The complexity of Boolean functions
The complexity of Boolean functions
Bent functions and random boolean formulas
Discrete Mathematics
Some typical properties of large AND/OR Boolean formulas
Random Structures & Algorithms
Coloring rules for finite trees, and probabilities of monadic second order sentences
Random Structures & Algorithms
Improved Boolean formulas for the Ramsey graphs
Random Structures & Algorithms
The number of Boolean functions computed by formulas of a given size
proceedings of the eighth international conference on Random structures and algorithms
Combinatorics, Probability and Computing
The Boolean functions computed by random Boolean formulas or how to grow the right function
Random Structures & Algorithms
On the properties of asymptotic probability for random boolean expression values in binary bases
SAGA'05 Proceedings of the Third international conference on StochasticAlgorithms: foundations and applications
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For any Boolean function f, let L(f) be its formula size complexity in the basis {∧, ⊕ 1}. For every n and every k≤n/2, we describe a probabilistic distribution on formulas in the basis {∧, ⊕ 1} in some given set of n variables and of size at most 𝓁(k)=4k. Let pn,k(f) be the probability that the formula chosen from the distribution computes the function f. For every function f with L(f)≤𝓁(k)α, where α=log4(3/2), we have pn,k(f)0. Moreover, for every function f, if pn,k(f)0, then***** Insert equation here *****where c1 is an absolute constant. Although the upper and lower bounds are exponentially small in 𝓁(k), they are quasi-polynomially related whenever 𝓁(k)≥lnΩ(1)n. The construction is a step towards developing a model appropriate for investigation of the properties of a typical (random) Boolean function of some given complexity.