{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:39:32Z","timestamp":1787341172052,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2013,1]]},"abstract":"<jats:p>We study the problem of testing isomorphism (equivalence up to relabeling of the input variables) between Boolean functions. We prove the following: (1) For most functions $f:\\{0,1\\}^n \\to \\{0,1\\}$, the query complexity of testing isomorphism to $f$ is $\\Omega(n)$. Moreover, the query complexity of testing isomorphism to most $k$-juntas $f:\\{0,1\\}^n \\to \\{0,1\\}$ is $\\Omega(k)$. (2) Isomorphism to any $k$-junta $f:\\{0,1\\}^n \\to \\{0,1\\}$ can be tested with $O(k \\log k)$ queries. (3) For some $k$-juntas $f:\\{0,1\\}^n \\to \\{0,1\\}$, testing isomorphism to $f$ with one-sided error requires $\\Omega(k\\log(n\/k))$ queries. In particular, testing whether $f:\\{0,1\\}^n \\to \\{0,1\\}$ is a $k$-parity with one-sided error requires $\\Omega(k\\log(n\/k))$ queries. (4) The query complexity of testing isomorphism between two unknown functions $f,g:\\{0,1\\}^n \\to \\{0,1\\}$ is $\\widetilde{\\Theta}(2^{n\/2})$. These bounds are tight up to logarithmic factors, and they significantly strengthen the bounds proved by Fischer, Kindler, Ron, Safra, and Samorodnitsky [J. Comput. System Sci., 68 (2004), pp. 753--787] and Blais and O'Donnell [Proceedings of the IEEE Conference on Computational Complexity, 2010, pp. 235--246]. We also obtain results closely related to isomorphism testing, answering a question posed by Diakonikolas, Lee, Matulef, Onak, Rubinfeld, Servedio, and Wan [Proceedings of the IEEE Symposium on Foundations of Computer Science, 2007, pp. 549--558]: testing whether a function $f:\\{0,1\\}^n \\to \\{0,1\\}$ can be computed by a circuit of size $\\le s$ requires $s^{\\Omega(1)}$ queries. All of our lower bounds apply to general (adaptive) testers.<\/jats:p>","DOI":"10.1137\/110832677","type":"journal-article","created":{"date-parts":[[2013,3,12]],"date-time":"2013-03-12T09:29:12Z","timestamp":1363080552000},"page":"459-493","source":"Crossref","is-referenced-by-count":3,"title":["Nearly Tight Bounds for Testing Function Isomorphism"],"prefix":"10.1137","volume":"42","author":[{"given":"Noga","family":"Alon","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eric","family":"Blais","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sourav","family":"Chakraborty","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David","family":"Garc\u00eda-Soriano","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arie","family":"Matsliah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2013,3,12]]},"reference":[{"key":"atypb1","first-page":"394","author":"Alon N.","year":"2010","journal-title":"Berlin"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s004930070001"},{"key":"atypb3","unstructured":"N. Alon and J. H. Spencer,\n                      The Probabilistic Method\n                      , Wiley, New York, 1992."},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"E. Blais, J. Brody, and K. Matulef,\n                      Property testing lower bounds via communication complexity\n                      , in Proceedings of the 26th Annual IEEE Conference on Computational Complexity, 2011, pp. 210-220.","DOI":"10.1109\/CCC.2011.31"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s00037-011-0020-6"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"T. Batu, E. Fischer, L. Fortnow, R. Kumar, R. Rubinfeld, and P. White,\n                      Testing random variables for independence and identity\n                      , in Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, 2001, pp. 442-451.","DOI":"10.1109\/SFCS.2001.959920"},{"key":"atypb8","first-page":"151","author":"Blais E.","year":"2009","journal-title":"New York"},{"key":"atypb9","first-page":"73","author":"Blum M.","year":"1990","journal-title":"New York"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"E. Blais and R. O'Donnell,\n                      Lower bounds for testing function isomorphism\n                      , in Proceedings of the 25th Annual IEEE Conference on Computational Complexity, 2010, pp. 235-246.","DOI":"10.1109\/CCC.2010.30"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2004.01.023"},{"key":"atypb12","first-page":"545","author":"Chakraborty S.","year":"2011","journal-title":"Heidelberg"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973082.130"},{"key":"atypb14","first-page":"97","author":"Dodis Y.","year":"1999","journal-title":"Berlin"},{"key":"atypb15","first-page":"549","author":"Diakonikolas I.","year":"2007","journal-title":"Proceedings of the IEEE Symposium on Foundations of Computer Science"},{"key":"atypb16","unstructured":"E. Fischer,\n                      The art of uninformed decisions: A primer to property testing\n                      , Bull. Eur. Assoc. Theor. Comput. Sci., No. 75 (2001), pp. 97-126."},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/S009753970445603"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2003.11.004"},{"key":"atypb19","first-page":"474","author":"Fischer E.","year":"2002","journal-title":"New York"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/070680795"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10110"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1987-0871675-6"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02579457"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s004930070011"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1145\/285055.285060"},{"key":"atypb26","first-page":"574","author":"Goldreich O.","year":"2010","journal-title":"Berlin"},{"key":"atypb27","first-page":"601","author":"Hajnal A.","year":"1970","journal-title":"Amsterdam"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"S. Jukna,\n                      Extremal Combinatorics. With Applications in Computer Science\n                      , Springer, Berlin, 2001.","DOI":"10.1007\/978-3-662-04650-0"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480103434634"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973068.29"},{"key":"atypb31","first-page":"646","author":"Matulef K.","year":"2009","journal-title":"Berlin"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480101407444"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1561\/2200000004"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1561\/0400000029"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539793255151"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1137\/100791075"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/S089548019223872X"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/110832677","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:08:28Z","timestamp":1787339308000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/110832677"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,1]]},"references-count":36,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2013,1]]}},"alternative-id":["10.1137\/110832677"],"URL":"https:\/\/doi.org\/10.1137\/110832677","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,1]]}}}