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Non-commutative arithmetic circuits: depth reduction and size lower bounds

Allender, Eric and Jiao, Jia and Mahajan, Meena and Vinay, V (1998) Non-commutative arithmetic circuits: depth reduction and size lower bounds. In: Theoretical Computer Science, 209 (1-2). pp. 47-86.

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We investigate the phenomenon of depth-reduction in commutative and non-commutative arithmetic circuits. We prove that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting. This also provides a unified proof of the circuit characterizations of the class LOGCFL and its counting variant LOGCFL. We show that AC(1) has no more power than arithmetic circuits of polynomial size and degree n(O(log log n)) (improving the trivial bound of n(O(log n))). Connections sire drawn between TC1 and arithmetic circuits of polynomial size and degree. Then we consider non-commutative computation. We show that over the algebra (Sigma*, max, concat), arithmetic circuits of polynomial size and polynomial degree can be reduced to O(log(2) n) depth land even to O(log n) depth if unbounded-fanin gates are allowed). This establishes that OptLOGCFL is in AC(1). This is the first depth-reduction result for arithmetic circuits over a non-commutative semiring, and it complements the lower bounds of Kosaraju and Nisan showing that depth reduction cannot be done in the general non-commutative setting. We define new notions called "short-left-paths" and "short-right-paths'' and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible. This class also can be characterized using the AuxPDA model.n Finally, we characterize the languages generated by efficient circuits over the semiring (2(Sigma*), union, concat) in terms of simple one-way machines, and we investigate and extend earlier lower bounds on non-commutative circuits.

Item Type: Journal Article
Publication: Theoretical Computer Science
Publisher: Elsevier Science
Additional Information: Copyright of this article belongs to Elsevier Science.
Keywords: arithmetic circuits;depth complexity;non-commutative computation;AuxPDAs;skew circuits.
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Date Deposited: 18 Dec 2009 07:37
Last Modified: 19 Sep 2010 05:26
URI: http://eprints.iisc.ac.in/id/eprint/19000

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