Volume 27, Issue 3

We consider average-case complexity of computing monotone Boolean functions by straight-line programs with a conditional stop over the basis of all Boolean functions of at most two variables. For the set of all n -ary monotone Boolean functions new Shannon-type upper and lower bounds for the average-case complexity as n → ∞ are established.

The paper gives a complete justification of the modular algorithm for reducing matrices to the Hermitian normal form, which enables one to construct a new modular algorithm for reducing to the Smith normal form that may simultaneously calculate the left matrix of the transformations. The main term in the estimate of the number of operations is 2( n 3 log D ), where n is the size and D is the determinant (or a multiple of it) of the matrix under consideration.

The paper is concerned with estimating the number ξ of ones in triangular arrays consisting of elements of the field GF (2) which are defined by the bottom row of s elements. The elements of each higher row are obtained (as in Pascal triangles) by the summation of pairs of elements from the corresponding lower row. It is shown that there exists a monotone unbounded sequence 0 = k 0 < k 1 < k 2 < ... of rational numbers such that, for any k > 0, for sufficiently large s the admissible values of ξ which are smaller than ks or larger than s ( s + 1)/3 − sk /3 are concentrated in neighbourhoods of points k i s and s ( s + 1)/3 − sk i /3, i ⩾ 0. The resulting estimates of the neighbourhoods are functions of i for each i ⩾ 0 and do not depend on s . The distributions of the numbers of triangles with values ξ in these neighbourhoods depend only on the residues of s with respect to moduli that depend on i ⩾ 0.

An improvement of the algebraic method of computing the Hadamard product of rational functions is given. Some applications of the Hadamard product to combinatorial and probabilistic problems are considered.

The paper is concerned with the subsets I ⊂ {0, ..., d – 1} for which gcd( n – m , d ) ≠ 1 for any n , m ∈ I . Such subsets are called sets of nontrivial differences. Let d > 1 and d 1 be the least prime divisor of d . We prove that the largest cardinality of a set of nontrivial differences is d / d 1 . Sets of nontrivial differences in which not all differences of elements are multiples of the same prime factor d are called nonelementary. Let t be the number of prime factors of d . We show that there are no nonelementary sets for t ⩽ 2. It is shown that a minimal nonelementary set may have arbitrary order in the interval 3,t¯ $\overline {3,\;t}$ . The largest cardinality of nonelementary sets is estimated from below and above.