Volume 13, Issue 4

We define the operator of Boolean reducibility on the set of all infinite binary sequences. This operator is a variant of the operator of finite-automaton transformability when automata with several inputs and one state are considered. Each set Q of Boolean functions containing a selector function and closed with respect to the operation of superposition of a special form defines the Q -reducibility and Q -degrees, that is, the sets of Q -equivalent sequences. We study properties of the partially ordered set ℒ Q of all Q -degrees, namely, the existence of maximal, minimal and the greatest elements, infinite chains and antichains, and upper bounds.

In this paper, it is shown that under Turing coding of natural numbers primality of a number is tested by homogeneous structures in time asymptotically equal to the half of the length of the code.

We investigate variants of the notion of distinguishability of automata. The distinguishability in the sense of a given metric on the set of output symbols, the k -distinguishability and the ∞-distinguishability are considered. For each variant the exact value of the corresponding Shannon function is obtained.We find the minimum value of the parameter k for which the k -distinguishability implies the ∞-distinguishability.

We consider the problem on glueing states of an automaton which often arises in investigations of adaptive experiments.We give several estimates of time of glueing r states of an automaton with n states.

We prove the following theorem which characterises the class of groups without involutions which have an almost layer-finite periodic part: if the normaliser of any non-trivial finite subgroup of a Shunkov group without involutions has an almost layer-finite periodic part, then the group also has an almost layer-finite periodic part.

We consider a functional approach to a system of graphs with operations: we study the structure of classes of graphs closed with respect to some operations, the total number of closed classes, the number of precomplete classes and methods of generating the classes. We indicate the systems of operations with respect to which the sets of closed classes of graphs are continual, countable or finite. It is shown that systems of operations realise all possibilities of generation of closed classes of graphs: there exist classes with finite, countable bases, and classes which have no bases.

We obtain explicit formulas for the coefficients of asymptotic expansions in the domain of large deviations for the distributions of the number of cycles ν n in a random permutation of degree n , that is, for the probability P {ν n = N } under the condition that n , N → ∞ in such a way that 1 < α 0 ≤ α = n / N ≤ α 1 < ∞, where α 0 , α 1 are constants. These formulas express the coefficients in terms of cumulants of the random variable which has the distribution of the logarithmic series with specially chosen parameter. For the cumulants of the third and fourth orders we give the corresponding values. We discuss the accuracy of the obtained approximations. If n , N → ∞ so that 0 < γ 0 ≤ γ = N / ln n ≤ γ 1 < ∞, where γ 0 , γ 1 are constants, we give asymptotic estimates of the probabilities P {ν n = N }, P {ν n ≤ N }, P {ν n ≥ N } with the remainder terms of order O ((ln n ) -2 ) uniform in γ ∈ [γ 0 ,γ 1 ]. The corresponding estimate for the probability P {ν n = N } refines the previously known results for the case N = β ln n + o (ln n ), where β is a positive constant.