On the Depth of a Multiplexer Function with a Small Number of Select Lines

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes Pub Date : 2024-07-15 DOI:10.1134/s0001434624050092

S. A. Lozhkin

引用次数: 0

Abstract

This paper continues the research on the circuit synthesis problem for a multiplexer function of logic algebra, which is a component of many integrated circuits and is also used in theoretical study. The exact value of the depth of a multiplexer with \(n\) select lines in the standard basis is found under the assumption that the conjunction and disjunction gates are of depth 1 and the negation gate is of depth 0; the depth equals \(n+2\) if \(10 \le n \le 19\). Thus, it follows from previous results that the exact depth value equals \(n+2\) for all positive integers \(n\) such that either \(2 \le n \le 5\) or \(n \ge 10\). Moreover, for \(n=1\), this value equals 2, and for \(6 \le n \le 9\), it equals either \(n+2\) or \(n+3\). Similar results are also obtained for a basis consisting of all elementary conjunctions and elementary disjunctions of two variables.

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关于具有少量选择行的多路复用器函数深度

摘要本文继续研究逻辑代数的多路复用器函数的电路合成问题，多路复用器是许多集成电路的组成部分，也用于理论研究。在联结门和析取门的深度为 1，否定门的深度为 0 的假设下，求出了在标准基础上具有 \(n\) 条选择线的多路复用器深度的精确值；如果 \(10 \le n \le 19\) ，深度等于 \(n+2\)。因此，从前面的结果可以得出，对于所有正整数 \(n)，要么是 \(2 \le n \le 5\) 要么是 \(n \ge 10\) ，精确深度值等于 \(n+2\)。此外，对于（n=1），这个值等于 2，而对于（6），它要么等于（n+2），要么等于（n+3）。对于由两个变量的所有基本连词和基本断词组成的基础，也可以得到类似的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.