文章基本信息

标题：Uniform, Integral and Feasible Proofs for the Determinant Identities
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作者：Iddo Tzameret ; Stephen Cook
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2018
卷号：2018
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over G F (2) in Hrubes-Tzameret [SICOMP'15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC 2 ; the latter is a first-order theory corresponding to the complexity class NC 2 consisting of problems solvable by uniform families of polynomial-size circuits and O ( log 2 n ) -depth. This also establishes the existence of uniform polynomial-size NC 2 -Frege proofs of the basic determinant identities over the integers (previous propositional proofs hold only over the two element field).
关键词：algebraic circuit complexity ; Algebraic circuits ; algebraic proofs ; bounded arithmetic ; circuit depth ; Proof Complexity