{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T23:40:09Z","timestamp":1748562009665,"version":"3.41.0"},"publisher-location":"Berlin, Heidelberg","reference-count":18,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"type":"print","value":"9783662480533"},{"type":"electronic","value":"9783662480540"}],"license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2015]]},"DOI":"10.1007\/978-3-662-48054-0_2","type":"book-chapter","created":{"date-parts":[[2015,8,10]],"date-time":"2015-08-10T11:57:29Z","timestamp":1439207849000},"page":"14-25","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Dual VP Classes"],"prefix":"10.1007","author":[{"given":"Eric","family":"Allender","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anna","family":"G\u00e1l","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ian","family":"Mertz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2015,8,11]]},"reference":[{"key":"2_CR1","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1006\/jcss.1999.1675","volume":"60","author":"M Agrawal","year":"2000","unstructured":"Agrawal, M., Allender, E., Datta, S.: On TC$$^0$$, AC$$^0$$, and arithmetic circuits. J. Comput. Syst. Sci. 60, 395\u2013421 (2000)","journal-title":"J. Comput. Syst. Sci."},{"key":"2_CR2","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1016\/S0304-3975(97)00227-2","volume":"209","author":"J Allender","year":"1998","unstructured":"Allender, J., Jiao, J., Mahajan, M., Vinay, V.: Non-commutative arithmetic circuits. Theoret. Comp. Sci. 209, 47\u201386 (1998)","journal-title":"Theoret. Comp. Sci."},{"key":"2_CR3","doi-asserted-by":"crossref","unstructured":"Allender, E., G\u00e1l, A., Mertz, I.: Dual VP classes. In: ECCC (2014). TR14-122","DOI":"10.1007\/978-3-662-48054-0_2"},{"issue":"2","key":"2_CR4","doi-asserted-by":"publisher","first-page":"164","DOI":"10.1006\/jcss.1999.1646","volume":"59","author":"E Allender","year":"1999","unstructured":"Allender, E., Reinhardt, K., Zhou, S.: Isolation, matching, and counting: Uniform and nonuniform upper bounds. J. Comput. Syst. Sci. 59(2), 164\u2013181 (1999)","journal-title":"J. Comput. Syst. Sci."},{"key":"2_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0701-6","volume-title":"Complexity and Real Computation","author":"L Blum","year":"1998","unstructured":"Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)"},{"key":"2_CR6","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1137\/0218038","volume":"18","author":"A Borodin","year":"1989","unstructured":"Borodin, A., Cook, S.A., Dymond, P.W., Ruzzo, W.L., Tompa, M.: Two applications of inductive counting for complementation problems. SIAM J. Comput. 18, 559\u2013578 (1989). See Erratum in SIAM. J. Comput. 18, 1283 (1989)","journal-title":"SIAM J. Comput."},{"key":"2_CR7","doi-asserted-by":"crossref","unstructured":"Buhrman, H., Cleve, R., Kouck\u00fd, M., Loff, B., Speelman, F.: Computing with a full memory: catalytic space. In: STOC, pp. 857\u2013866 (2014)","DOI":"10.1145\/2591796.2591874"},{"issue":"2","key":"2_CR8","doi-asserted-by":"publisher","first-page":"276","DOI":"10.1006\/jnth.1997.2114","volume":"64","author":"C Corrales-Rodrig\u00e1\u00f1ez","year":"1997","unstructured":"Corrales-Rodrig\u00e1\u00f1ez, C., Schoof, R.: The support problem and its elliptic analogue. J. Num. Theor. 64(2), 276\u2013290 (1997)","journal-title":"J. Num. Theor."},{"issue":"1\u20132","key":"2_CR9","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1002\/(SICI)1098-2418(199608\/09)9:1\/2<99::AID-RSA7>3.0.CO;2-6","volume":"9","author":"A G\u00e1l","year":"1996","unstructured":"G\u00e1l, A., Wigderson, A.: Boolean complexity classes vs. their arithmetic analogs. Random Struct. Algorithms 9(1\u20132), 99\u2013111 (1996)","journal-title":"Random Struct. Algorithms"},{"issue":"2","key":"2_CR10","doi-asserted-by":"publisher","first-page":"211","DOI":"10.1007\/s00037-010-0287-z","volume":"19","author":"KA Hansen","year":"2010","unstructured":"Hansen, K.A., Kouck\u00fd, M.: A new characterization of ACC$$^0$$ and probabilistic CC$$^0$$. Comput. Complex. 19(2), 211\u2013234 (2010)","journal-title":"Comput. Complex."},{"key":"2_CR11","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1006\/inco.1995.1007","volume":"116","author":"N Immerman","year":"1995","unstructured":"Immerman, N., Landau, S.: The complexity of iterated multiplication. Inf. Comput. 116, 103\u2013116 (1995)","journal-title":"Inf. Comput."},{"key":"2_CR12","doi-asserted-by":"publisher","first-page":"896","DOI":"10.1137\/0221053","volume":"21","author":"J Reif","year":"1992","unstructured":"Reif, J., Tate, S.: On threshold circuits and polynomial computation. SIAM J. Comput. 21, 896\u2013908 (1992)","journal-title":"SIAM J. Comput."},{"key":"2_CR13","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1137\/S0097539798339041","volume":"29","author":"K Reinhardt","year":"2000","unstructured":"Reinhardt, K., Allender, E.: Making nondeterminism unambiguous. SIAM J. Comput. 29, 1118\u20131131 (2000)","journal-title":"SIAM J. Comput."},{"key":"2_CR14","doi-asserted-by":"crossref","unstructured":"Smolensky, R.: Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In: STOC, pp. 77\u201382 (1987)","DOI":"10.1145\/28395.28404"},{"key":"2_CR15","doi-asserted-by":"crossref","unstructured":"Valiant, L.G.: Completeness classes in algebra. In: Proceedings of the 11th ACM STOC, pp. 249\u2013261 (1979)","DOI":"10.1145\/800135.804419"},{"issue":"4","key":"2_CR16","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1137\/0212043","volume":"12","author":"LG Valiant","year":"1983","unstructured":"Valiant, L.G., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12(4), 641\u2013644 (1983)","journal-title":"SIAM J. Comput."},{"key":"2_CR17","doi-asserted-by":"crossref","unstructured":"Vinay, V.: Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In: Proceedings of 6th Structure in Complexity Theory Conference, pp. 270\u2013284 (1991)","DOI":"10.1109\/SCT.1991.160269"},{"key":"2_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03927-4","volume-title":"Introduction to Circuit Complexity: A Uniform Approach","author":"H Vollmer","year":"1999","unstructured":"Vollmer, H.: Introduction to Circuit Complexity: A Uniform Approach. Springer, Heidelberg (1999)"}],"container-title":["Lecture Notes in Computer Science","Mathematical Foundations of Computer Science 2015"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-662-48054-0_2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T23:07:53Z","timestamp":1748560073000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-662-48054-0_2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015]]},"ISBN":["9783662480533","9783662480540"],"references-count":18,"URL":"https:\/\/doi.org\/10.1007\/978-3-662-48054-0_2","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2015]]},"assertion":[{"value":"11 August 2015","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}