Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
The average sensitivity of bounded-depth circuits
Information Processing Letters
An improved exponential-time algorithm for k-SAT
Journal of the ACM (JACM)
Some topics in analysis of boolean functions
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Variable Influences in Conjunctive Normal Forms
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
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The average sensitivity of a Boolean function is the expectation, given a uniformly random input, of the number of input bits which when flipped change the output of the function. A k-CNF is a CNF in which every clause contains at most k literals. It has recently been shown by the author Amano (2011) [1] that the average sensitivity of a k-CNF is at most k. This bound is tight since the parity function on k variables has the average sensitivity k. In this paper, we consider the problem to determine the extremal formulas achieving this bound. We give a class of such formulas that contains a double exponential (in k) number of non-isomorphic ones. This class captures all formulas, with only one exception, that we have obtained so far. We also give the complete list for k=2 and 3 as well as several structural properties of such extremal formulas.