Boolean Circuits, Tensor Ranks, and Communication Complexity

We investigate two methods for proving lower bounds on the size of small-depth circuits, namely the approaches based on multiparty communication games and algebraic characterizations extending the concepts of the tensor rank and rigidity of matrices. Our methods are combinatorial, but we think that our main contribution concerns the algebraic concepts used in this area (tensor ranks and rigidity). Our main results are following. (i) An $o(n)$-bit protocol for a communication game for computing shifts, which also gives an upper bound of $o(n^2)$ on the contact rank of the tensor of multiplication of polynomials; this disproves some earlier conjectures. A related probabilistic construction gives an $o(n)$ upper bound for computing all permutations and an $O(n\log\log n)$ upper bound on the communication complexity of pointer jumping with permutations. (ii) A lower bound on certain restricted circuits of depth 2 which are related to the problem of proving a superlinear lower bound on the size of logarithmic-depth circuits; this bound has interpretations both as a lower bound on the rigidity of the tensor of multiplication of polynomials and as a lower bound on the communication needed to compute the shift function in a restricted model. (iii) An upper bound on Boolean circuits of depth 2 for computing shifts and, more generally, all permutations; this shows that such circuits are more efficient than the model based on sending bits along vertex-disjoint paths.