Properties of complexity measures for prams and wrams
Theoretical Computer Science
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
On the degree of Boolean functions as real polynomials
Computational Complexity - Special issue on circuit complexity
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Polynomial Degree vs. Quantum Query Complexity
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Learning functions of k relevant variables
Journal of Computer and System Sciences - Special issue: STOC 2003
Negative weights make adversaries stronger
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Probabilistic Boolean decision trees and the complexity of evaluating game trees
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Generic oracles and oracle classes
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
On rank vs. communication complexity
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
Simulating independence: New constructions of condensers, ramsey graphs, dispersers, and extractors
Journal of the ACM (JACM)
On linear and semidefinite programming relaxations for hypergraph matching
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Hi-index | 0.00 |
For Boolean functions f:{0,1}n - {0,1} and g:{0,1}m - {0,1}, the function composition of f and g denoted by f O g : {0,1}nm - {0,1} is the value of f on n inputs, each of them is the calculation of g on a distinct set of m Boolean variables. Motivated by previous works that achieved some of the best separations between complexity measures such as sensitivity, block-sensitivity, degree, certificate complexity and decision tree complexity we show that most of these complexity measures behave multiplicatively under composition. We use this multiplicative behavior to establish several applications. First, we give a negative answer for Adam Kalai's question from [MOS04]: "Is it true that every Boolean function f:{0,1}n - {0,1} with degree as a polynomial over the reals (denoted by deg(f)) at most n/3, has a restriction fixing 2n/3 of its variables under which f becomes a parity function?" This question was motivated by the problem of learning juntas as it suggests a simple algorithm, faster than that of Mossel et al. We give a counterexample for the question using composition of functions strongly related to the Walsh-Hadamard code. In fact, we show that for every constants ε,δ0 there are (infinitely many) Boolean functions f: {0,1}n - {0,1} such that deg(f) ≤ ε ⋅ n and under any restriction fixing less than (1-δ) ⋅ n variables, f does not become a parity function. Second, we show that for composition, the block sensitivity (denoted by bs) property has an unusual behavior - namely that bs(f O g) can be larger than bs (f) ⋅ bs(g). We show that the ratio between these two has a strong connection to the integrality gap of the Set Packing problem. In addition, we obtain the best known separation between block-sensitivity and certificate complexity (denoted by C) giving infinitely many functions f such that C(f) ≥ bs (f){log(26)/log(17) = bs (f)1.149.... Last, we present a factor 2 improvement of a result by Nisan and Szegedy [NS94], by showing that for all Boolean functions bs f ≤ deg(f)2.