{"title":"On the Depth of a Multiplexer Function with a Small Number of Select Lines","authors":"S. A. Lozhkin","doi":"10.1134/s0001434624050092","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> 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 <span>\\(n\\)</span> 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 <span>\\(n+2\\)</span> if <span>\\(10 \\le n \\le 19\\)</span>. Thus, it follows from previous results that the exact depth value equals <span>\\(n+2\\)</span> for all positive integers <span>\\(n\\)</span> such that either <span>\\(2 \\le n \\le 5\\)</span> or <span>\\(n \\ge 10\\)</span>. Moreover, for <span>\\(n=1\\)</span>, this value equals 2, and for <span>\\(6 \\le n \\le 9\\)</span>, it equals either <span>\\(n+2\\)</span> or <span>\\(n+3\\)</span>. Similar results are also obtained for a basis consisting of all elementary conjunctions and elementary disjunctions of two variables. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624050092","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
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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.
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.
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