Volume 32, Issue 2

Compositions of n are finite sequences of positive integers (σ i)i=1k $\begin{array}{} (\sigma_i)_{i = 1}^k \end{array} $ such that σ 1+σ 2+⋯ +σ k=n. $$\begin{array}{} \sigma_1+\sigma_2+\cdots +\sigma_k = n. \end{array} $$ We represent a composition of n as a bargraph with area n such that the height of the i -th column of the bargraph equals the size of the i -th part of the composition. We consider the site-perimeter which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the total site-perimeter of compositions is obtained. In addition, we rederive the average site-perimeter of a composition by direct counting. Finally we determine the average site-perimeter of a bargraph with a given semi-perimeter.

Estimates for the cardinality of the set of correlation-immune n -ary Boolean functions with fixed weight are obtained.

We show that in multivalued logic there exist a continual family of pairwise incomparable closed sets with minimal logarithmic growth rate and a continual chain of nested closed sets with minimal logarithmic growth rate. As a corollary we prove that any subset-preserving class in multivalued logic contains a continual chain of nested closed sets and a continual family of pairwise incomparable closed sets such that none of the sets is a subset of any other precomplete class.

The enumeration closure operator (the Π -operator) is considered on the set P k of functions of the k -valued logic. It is proved that, for any k ⩾ 2, any positively precomplete class in P k is also Π -precomplete. It is also established that there are no other Π -precomplete classes in the three-valued logic.

We analyse closed classes in k -valued logics containing all linear functions modulo k . The classes are determined by divisors d of a number k and canonical formulas for functions. We construct the lattice of all such classes for k = p 2 , where p is a prime, and construct fragments of the lattice for other composite k .

A period of a Boolean function f ( x 1 , …, x n ) is a binary n -tuple a = ( a 1 , …, a n ) that satisfies the identity f ( x 1 + a 1 , …, x n + a n ) = f ( x 1 , …, x n ). A Boolean function is periodic if it admits a nonzero period. We propose an algorithm that takes the Zhegalkin polynomial of a Boolean function f ( x 1 , …, x n ) as the input and finds a basis of the space of all periods of f ( x 1 , …, x n ). The complexity of this algorithm is n O ( d ) , where d is the degree of the function f . As a corollary we show that a basis of the space of all periods of a Boolean function specified by the Zhegalkin polynomial of a bounded degree may be found with complexity which is polynomial in the number of variables.

Some results of classical statistical decision theory are generalized by means of the theory of fuzzy sets. The concepts of an admissible decision in the restricted sense, an admissible decision in the broad sense, a Bayes decision in the restricted sense, and a Bayes decision in the broad sense are introduced. It is proved that any Bayes decision in the broad sense with positive prior discrete density is admissible in the restricted sense. The class of Bayes decisions is shown to be complete under certain conditions on the loss function. Problems with a finite set of possible states are considered.