28,688 research outputs found

    Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas

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    In this paper we study the identity testing problem of arithmetic read-once formulas (ROF) and some related models. A read-once formula is formula (a circuit whose underlying graph is a tree) in which the operations are {+,x} and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for read-once formulas, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of [Shpilka-Volkovich, 2015

    Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs

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    Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs). In this work, we give an exponential lower bound of exp(n/k^{O(k)}) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2^{~O(n^{1-1/2^{k-1}})} and needs white box access only to know the order in which the variables appear in the ABP

    Deterministic Black-Box Identity Testing π\pi-Ordered Algebraic Branching Programs

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    In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. Given a permutation π\pi of nn variables, for a π\pi-ordered ABP (π\pi-OABP), for any directed path pp from source to sink, a variable can appear at most once on pp, and the order in which variables appear on pp must respect π\pi. An ABP AA is said to be of read rr, if any variable appears at most rr times in AA. Our main result pertains to the identity testing problem. Over any field FF and in the black-box model, i.e. given only query access to the polynomial, we have the following result: read rr π\pi-OABP computable polynomials can be tested in \DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]. Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size Ω(2n/n)\Omega(2^n/n) and read Ω(2n/n2)\Omega(2^n/n^2). We give a multilinear polynomial pp in 2n+12n+1 variables over some specifically selected field GG, such that any OABP computing pp must read some variable at least 2n2^n times. We show that the elementary symmetric polynomial of degree rr in nn variables can be computed by a size O(rn)O(rn) read rr OABP, but not by a read (r−1)(r-1) OABP, for any 0<2r−1≤n0 < 2r-1 \leq n. Finally, we give an example of a polynomial pp and two variables orders π≠π′\pi \neq \pi', such that pp can be computed by a read-once π\pi-OABP, but where any π′\pi'-OABP computing pp must read some variable at least $2^n

    Progress on Polynomial Identity Testing - II

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    We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve
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