Volume 30, Issue 5

We consider exact attribute-efficient learning of functions from Post closed classes using membership queries and obtain bounds on learning complexity.

We consider Boolean circuits in Zhegalkin basis and describe all Boolean functions that can be implemented by a circuit admitting a complete fault detection test of length 1 in case of constant faults of type “1” at gate outputs.

We use an analogue of Gluskin–Hosszú theorem for strongly dependent finite n -ary semigroups to prove an analogue of Post coset theorem. We show that in fact these theorems are equivalent. Thus it turns our that the case of n -ary semigroups is fully similar to the case of n -groups.

An easily implementable recursive parallelization strategy for solving the subset sum problem by the branch-and-bound method is proposed. Two different frontal and balanced variants of this strategy are compared. On an example of a particular case of the subset sum problem we show that the balanced variant is more effective than the frontal one. Moreover, we show that, for the considered particular case of the subset sum problem, the balanced variant is also time optimal.

Formulas for distributions of number of ones (non-zeroes) in the cycle of the output sequence of generalized binary Pohl generator are obtained. Limit theorems for these distributions are derived in the case when the lengths of registers are coprime and tend to infinity, the contents of different registers are independent, but cell contents within each register may be dependent. The consequences of these theorems are given for the case when the contents of cells are independent random variables having equiprobable distribution on {0, 1}.

Asymptotic upper and lower bounds for the numbers of distinct subsets A + B in Abelian group of order n are derived, where | A |, | B | ≥ n (log n ) −1/8 .

We consider a generalized scheme of allocation of n particles (elements) over unordered cells (components) under the condition that the number of particles in each cell belongs to a fixed finite set A of positive integers. A new asymptotic estimates for the total number I n ( A ) of variants of allocations of n particles are obtained under some conditions on the set A ; these estimates have an explicit form (up to equivalence). Some examples of combinatorial-probabilistic character are given to illustrate by particular cases the notions introduced and results obtained. For previously known theorems on the convergence to the normal law of the total number of components and numbers of components with given cardinalities the norming parameters are obtained in the explicit form without using roots of algebraic or transcendent equations.

We consider universal function’s construction for classes of sums of two arguments modulo 2. We constructed functions with optimal domain cardinality O (log n ).