4 research outputs found
On the size of homogeneous and of depth four formulas with low individual degree
International audienceLet r ≥ 1 be an integer. Let us call a polynomial f (x_1,...,x_N) ∈ F[x] as a multi-r-ic polynomial if the degree of f with respect to any variable is at most r (this generalizes the notion of multilinear polynomials). We investigate arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. We prove lower bounds for several subclasses of such circuits. Specifically, first define the formal degree of a node α with respect to a variable x_i inductively as follows. For a leaf α it is 1 if α is labelled with x_i and zero otherwise; for an internal node α labelled with × (respectively +) it is the sum of (respectively the maximum of) the formal degrees of the children with respect to x_i. We call an arithmetic circuit as a multi-r-ic circuit if the formal degree of the output node with respect to any variable is at most r. We prove lower bounds for various subclasses of multi-r-ic circuits
Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity
We say that a circuit over a field functionally computes an
-variate polynomial if for every we have that . This is in contrast to syntactically computing , when as
formal polynomials. In this paper, we study the question of proving lower
bounds for homogeneous depth- and depth- arithmetic circuits for
functional computation. We prove the following results :
1. Exponential lower bounds homogeneous depth- arithmetic circuits for a
polynomial in .
2. Exponential lower bounds for homogeneous depth- arithmetic circuits
with bounded individual degree for a polynomial in .
Our main motivation for this line of research comes from our observation that
strong enough functional lower bounds for even very special depth-
arithmetic circuits for the Permanent imply a separation between and
. Thus, improving the second result to get rid of the bounded individual
degree condition could lead to substantial progress in boolean circuit
complexity. Besides, it is known from a recent result of Kumar and Saptharishi
[KS15] that over constant sized finite fields, strong enough average case
functional lower bounds for homogeneous depth- circuits imply
superpolynomial lower bounds for homogeneous depth- circuits.
Our proofs are based on a family of new complexity measures called shifted
evaluation dimension, and might be of independent interest
On the Size of Homogeneous and of Depth-Four Formulas with Low Individual Degree
International audienceLet r ≥ 1 be an integer. Let us call a polynomial f (x 1 , x 2 ,. .. , x N) ∈ F[x] a multi-r-ic polynomial if the degree of f with respect to any variable is at most r. (This generalizes the notion of multilinear polynomials.) We investigate the arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. We prove lower bounds for several subclasses of such circuits, including the following. 1. An N Ω(log N) lower bound against homogeneous multi-r-ic formulas (for an explicit multi-r-ic polynomial on N variables). 2. An (n/r 1.1) Ω √ d/r lower bound against depth-four multi-r-ic circuits computing the polynomial IMM n,d corresponding to the product of d matrices of size n × n each. √ N) lower bound against depth-four multi-r-ic circuits computing an explicit multi-r-ic polynomial on N variables