Volume 33, Issue 6

The paper studies the relationships between local affinities of a Boolean function and some notions of its degeneracy (differential degeneracy, algebraic degeneracy, affine splittability). New relations connecting the parameters of local affinities and degeneracies are obtained. Relationships between some types of degeneracy of Boolean functions are found.

A generalized stochastic modification of the Colonel Blotto game, also known as the game of gladiators, is considered. In the original model, each of two players has a set of gladiators with given strengths. The battle of gladiator teams takes place through individual gladiator battles. In each fight, the probability of gladiator winning is proportional to its strength. Kaminsky et al. in 1984 had obtained a formula for the probability of winning in terms of weighted sums of exponential random variables. Here we provide an interpretation of this result from the Markov chains with continuous time point of view, and a more general statement of the problem is considered, for which a similar expression is obtained.

In the present survey, we consider all logical extensions of the parametric closure operator. Such extensions are obtained by augmenting the parametric closure language with arbitrary logical connectives or the universal quantifier. This gives, in addition to the parametric closure operator, the operators of positive and implicative closure, and also the closure operator with complete system of logical connectives, and the conjunctive-quantifier closure operator. We present basic facts on classifications of the sets P k generated by these operators.

We prove that any (almost any) n -place Boolean function can be implemented by a logic circuit in the “conjunction, disjunction, implication, negation” basis admitting a conditional complete diagnostic test of length at most n (at most n − 1, respectively) relative to constant faults of type 0 on outputs of gates.

The nonlinearity of vectorial functions and of their restrictions to manifolds are defined as the Hamming distance to the set of affine mappings and of their restrictions to the manifold, respectively. Relations between the parameters of the nonlinearity of a vectorial function and their analogues for its coordinate functions and its restrictions to manifolds are established. An analogue of the Parseval identity for such parameters of vectorial functions is proved, which implies the upper bound ( q k − 1) q n − k − q n /2− k for the nonlinearity of a mapping over a q -element field of n variables with k coordinates. Attainability conditions for this estimate are found, and a class of Boolean vectorial functions with high value of nonlinearity is constructed. Estimates characterizing the distribution of the nonlinearity of a vectorial function and of its restrictions to manifolds are obtained.