On the Sensitivity Conjecture for Read-k Formulas

Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link". Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and functions describing graph properties. In this paper, we consider the conjecture for Boolean functions computable by read-k formulas. A read-k formula is a tree in which each variable appears at most k times among the leaves and has Boolean gates at its internal nodes. We show that the sensitivity conjecture holds for read-once formulas with gates computing symmetric functions. We next consider regular formulas with OR and AND gates. A formula is regular if it is a leveled tree with all gates at a given level having the same fan-in and computing the same function. We prove the sensitivity conjecture for constant depth regular read-k formulas for constant k

Using Elimination Theory to construct Rigid Matrices

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Omega(n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in this family are distinct primitive roots of unity of orders roughly exp(n^2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we use elimination theory to examine whether the rigidity function is semi-continuous.Comment: 25 Pages, minor typos correcte

On the Rigidity of Vandermonde Matrices

The rigidity function RA (r) of a matrix A is the minimum number of entries of A that must be changed to reduce the rank of A to less than or equal to r. While almost all matrices have rigidity close to (n r) 2 , proving strong lower bounds on the rigidity of explicit matrices is a fundamental open question with several consequences in complexity theory. A natural class of matrices expected to have high rigidity is that of Vandermonde matrices V = (x j 1 i ) 1i;jn . However, even when the x i are algebraically independent, it was not known if R V (r) = n 2 ) for nonconstant r. We prove that for any constant c < 1, there exists a constant > 0 such that if r p n, then R V (r) cn 2 , when the x i are algebraically independent. Although not explicit, this provides a natural n-dimensional manifold in the space of n n matrices with n 2 ) rigidity for nonconstant r. Our proof is based on a technique due to Shoup and Smolensky [11]. For explicit Vandermonde matrices..

Spectral Methods for Matrix Rigidity with Applications to Size-Depth Tradeoffs and Communication Complexity

The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have implications to complexity theory. We consider restricted variants of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. Two applications of such restricted variants are given. First, we show that our lower bound on a variant of rigidity implies lower bounds on size-depth tradeoffs for arithmetic circuits with bounded coefficients computing linear transformations. These bounds generalize a result of Nisan and Wigderson. The second application is conditional; we show that it would suffice to prove lower bounds on certain restricted forms of rigidity to conclude several separation results such as separating the analogs of PH and PSPACE in communication complexity theory. Our res..

Simultaneous Messages vs. Communication

. In the multiparty communication game introduced by Chandra, Furst, and Lipton [CFL] (1983), k players wish to evaluate collaboratively a function f(x0 ; : : : ; xk\Gamma1 ) for which player i sees all inputs except x i : The players have unlimited computational power. The objective is to minimize the amount of communication. We consider a restricted version of the multiparty communication game which we call the simultaneous messages model. The difference is that in this model, each of the k players simultaneously sends a message to a referee, who sees none of the input. The referee then announces the function value. We show demonstrate an exponential gap between the Simultaneous Messages and the Communication models for up to (log n) 1\Gammaffl players, for any ffl ? 0: The separation is obtained by comparing the respective complexities of the generalized addressing function, GAFn;k , in each model. This work is motivated by an approach suggested by the results of Hastad & Goldman..

Communication Complexity of Simultaneous Messages

In the multiparty communication game (CFL-game) of Chandra, Furst, and Lipton (Proc