{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T09:04:18Z","timestamp":1648717458187},"reference-count":3,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2010,12]]},"abstract":"<jats:p> A word circuit [1] is a directed acyclic graph in which each edge holds a w-bit word (i.e., some x \u2208 {0, 1}<jats:sup>w<\/jats:sup>) and each node is a gate computing some binary function g : {0, 1}<jats:sup>w<\/jats:sup> \u00d7 {0, 1}<jats:sup>w<\/jats:sup> \u2192 {0, 1}<jats:sup>w<\/jats:sup>. The following problem was studied in [1]: How many binary gates are needed to compute a ternary function f : ({0, 1}<jats:sup>w<\/jats:sup>)<jats:sup>3<\/jats:sup> \u2192 {0, 1}<jats:sup>w<\/jats:sup>. They proved that (2 + o(1))2<jats:sup>w<\/jats:sup> binary gates are enough for any ternary function, and there exists a ternary function which requires word circuits of size (1 - o(1))2<jats:sup>w<\/jats:sup>. One of the open problems in [1] is to get these bounds tight within a low order term. In this paper we solved this problem by constructing new word circuits for ternary functions of size (1 + o(1))2<jats:sup>w<\/jats:sup>. We investigate the problem in a general setting: How many k-input word gates are needed for computing an n-input word function f : ({0, 1}<jats:sup>w<\/jats:sup>)<jats:sup>n<\/jats:sup> \u2192 {0, 1}<jats:sup>w<\/jats:sup> (here n \u2265 k). We show that for any fixed n, (1 - o(1))2<jats:sup>(n - k)w<\/jats:sup> basic gates are necessary and (1 + o(1))2<jats:sup>(n - k)w<\/jats:sup> gates are sufficient (assume w is sufficiently large). Since word circuit is a natural generalization of boolean circuit, we also consider the case when w is a constant and the number of inputs n is sufficiently large. We show that [Formula: see text] basic gates are necessary and sufficient in this case. <\/jats:p>","DOI":"10.1142\/s1793830910000826","type":"journal-article","created":{"date-parts":[[2011,1,17]],"date-time":"2011-01-17T08:21:26Z","timestamp":1295252486000},"page":"483-492","source":"Crossref","is-referenced-by-count":0,"title":["THE COMPLEXITY OF WORD CIRCUITS"],"prefix":"10.1142","volume":"02","author":[{"given":"XUE","family":"CHEN","sequence":"first","affiliation":[{"name":"Department of Computer Science and Technology, Tsinghua University, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"GUANGDA","family":"HU","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Technology, Tsinghua University, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"XIAOMING","family":"SUN","sequence":"additional","affiliation":[{"name":"Institute for Theoretical Computer Science and Center for Advanced Study, Tsinghua University, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,4,6]]},"reference":[{"key":"rf2","first-page":"120","author":"Lupanov O. B.","journal-title":"Izvestiya VUZ, Radiofizika"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1949.tb03624.x"},{"key":"rf4","doi-asserted-by":"crossref","unstructured":"I.\u00a0Wegener, The Complexity of Boolean Functions (John-Wiley & Sons and B. G. Teubner, 1987)\u00a0pp. 87\u201392.","DOI":"10.1007\/3-540-18170-9_185"}],"container-title":["Discrete Mathematics, Algorithms and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S1793830910000826","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T17:15:53Z","timestamp":1565111753000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S1793830910000826"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,12]]},"references-count":3,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2012,4,6]]},"published-print":{"date-parts":[[2010,12]]}},"alternative-id":["10.1142\/S1793830910000826"],"URL":"https:\/\/doi.org\/10.1142\/s1793830910000826","relation":{},"ISSN":["1793-8309","1793-8317"],"issn-type":[{"value":"1793-8309","type":"print"},{"value":"1793-8317","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,12]]}}}