Revision #1 Authors: Oded Goldreich, Avi Wigderson

Accepted on: 25th February 2014 09:57

Downloads: 1813

Keywords:

We propose that multi-linear functions of relatively low degree

over GF(2) may be good candidates for obtaining exponential

lower bounds on the size of constant-depth Boolean circuits

(computing explicit functions).

Specifically, we propose to move gradually from linear functions

to multilinear ones, and conjecture that, for any $t\geq2$,

some explicit $t$-linear functions $F:(\{0,1\}^n)^t\to\{0,1\}$

require depth-three circuits of size $\exp(\Omega(tn^{t/(t+1)}))$.

Towards studying this conjecture,

we suggest to study two frameworks for the design of

depth-three Boolean circuits computing multilinear functions,

yielding restricted models for which lower bounds may be easier to prove.

Both correspond to constructing a circuit by expressing

the target polynomial as a composition of simpler polynomials.

The first framework corresponds to a direct composition,

whereas the second (and stronger) framework corresponds

to nested composition and yields depth-three Boolean circuits

via a "guess-and-verify" paradigm in the style of Valiant.

The corresponding restricted models of circuits are called D-canonical

and ND-canonical, respectively.

Our main results are

(1) a generic upper bound on the size of depth-three D-canonical

circuits for computing any $t$-linear function, and

(2) a lower bound on the size of any depth-three ND-canonical circuits

for computing some (in fact, almost all) $t$-linear functions.

These bounds match the foregoing conjecture

(i.e., they have the form of $\exp(tn^{t/(t+1)})$).

Another important result is separating the two models:

We prove that ND-canonical circuits can be super-polynomially

smaller than their D-canonical counterparts.

We also reduce proving lower bounds for the ND-model

to Valiant's matrix rigidity problem

(for parameters that were not the focus of previous works).

The study of the foregoing (Boolean) models calls for

an understanding of new types of arithmetic circuits,

which we define in this paper and may be of independent interest.

These circuits compute multilinear polynomials by

using arbitrary multilinear gates of some limited arity.

It turns out that a GF(2)-polynomial is computable by such circuits

with at most $s$ gates of arity at most $s$ if and only if

it can be computed by ND-canonical circuits of size $\exp(s)$.

A similar characterization holds for D-canonical circuits

if we further restrict the arithmetic circuits to have depth two.

We note that the new arithmetic model makes sense over any field, and

indeed all our results carry through to all fields. Moreover, it raises

natural arithmetic complexity problems which are independent of our

original motivation.

NOTE: Throughout this paper, when we say that a function $f$

is exponential, we mean that $f(n)=\exp(\Theta(n))$.

In addition to correcting a few typos, this version

contains two new subsections:

In Section 4.3, we introduce a relaxed notion rigidity,

which we call structured rigidity.

We observe that structured rigidity may replace standard

rigidity in our lower bound reductions, whereas the two

notions can be strictly separated.

In Section 5.1, we study arithmetic circuits that compute functions

without relying on cancellations. We show that such circuits are weaker

than the general arithmetic circuits considered in the bulk of the paper.

(The original Section 5 was moved to Section 5.2.)

TR13-043 Authors: Oded Goldreich, Avi Wigderson

Publication: 25th March 2013 11:22

Downloads: 2046

Keywords:

We propose that multi-linear functions of relatively low degree

over GF(2) may be good candidates for obtaining exponential

lower bounds on the size of constant-depth Boolean circuits

(computing explicit functions).

Specifically, we propose to move gradually from linear functions

to multilinear ones, and conjecture that, for any $t\geq2$,

some explicit $t$-linear functions $F:(\{0,1\}^n)^t\to\{0,1\}$

require depth-three circuits of size $\exp(\Omega(tn^{t/(t+1)}))$.

Towards studying this conjecture,

we suggest to study two frameworks for the design of

depth-three Boolean circuits computing multilinear functions,

yielding restricted models for which lower bounds may be easier to prove.

Both correspond to constructing a circuit by expressing

the target polynomial as a composition of simpler polynomials.

The first framework corresponds to a direct composition,

whereas the second (and stronger) framework corresponds

to nested composition and yields depth-three Boolean circuits

via a "guess-and-verify" paradigm in the style of Valiant.

The corresponding restricted models of circuits are called D-canonical

and ND-canonical, respectively.

Our main results are

(1) a generic upper bound on the size of depth-three D-canonical

circuits for computing any $t$-linear function, and

(2) a lower bound on the size of any depth-three ND-canonical circuits

for computing some (in fact, almost all) $t$-linear functions.

These bounds match the foregoing conjecture

(i.e., they have the form of $\exp(tn^{t/(t+1)})$).

Another important result is separating the two models:

We prove that ND-canonical circuits can be super-polynomially

smaller than their D-canonical counterparts.

We also reduce proving lower bounds for the ND-model

to Valiant's matrix rigidity problem

(for parameters that were not the focus of previous works).

The study of the foregoing (Boolean) models calls for

an understanding of new types of arithmetic circuits,

which we define in this paper and may be of independent interest.

These circuits compute multilinear polynomials by

using arbitrary multilinear gates of some limited arity.

It turns out that a GF(2)-polynomial is computable by such circuits

with at most $s$ gates of arity at most $s$ if and only if

it can be computed by ND-canonical circuits of size $\exp(s)$.

A similar characterization holds for D-canonical circuits

if we further restrict the arithmetic circuits to have depth two.

We note that the new arithmetic model makes sense over any field, and

indeed all our results carry through to all fields. Moreover, it raises

natural arithmetic complexity problems which are independent of our

original motivation.

NOTE: Throughout this paper, when we say that a function $f$

is exponential, we mean that $f(n)=\exp(\Theta(n))$.