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Complexity of realization of symmetric Boolean functions by switching circuits

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Title: Complexity of realization of symmetric Boolean functions by switching circuits
Authors: Grinchuk, M. I.
Publisher Information: Nauka, Moscow
Subject Terms: Analysis of algorithms and problem complexity, Switching theory, application of Boolean algebra, Boolean functions, contact schemes, symmetric Boolean functions, complexity, switching circuits
Description: This paper is a valuable contribution to Boolean function complexity theory, especially for the development of lower bound proof techniques. The author investigates the 25-year-old open problem of whether any symmetric Boolean function can be realized with linear complexity by contact schemes (switching circuits). Due to the almost linear \((O(n (\log n)^ 2/ \log\log n)\), \(O(n(\log n)^ 4/ (\log\log n)^ 2))\) upper bounds on the complexity of some subclasses of symmetric Boolean functions established in earlier papers by \textit{O. B. Lupanov} [Probl. Kibern. 15, 85-99 (1965)] and \textit{E. G. Krasulina} [Mat. Vopr. Kobern. 1, 140-167 (1988; Zbl 0668.94021)], the hypothesis that any symmetric function has linear complexity has been formulated. Here, it is shown that there exist symmetric Boolean functions which have nonlinear complexity (by contact schemes realization). The method of proof is interesting and is based on nontrivial combinatorial considerations. The paper should be of great interest to anyone dealing with lower bound proofs in complexity theory.
Document Type: Article
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Access URL: https://zbmath.org/23013
Accession Number: edsair.c2b0b933574d..e721b3b4efd3af1f7e9923e78540830d
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  Data: Complexity of realization of symmetric Boolean functions by switching circuits
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  Data: Nauka, Moscow
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  Data: <searchLink fieldCode="DE" term="%22Analysis+of+algorithms+and+problem+complexity%22">Analysis of algorithms and problem complexity</searchLink><br /><searchLink fieldCode="DE" term="%22Switching+theory%2C+application+of+Boolean+algebra%22">Switching theory, application of Boolean algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Boolean+functions%22">Boolean functions</searchLink><br /><searchLink fieldCode="DE" term="%22contact+schemes%22">contact schemes</searchLink><br /><searchLink fieldCode="DE" term="%22symmetric+Boolean+functions%22">symmetric Boolean functions</searchLink><br /><searchLink fieldCode="DE" term="%22complexity%22">complexity</searchLink><br /><searchLink fieldCode="DE" term="%22switching+circuits%22">switching circuits</searchLink>
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  Data: This paper is a valuable contribution to Boolean function complexity theory, especially for the development of lower bound proof techniques. The author investigates the 25-year-old open problem of whether any symmetric Boolean function can be realized with linear complexity by contact schemes (switching circuits). Due to the almost linear \((O(n (\log n)^ 2/ \log\log n)\), \(O(n(\log n)^ 4/ (\log\log n)^ 2))\) upper bounds on the complexity of some subclasses of symmetric Boolean functions established in earlier papers by \textit{O. B. Lupanov} [Probl. Kibern. 15, 85-99 (1965)] and \textit{E. G. Krasulina} [Mat. Vopr. Kobern. 1, 140-167 (1988; Zbl 0668.94021)], the hypothesis that any symmetric function has linear complexity has been formulated. Here, it is shown that there exist symmetric Boolean functions which have nonlinear complexity (by contact schemes realization). The method of proof is interesting and is based on nontrivial combinatorial considerations. The paper should be of great interest to anyone dealing with lower bound proofs in complexity theory.
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      – SubjectFull: Analysis of algorithms and problem complexity
        Type: general
      – SubjectFull: Switching theory, application of Boolean algebra
        Type: general
      – SubjectFull: Boolean functions
        Type: general
      – SubjectFull: contact schemes
        Type: general
      – SubjectFull: symmetric Boolean functions
        Type: general
      – SubjectFull: complexity
        Type: general
      – SubjectFull: switching circuits
        Type: general
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      – TitleFull: Complexity of realization of symmetric Boolean functions by switching circuits
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