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Abstract:
We prove in this paper that it is much harder to evaluate depth--2,
size--$N$ circuits with MOD $m$ gates than with MOD $p$ gates by
$k$--party communication protocols: we show a $k$--party protocol
which communicates $O(1)$ bits to evaluate circuits with MOD $p$ gates,
while evaluating circuits with MOD $m$ gates needs $\Omega(N)$ bits,
where $p$ denotes a prime, and $m$ a composite, non-prime power number.
Let us note that using $k$--party protocols with $k\geq p$ is crucial
here, since there are depth--2, size--$N$ circuits with MOD $p$ gates
with $p>k$, whose $k$--party evaluation needs $\Omega(N)$ bits. As a
corollary, for all $m$, we show a function, computable with a depth--2
circuit with MOD $m$ gates, but not with any depth--2 circuit with MOD
$p$ gates.
It is easy to see that the $k$--party protocols are not weaker than the
$k'$--party protocols, for $k'>k$. Our results imply that if there is a
prime $p$ between $k$ and $k'$: $k<p\leq k'$, then there exists a
function which can be computed by a $k'$--party
protocol with a constant number of communicated bits, while any
$k$--party protocol needs linearly many bits of communication. This
result gives a hierarchy theorem for multi--party protocols.
