Abstract
As is well known, any function of the two-valued algebra of logic (FAL) can be realized in the basis &, V, ˥ by a scheme of depth 2 (if by depth we understand the maximal number of alternations of the operators & and V). Such a scheme is obtained in modeling normal forms of FAL [1]. With this, an asymptotic bound on the complexity of the scheme equals n·2n−1, where n is the number of variables. Lupanov [2] showed that any FAL is realized in basis &, V, ˥ by a scheme of depth 3 with asymptotic bound on its complexity of 2n/log2n. With a further increase in depth, this bound is not changed.
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Literature Cited
N. E. Kobrinskii and B. A. Trakhtenbrot, Introduction to the Theory of Finite Automata, Fizmatgiz (1962).
O. B. Lupanov, “On the realization of functions of the algebra of logic by formulas of finite class (formulas of bounded depth) in the basis &, V,” in: Problems of Cybernetics, Vol. 6, Fizmatgiz (1961).
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Fet, Y.I. (1969). Some Algorithms for Synthesizing Schemes of Minimal Depth. In: Lazarev, V.G., Zakrevskii, A.V. (eds) Synthesis of Digital Automata / Problemy Sinteza Tsifrovykh Avtomatov / Проƃлемы Синтеза Цифровых Автоматов. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-9033-6_4
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DOI: https://doi.org/10.1007/978-1-4684-9033-6_4
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