Using multivalued logic for qualitative data analysis

The paper considers a logical approach to data analysis to solve the classification problem. The studied data is a set of objects and their features. As a rule, this is scattered heterogeneous information and it is not enough for a reasonable application of probabilistic models. Therefore, logical algorithms are considered, which under certain conditions may be more adequate. For an expressive formal representation of the relationship between objects and their attributes, multivalued logic is used, and the number of values depends on a specific attribute. Therefore, a system of operations is proposed for variables with different domains of definition. As a result, a decisive function is built, which is a classifier of objects that are present in the studied data. The properties and capabilities of this function are analyzed. It is shown that the logical function, which is a conjunction in the space of rules connecting given objects with their characteristic features, uniquely characterizes the initial data, divides the subject area into classes, possesses modifiability properties, meets the requirements of completeness and consistency in a given area. The paper also proposes an algorithm for its implementation.

On the action of the implicative closure operator on the set of partial functions of the multivalued logic

On the set P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ of partial functions of the k-valued logic, we consider the implicative closure operator, which is the extension of the parametric closure operator via the logical implication. It is proved that, for any k ⩾ 2, the number of implicative closed classes in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ is finite. For any k ⩾ 2, in P k ∗ $\begin{array}{} \displaystyle P_k^* \end{array}$ two series of implicative closed classes are defined. We show that these two series exhaust all implicative precomplete classes. We also identify all 8 atoms of the lattice of implicative closed classes in P 3 ∗ $\begin{array}{} \displaystyle P_3^* \end{array}$ .

Cryptographic constructions on the basis of functions of multivalued logic

Symmetric encryption algorithms have been successfully used to protect information during transmission on an open channel. The classical approach to the synthesis of modern cryptographic algorithms and cryptographic primitives on which they are based, is the use of mathematical apparatus of Boolean functions. The authors demonstrate that the use to solve this problem of functions of multivalued logic (FML) allows to largely improve the durability of the cryptographic algorithms and to extend the used algebraic structures. On the other hand, the study of functions of multivalued logic in cryptography leads to a better understanding of the principles of cryptographic primitives and the emergence of new methods of describing cryptographic constructions. In the monograph the results of theoretical and experimental studies of the properties of the FML, the presented algorithms for generating high-quality S-blocks for the symmetric encryption algorithms, as well as full-working samples of the cryptographic algorithms ready for practical implementation. For students and teachers and all those interested in issues of information security.