. Savage, The Complexity of Computing (Wiley, New York, 1976; Faktorial, Moscow, 1998).

    MATH  Google Scholar 

  15. E. Blais, C. L. Canonne, I. C. Oliveira, R. A. Servedio, and L.-Y. Tan, “Learning Circuits with Few Negations,” Electron. Colloq. Comput. Complex., Rep. No. 144 (2014) [Available at http://eccc.weizmann.ac.il/report/2014/144/download (accessed on November 16, 2017)].

  16. E. N. Gilbert, “Lattice Theoretic Properties of Frontal Switching Functions,” J. Math. Phys. 33 (1), 57–67 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Guo, T. Malkin, I. C. Oliveira, and A. Rosen, “The Power of Negations in Cryptography,” in Theory of Cryptography (Springer, Heidelberg, 2015), pp. 36–65.

    Google Scholar 

  18. M. J. Fischer, “The Complexity of Negation-Limited Networks—A Brief Survey,” in Automata Theory and Formal Languages (Springer, Berlin, 1975), pp. 71–82.

    Google Scholar 

  19. S. Jukna, Boolean Function Complexity: Advances and Frontiers (Springer, Heidelberg, 2012).

    Book  MATH  Google Scholar 

  20. V. V. Kochergin and A. V. Mikhailovich, “Some Extensions of the Inversion Complexity of Boolean Functions” (Cornell Univ. Libr. e-Print Archive, arXiv:1506.04485, 2015).

    Google Scholar 

  21. V. V. Kochergin and A. V. Mikhailovich, “Inversion Complexity of Functions of Multi-Valued Logic” (Cornell Univ. Libr. e-Print Archive, arXiv:1510.05942, 2015).

    Google Scholar 

  22. H. Morizumi, “A Note on the Inversion Complexity of Boolean Functions in Boolean Formulas” (Cornell Univ. Libr. e-Print Archive, arXiv:0811.0699, 2008).

    Google Scholar 

  23. N. Pippenger, “The Minimum Number of Edges in Graphs with Prescribed Paths,” Math. Systems Theory 12 (4), 325–346 (1979).

    MathSciNet  MATH  Google Scholar 

  24. S. Sung and K. Tanaka, “Limiting Negations in Bounded-Depth Circuits: An Extension of Markov’s Theorem,” in Algorithms and Computation (Springer, Heidelberg, 2003), pp. 108–116.

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Lomonosov Moscow State University, Leninskie gory 1, Moscow, 119991, Russia

    V. V. Kochergin

  2. National Research University Higher School of Economics, ul. Myasnitskaya 20, Moscow, 101000, Russia

    A. V. Mikhailovich

Authors
  1. V. V. Kochergin
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  2. A. V. Mikhailovich
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Correspondence to V. V. Kochergin.

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Original Russian Text © V.V. Kochergin, A.V. Mikhailovich, 2018, published in Diskretnyi Analiz i Issledovanie Operatsii, 2018, Vol. 25, No. 1, pp. 42–74.

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Kochergin, V.V., Mikhailovich, A.V. On the Complexity of Multivalued Logic Functions over Some Infinite Basis. J. Appl. Ind. Math. 12, 40–58 (2018). https://doi.org/10.1134/S1990478918010052

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  • DOI: https://doi.org/10.1134/S1990478918010052

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