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ACM"],"published-print":{"date-parts":[[2016,5,4]]},"abstract":"<jats:p>We connect both discrete and algebraic complexity theory with the satisfiability problem in certain non-Boolean lattices.<\/jats:p>\n          <jats:p>Specifically, quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables: in the 1D case it coincides with Boolean propositional logic but, starting with dimension two, violates the distributive law.<\/jats:p>\n          <jats:p>\n            We introduce the weak and strong satisfiability problem for quantum logic propositional formulae. It turns out that in dimension two, both are also\n            <jats:italic>NP<\/jats:italic>\n            --complete.\n          <\/jats:p>\n          <jats:p>\n            For higher-dimensional spaces \u211d\n            <jats:sup>\n              <jats:italic>d<\/jats:italic>\n            <\/jats:sup>\n            and \u2102\n            <jats:sup>\n              <jats:italic>d<\/jats:italic>\n            <\/jats:sup>\n            with\n            <jats:italic>d<\/jats:italic>\n            \u2265 3 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale (BSS) model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and extends the (still relatively scarce) list of problems established\n            <jats:italic>NP<\/jats:italic>\n            <jats:sub>\u211d<\/jats:sub>\n            --complete with one, perhaps, closest in spirit to the classical Cook-Levin Theorem. More precisely, strong satisfiability of \u2227 \u2228 \u2227 --terms is complete, while that of \u2227 \u2228--terms (i.e., those in conjunctive form) can be decided in polynomial time in dimensions\n            <jats:italic>d<\/jats:italic>\n            \u2265 2.\n          <\/jats:p>\n          <jats:p>\n            The decidability of the infinite-dimensional case being still open, we proceed to investigate the case of indefinite finite dimensions. Here, weak satisfiability still belongs to\n            <jats:italic>NP<\/jats:italic>\n            <jats:sub>R<\/jats:sub>\n            and strong satisfiability is still hard; the latter, in fact, turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations over matrix rings.\n          <\/jats:p>","DOI":"10.1145\/2869073","type":"journal-article","created":{"date-parts":[[2016,5,5]],"date-time":"2016-05-05T13:23:22Z","timestamp":1462454602000},"page":"1-31","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["Computational Complexity of Quantum Satisfiability"],"prefix":"10.1145","volume":"63","author":[{"given":"Christian","family":"Herrmann","sequence":"first","affiliation":[{"name":"TU Darmstadt and KAIST, Darmstadt, Germany"}]},{"given":"Martin","family":"Ziegler","sequence":"additional","affiliation":[{"name":"TU Darmstadt and KAIST, Daejeon, Rep. of Korea"}]}],"member":"320","published-online":{"date-parts":[[2016,5,4]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.5555\/1018438.1021878"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/070697926"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.5555\/1832876.1832877"},{"key":"e_1_2_1_4_1","volume-title":"Encyclopedia of Mathematics and Its Applications. 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Computational approach to polynomial identities of matrices\u2014a survey. In Polynomial Identities And Combinatorial Methods, Giambruno, Regev, and Zaicev (Eds.). Lectures Notes in Pure and Applied Mathematics, Vol. 234. Dekker, 141--178."},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.2307\/1968621"},{"key":"e_1_2_1_7_1","volume-title":"Oriented matroids. Encyclopedia of Mathematics and its Applications","author":"Bj\u00f6rner A","unstructured":"A Bj\u00f6rner , M Las Vergnas , B Sturmfels , Neil White , and GM Ziegler . 1999. Oriented matroids. Encyclopedia of Mathematics and its Applications , Vol. 46 . A Bj\u00f6rner, M Las Vergnas, B Sturmfels, Neil White, and GM Ziegler. 1999. Oriented matroids. 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