28,688 research outputs found
Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas
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
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 -Ordered Algebraic Branching Programs
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 of variables, for a -ordered ABP (-OABP), for
any directed path from source to sink, a variable can appear at most once
on , and the order in which variables appear on must respect . An
ABP is said to be of read , if any variable appears at most times in
. Our main result pertains to the identity testing problem. Over any field
and in the black-box model, i.e. given only query access to the polynomial,
we have the following result: read -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
and read . We give a multilinear polynomial
in variables over some specifically selected field , such that
any OABP computing must read some variable at least times. We show
that the elementary symmetric polynomial of degree in variables can be
computed by a size read OABP, but not by a read OABP, for
any . Finally, we give an example of a polynomial and two
variables orders , such that can be computed by a read-once
-OABP, but where any -OABP computing must read some variable at
least $2^n
Progress on Polynomial Identity Testing - II
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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