Monotone bipartite graph properties are evasive
SIAM Journal on Computing
Learning decision trees using the Fourier spectrum
SIAM Journal on Computing
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
On the degree of Boolean functions as real polynomials
Computational Complexity - Special issue on circuit complexity
Some results related to the evasiveness conjectures
Journal of Combinatorial Theory Series B
Evasiveness of Subgraph Containment and Related Properties
SIAM Journal on Computing
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Every decision tree has an in.uential variable
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Probabilistic Boolean decision trees and the complexity of evaluating game trees
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
On the parity complexity measures of Boolean functions
Theoretical Computer Science
Gems in decision tree complexity revisited
ACM SIGACT News
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A function f : {0, 1}n - {0, 1} is called evasive if its decision tree complexity is maximal, i.e., D(f) = n. The long-standing Anderaa-Rosenberg-Karp (ARK) Conjecture asserts that every non-trivial monotone graph property is evasive. The Evasiveness Conjecture (EC) is a generalization of ARK Conjecture from monotone graph properties to arbitrary monotone transitive Boolean functions. In this paper we study a weakening of the Evasiveness Conjecture called Weak Evasivenss Conjecture (weak-EC). The weak-EC asserts that every non-trivial monotone transitive Boolean function must have D(f) ≥ n1- ε, for every ε 0. The purpose of this note is to make some remarks on weak-EC that hint towards a plausible attack on EC. First we observe that weak-EC is equivalent to EC. Further we observe that ruling out only certain simple (monotone-NC1) counter-examples to weak-EC suffices to confirm EC in its whole generality. Finally we rule out some simple counter-examples to weak-EC (AC0 : unconditionally; and monotone-TC0 : under a conjecture of Benjamini, Kalai, and Schramm on their noise stability). We also investigate an analogue of weak-EC for the stronger model of parity decision trees and provide a counter-example to this seemingly stronger version under a conjecture of Montanaro and Osborne.