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

PDF
acm_tra_com_the_1201_2020.pdf  Published Version Download (1MB)  Preview 
Abstract
We showan exponential separation between twowellstudied models of algebraic computation, namely, readonce oblivious algebraic branching programs (ROABPs) and multilinear depththree circuits. In particular, we show the following: (1) There exists an explicit nvariate polynomial computable by linear sized multilinear depththree circuits (with only two product gates) such that every ROABP computing it requires 2O(n) size. (2) Any multilinear depththree circuit computing IMMn,d (the iterated matrix multiplication polynomial formed by multiplying d, n Ã� 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 depthfour and multilinear depththree circuits: There is an explicit nvariate, degree d polynomial computable by a poly(n) sized multilinear depthfour circuit such that any multilinear depththree circuit computing it has size nO(d) . This improves upon the quasipolynomial 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. Â© 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 
Actions (login required)
View Item 