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Separation between read-once oblivious algebraic branching programs (ROABPs) and multilinear depth-three circuits

Kayal, N and Nair, V and Saha, C (2020) Separation between read-once oblivious algebraic branching programs (ROABPs) and multilinear depth-three circuits. In: ACM Transactions on Computation Theory, 12 (1).

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Official URL: https://dx.doi.org/10.1145/3369928


We showan exponential separation between twowell-studied models of algebraic computation, namely, readonce oblivious algebraic branching programs (ROABPs) and multilinear depth-three circuits. In particular, we show the following: (1) There exists an explicit n-variate polynomial computable by linear sized multilinear depth-three circuits (with only two product gates) such that every ROABP computing it requires 2O(n) size. (2) Any multilinear depth-three circuit computing IMMn,d (the iterated matrix multiplication polynomial formed by multiplying d, n � n symbolic matrices) has nO(d) size. IMMn,d can be easily computed by a poly(n,d) sized ROABP. (3) Further, the proof of (2) yields an exponential separation between multilinear depth-four and multilinear depth-three circuits: There is an explicit n-variate, degree d polynomial computable by a poly(n) sized multilinear depth-four circuit such that any multilinear depth-three circuit computing it has size nO(d) . This improves upon the quasi-polynomial separation of Reference 36 between these two models. The hard polynomial in (1) is constructed using a novel application of expander graphs in conjunction with the evaluation dimension measure 15, 33, 34, 36, while (2) is proved via a new adaptation of the dimension of the partial derivatives measure of Reference 32. Our lower bounds hold over any field. © 2020 Association for Computing Machinery.

Item Type: Journal Article
Publication: ACM Transactions on Computation Theory
Publisher: Association for Computing Machinery
Additional Information: The copyright of this article belongs to Association for Computing Machinery
Keywords: Computer circuits; Multiplying circuits; Polynomials; Separation; Timing circuits, Algebraic branching programs; Depth three circuits; Evaluation dimension; Expander graphs; MAtrix multiplication; Partial derivatives, Matrix algebra
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Date Deposited: 10 Aug 2020 12:30
Last Modified: 10 Aug 2020 12:30
URI: http://eprints.iisc.ac.in/id/eprint/64877

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