{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:36:51Z","timestamp":1753889811262,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Discrete Algorithms","license":[{"start":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T00:00:00Z","timestamp":1574726400000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T00:00:00Z","timestamp":1574726400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,11,26]],"date-time":"2019-11-26T00:00:00Z","timestamp":1574726400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"funder":[{"name":"National Science Foundation","award":["1916805"],"award-info":[{"award-number":["1916805"]}]},{"name":"National Science Foundation","award":["1745207"],"award-info":[{"award-number":["1745207"]}]},{"name":"National Science Foundation","award":["1908530"],"award-info":[{"award-number":["1908530"]}]},{"name":"National Science Foundation","award":["1443054"],"award-info":[{"award-number":["1443054"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>Many researchers have studied symmetry properties of various Boolean functions. A class of Boolean functions, called nested canalyzing functions (NCFs), has been used to model certain biological phenomena. We identify some interesting relationships between NCFs, symmetric Boolean functions and a generalization of symmetric Boolean functions, which we call $r$-symmetric functions (where $r$ is the symmetry level). Using a normalized representation for NCFs, we develop a characterization of when two variables of an NCF are symmetric. Using this characterization, we show that the symmetry level of an NCF $f$ can be easily computed given a standard representation of $f$. We also present an algorithm for testing whether a given $r$-symmetric function is an NCF. Further, we show that for any NCF $f$ with $n$ variables, the notion of strong asymmetry considered in the literature is equivalent to the property that $f$ is $n$-symmetric. We use this result to derive a closed form expression for the number of $n$-variable Boolean functions that are NCFs and strongly asymmetric. We also identify all the Boolean functions that are NCFs and symmetric.<\/jats:p><jats:p>Comment: 17 pages<\/jats:p>","DOI":"10.23638\/dmtcs-21-4-19","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:57:16Z","timestamp":1743699436000},"source":"Crossref","is-referenced-by-count":0,"title":["Symmetry Properties of Nested Canalyzing Functions"],"prefix":"10.23638","volume":"vol. 21 no. 4","author":[{"given":"Daniel J.","family":"Rosenkrantz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Madhav V.","family":"Marathe","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S. S.","family":"Ravi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Richard E.","family":"Stearns","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2019,11,26]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1906.03752v3","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1906.03752v3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:57:18Z","timestamp":1743699438000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/5565"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,26]]},"references-count":0,"journal-issue":{"issue":"Discrete Algorithms","published-online":{"date-parts":[[2019,11,26]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-21-4-19","relation":{"has-preprint":[{"id-type":"arxiv","id":"1906.03752v2","asserted-by":"subject"},{"id-type":"arxiv","id":"1906.03752v1","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1906.03752","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1906.03752","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2019,11,26]]},"article-number":"5565"}}