Separation Between Read-once Oblivious Algebraic Branching Programs (ROABPs) and Multilinear Depth Three Circuits
نویسندگان
چکیده
We show an exponential separation between two well-studied models of algebraic computation, namely read-once 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 2Ω(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 nΩ(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, d) sized multilinear depth four circuit such that any multilinear depth three circuit computing it has size nΩ(d). This improves upon the quasi-polynomial separation result by Raz and Yehudayoff [2009] 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 used previously in Nisan [1991], Raz [2006,2009], Raz and Yehudayoff [2009], and Forbes and Shpilka [2013], while 2 is proved via a new adaptation of the dimension of the partial derivatives measure used by Nisan and Wigderson [1997]. Our lower bounds hold over any field. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes
منابع مشابه
Polynomial Identity Testing and Lower Bounds for Sum of Special Arithmetic Branching Programs
A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. In this chapter, we give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity nO(logn). In both the case...
متن کاملPseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order
We give deterministic black-box polynomial identity testing algorithms for multilinear read-once oblivious algebraic branching programs (ROABPs), in nO(lg 2 n) time.1 Further, our algorithm is oblivious to the order of the variables. This is the first sub-exponential time algorithm for this model. Furthermore, our result has no known analogue in the model of read-once oblivious boolean branchin...
متن کاملQuasi-polynomial Hitting Sets for Circuits with Restricted Parse Trees
We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [LMP16] and Lagarde, Limaye and Srinivasan [LLS17]) and give the following constructions: • An explicit hitting set of quasipolynomial size for UPT circuits, • An explicit hitting set...
متن کاملSome Lower Bound Results for Set-Multilinear Arithmetic Computations
In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of showing lower bound results. We define some natural restrictions of these models for which we are able to show lower bound results. Some of our results extend existing lower bounds, while others are new and raise open questions. Specifically, our main results are t...
متن کاملDeterministic Identity Testing for Sum of Read Once ABPs
A read once ABP is an arithmetic branching program with each variable occurring in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity, i.e. n. The motivating special case of this model is sum of constantly many set-multiline...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electronic Colloquium on Computational Complexity (ECCC)
دوره 22 شماره
صفحات -
تاریخ انتشار 2015