Volume 12, Issue 3

For a property P of Boolean functions, a Boolean function f(x), x ∈ V n , and a subspace H of the space V n of all n-tuples of zeros and ones, we consider the set of all restrictions of the Boolean function f(x) onto the cosets of V n with respect to H. If the function f(x) itself and all its 2 n-dim H restrictions possess the property P, we say that the property P is inherited under the restrictions of the Boolean function f(x) and consider it as a new derived property. In this paper, this approach is applied to the following property of Boolean functions: the value f̂(α)/2 n , where f̂(α) is the Walsh-Hadamard coefficient, is fixed; the corresponding derived property is called the (H, α)-stability. We give convenient criteria for (H, α)-stability in terms of zeros of the Walsh-Hadamard coefficients, and establish relations between the (H, α)-stability, correlation immunity, and m-resiliency.

The class of bijunctive Boolean functions consists of the functions representable by the 2-CNF. The problem of enumeration of such function of arbitrary number of variables has not been solved. In the paper, we consider properties of inertia groups ofbijunctive functions in several groups and give an inductive method of generating all distinct representatives of the classes of geometric equivalence of bijunctive functions. By this method we calculate the numbers of bijunctive functions of 5, 6, and 7 variables.

The notion of a polarised polynomial form is extended to the case of multiple-valued logic functions. We introduce the Shannon functions of weight and length of polarised polynomial forms of multiple-valued logic functions and give some bounds for them.

We introduce a formalisation of the intuitive notion of almost periodicity of elements of a finite alphabet, the measure of approximate period of this sequence. We obtain a lower bound of the measure of approximate period of the sequence of states of an automaton for a given initial state and a given periodic input sequence. On the base of this estimate, we obtain a lower bound for the measures of approximate periods of output sequences of automata modelling the functioning of shift registers.

We suggest a mathematical model of information search with fuzzy logic. We give elementary properties of this model such as a criterion for admissibility of an information graph and a criterion for completeness of a basic set. We describe a method to go from a precise search problem to a fuzzy one and give conditions for a solution of the precise problem to be a solution of the corresponding fuzzy problem.

We consider the lattices of Ω-canonical and Ω-bicanonical formations and prove that all n-fold Ω-canonical and n-fold Ω-bicanonical formations are inductive and modular.

It is a well-known idea to embed the intuitionistic logic into the modal logic in order to interprete the modality 'provable' as the deducibility in the Peano arithmetics with also well-known difficulties. P. M. Solovay and A. V. Kuznetsov introduced into consideration the Gödel-Löb provability logic whose formulas are constructed from propositional variables with the use of the connectives &, v, ⊃, ¬, and Δ (Gödelised provability). This logic is defined by the classic propositional calculus complemented by the three Δ-axioms Δ (p ⊃ q) ⊃ (Δp ⊃ Δq), Δ(Δp ⊃ Δp) Δp, Δp ⊃ ΔΔp, and the extension rule (the Gödel rule). A formula F is called (functionally) expressible in terms of a system of formulas ∑ in logic L if, on the base of ∑ and variables, it is possible to obtain F with the use of the weakened substitution rule and the rule of change by equivalent in L. The general problem of expressibility in a logic L requires to give an algorithm which for any formula F and any finite system of formulas ∑ recognises whether F is expressible in terms of ∑ in L. In this paper, it is proved that in the Gödel-Löb provability logic and in many of extensions of this logic there is no algorithm which recognises the expressibility. In other words, the general expressibility problem is algorithmically undecidable in these logics.

We prove two theorems which give conditions for absence of a giant component in the generalised allocation scheme and present a series of examples.

We say that a digraph is 2-regular (dichotomous) if the out-degrees d 0 (j) and in-degrees d 1 (j) of any its vertex j ∈ V satisfy the equality d 0 (j) = d 1 (j) = 2. A graph Г is said to be primitive if for any pair i and j of its vertices in Г there exists a path from i to j of length m > 0. The least m is denoted y(Г) and called the exponent of Г. Let G(n, 2, p) stand for the class of strongly connected 2-regular graphs with n vertices of girth (the length of the shortest circuit) p, and let P(n, 2, p) denote the class of primitive 2-regular graphs of girth p with n vertices. The girth of a 2-regular graph with n vertices does not exceed ]n/2[, where ]x[ is the least integer no smaller than x. Earlier, the author proved that any primitive 2-regular graph with n vertices and with the maximal possible girth ]n/2[ had the exponent equal exactly to n - 1. In this paper we prove that for odd n ≥ 13 G(n, 2, (n- 1)/2) = P(n, 2, (n- 1)/2), any graph in G(n, 2, (n -1)/2) has a circuit of length (n + 1)/2, and for any Г ∈ G(n, 2, (n- 1)/2) the inequality is true.