This paper presents a new type of neuron, called Boolean neuron. We suggest algorithms for decomposition of Boolean functions sets based on Boolean neural networks that include only Boolean neurons. The advantages of these neural networks consist in the reduction of memory space and computation time in comparison to the representation of Boolean functions by usual neural networks. The Boolean n...

We consider the problem of compression for “easy” Boolean functions: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2/n. We get both positive and negative result...

We study the volatility of the output of a Boolean function when the input bits undergo a natural dynamics. For n = 1, 2, . . ., let fn : {0, 1}mn → {0, 1} be a Boolean function and X(n)(t) = (X1(t), . . . , Xmn(t))t∈[0,∞) be a vector of i.i.d. stationary continuous time Markov chains on {0, 1} that jump from 0 to 1 with rate pn ∈ [0, 1] and from 1 to 0 with rate qn = 1−pn. Our object of study ...

This paper is a contribution to the study of a quasi-order on the set Ω of Boolean functions, the simple minor quasi-order. We look at the join-irreducible members of the resulting poset˜Ω. Using a two-way correspondence between Boolean functions and hypergraphs, join-irreducibility translates into a combinatorial property of hypergraphs. We observe that among Steiner systems, those which yield...

We investigate the distribution of performance of the Boolean functions of 3 Boolean inputs (particularly that of the parity functions), the always-on-6 and even-6 parity functions. We us enumeration, uniform Monte-Carlo random sampling and sampling random full trees. As expected XOR dramatically changes the fitness distributions. In all cases once some minimum size threshold has been exceeded,...