{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T17:17:17Z","timestamp":1760203037840,"version":"3.41.0"},"reference-count":23,"publisher":"Association for Computing Machinery (ACM)","issue":"1","license":[{"start":{"date-parts":[[2016,10,13]],"date-time":"2016-10-13T00:00:00Z","timestamp":1476316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Comput. Theory"],"published-print":{"date-parts":[[2017,3,31]]},"abstract":"\n In this article, we explore the noncommutative analogues, VP\n nc<\/jats:sub>\n and VNP\n \n nc<\/jats:italic>\n <\/jats:sub>\n , of Valiant\u2019s algebraic complexity classes and show some striking connections to classical formal language theory. Our main results are the following:\n <\/jats:p>\n \n \u2014 We show that Dyck polynomials (defined from the Dyck languages of formal language theory) are complete for the class VP\n \n nc<\/jats:italic>\n <\/jats:sub>\n under \u2264\n \n abp<\/jats:italic>\n <\/jats:sub>\n reductions. To the best of our knowledge, these are the first natural polynomial families shown to be VP\n \n nc<\/jats:italic>\n <\/jats:sub>\n -complete. Likewise, it turns out that PAL (palindrome polynomials defined from palindromes) are complete for the class VSKEW\n \n nc<\/jats:italic>\n <\/jats:sub>\n (defined by polynomial-size skew circuits) under \u2264\n \n abp<\/jats:italic>\n <\/jats:sub>\n reductions. The proof of these results is by suitably adapting the classical Chomsky-Sch\u00fctzenberger theorem showing that Dyck languages are the hardest CFLs.\n <\/jats:p>\n \n \u2014 Assuming that VP\n \n nc<\/jats:italic>\n <\/jats:sub>\n \u2260 VNP\n \n nc<\/jats:italic>\n <\/jats:sub>\n , we exhibit a strictly infinite hierarchy of p-families, with respect to the projection reducibility, between the complexity classes VP\n \n nc<\/jats:italic>\n <\/jats:sub>\n and VNP\n \n nc<\/jats:italic>\n <\/jats:sub>\n (analogous to Ladner\u2019s theorem [Ladner 1975]).\n <\/jats:p>\n \n \u2014 Additionally, inside VP\n \n nc<\/jats:italic>\n <\/jats:sub>\n , we show that there is a strict hierarchy of p-families (based on the nesting depth of Dyck polynomials) with respect to the \u2264\n \n abp<\/jats:italic>\n <\/jats:sub>\n reducibility (defined explicitly in this article).\n <\/jats:p>","DOI":"10.1145\/2956230","type":"journal-article","created":{"date-parts":[[2016,10,13]],"date-time":"2016-10-13T19:28:22Z","timestamp":1476386902000},"page":"1-29","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Noncommutative Valiant's Classes"],"prefix":"10.1145","volume":"9","author":[{"given":"V.","family":"Arvind","sequence":"first","affiliation":[{"name":"Institute of Mathematical Sciences, Chennai, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P. S.","family":"Joglekar","sequence":"additional","affiliation":[{"name":"Vishwakarma Institute of Technology, Pune, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Raja","sequence":"additional","affiliation":[{"name":"Institute of Mathematical Sciences, Chennai, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2016,10,13]]},"reference":[{"volume-title":"Proceedings of the IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'09)","year":"2009","author":"Arvind Vikraman","key":"e_1_2_1_1_1"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-48054-0_4"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/1806689.1806782"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01077707"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/337244.337257"},{"key":"e_1_2_1_6_1","unstructured":"J. Berstel and C. Reutenauer. 2011. Noncommutative Rational Series with Applications. Cambridge University Press. https:\/\/books.google.co.in\/books?id=LL8Nhn72I_8C. J. Berstel and C. Reutenauer. 2011. Noncommutative Rational Series with Applications. Cambridge University Press. https:\/\/books.google.co.in\/books?id=LL8Nhn72I_8C."},{"key":"e_1_2_1_7_1","first-page":"73","article-title":"On the structure of Valiant's complexity classes","volume":"3","author":"B\u00fcrgisser Peter","year":"1999","journal-title":"Discrete Mathematics and Theoretical Computer Science"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539798367880"},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539798367892"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)72023-8"},{"volume-title":"Weyuker","year":"1994","author":"Davis Martin D.","key":"e_1_2_1_11_1"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1080\/03081088508817680"},{"volume-title":"On the Complexity of Computing Characters of a Finite Group. Master's Thesis","author":"Hepler Charles Thomas","key":"e_1_2_1_13_1"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1145\/1806689.1806781"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/CCC.2010.34"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1145\/321864.321877"},{"volume-title":"Electronic Colloquium on Computational Complexity. Retrieved","year":"2015","author":"Limaye Nutan","key":"e_1_2_1_17_1"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.4086\/toc.2013.v009a006"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/103418.103462"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-005-0188-8"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02165411"},{"key":"e_1_2_1_22_1","unstructured":"Seinosuke Toda. 1992. Classes of arithmetic circuits capturing the complexity of computing the determinant. IEICE Transactions on Informations and Systems E75-D 1 116--124. Seinosuke Toda. 1992. Classes of arithmetic circuits capturing the complexity of computing the determinant. IEICE Transactions on Informations and Systems E75-D 1 116--124."},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/800135.804419"}],"container-title":["ACM Transactions on Computation Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2956230","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2956230","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T03:39:43Z","timestamp":1750217983000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2956230"}},"subtitle":["Structure and Complete Problems"],"short-title":[],"issued":{"date-parts":[[2016,10,13]]},"references-count":23,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,3,31]]}},"alternative-id":["10.1145\/2956230"],"URL":"https:\/\/doi.org\/10.1145\/2956230","relation":{},"ISSN":["1942-3454","1942-3462"],"issn-type":[{"type":"print","value":"1942-3454"},{"type":"electronic","value":"1942-3462"}],"subject":[],"published":{"date-parts":[[2016,10,13]]},"assertion":[{"value":"2015-10-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-06-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2016-10-13","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}