MPI-I-93-161

In this paper we prove several fundamental theorems, concerning the multi--party communication complexity of Boolean functions. Let $g$ be a real function which approximates Boolean function $f$ of $n$ variables with error less than $1/5$. Then --- from our Theorem 1 --- there exists a $k=O(\log (n\L_1(g)))$--party protocol which computes $f$ with a communication of $O(\log^3(n\L_1(g)))$ bits, where $\L_1(g)$ denotes the $\L_1$ spectral norm of $g$. We show an upper bound to the symmetric $k$--party communication complexity of Boolean functions in terms of their $\L_1$ norms in our Theorem 3. For $k=2$ it was known that the communication complexity of Boolean functions are closely related with the {\it rank} of their communication matrix [Ya1]. No analogous upper bound was known for the k--party communication complexity of {\it arbitrary} Boolean functions, where $k>2$. For a Boolean function of exponential $\L_1$ norm our protocols need $n^{\Omega(1)}$ bits of communication. However, if the {\it Fourier--coefficients} of a Boolean function $f$ are {\it unevenly} distributed, more exactly, if they can be divided into two groups: one with small $\L_1$ norm (say, $L$), and the other with small enough $\L_2$ norm (say, $\varepsilon$), then there exists a $O(\log(nL))$--party protocol which computes $f$ with $O(\log^3(Ln))$ communication on the $(1-\varepsilon^2)$ fraction of all inputs. In contrast, we prove that almost all Boolean functions of $n$ variables has a $k$--party communication complexity of at least ${n/k}-4\log n$. This result, along with our upper bounds, shows that for almost all Boolean function no real approximating function of small $\L_1$ norm can be found, or: almost all Boolean function has exponential $\L_1$ norm, or: for almost all Boolean function the distribution of the Fourier--coefficients is ``even'': they cannot be divided into two classes: one with small $\L_1$, the other with small $\L_2$ norms. Our results suggest that in the multi--party communication theory, instead of the well--studied {\it degree} of a polynomial representation of a Boolean function, its $\L_1$ norm can be an important measure of complexity.