We consider a Hamiltonian decomposition problem of partitioning a regular graph into edge-disjoint Hamiltonian cycles. It is known that verifying vertex non-adjacency in the 1-skeleton of the symmetric and asymmetric traveling salesperson polytopes is an NP-complete problem. On the other hand, a suffcient condition for two vertices to be non-adjacent can be formulated as a combinatorial problem of finding a Hamiltonian decomposition of a 4-regular multigraph. We present two backtracking algorithms for verifying vertex non-adjacency in the 1-skeleton of the traveling salesperson polytope and constructing a Hamiltonian decomposition: an algorithm based on a simple path extension and an algorithm based on the chain edge fixing procedure. Based on the results of the computational experiments for undirected multigraphs, both backtracking algorithms lost to the known heuristic general variable neighborhood search algorithm. However, for directed multigraphs, the algorithm based on chain fixing of edges showed comparable results with heuristics on instances with existing solutions, and better results on instances of the problem where the Hamiltonian decomposition does not exist.

In this paper, we study undirected multiple graphs of any natural multiplicity k > 1. There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of k linked edges, which connect 2 or (k + 1) vertices correspondingly. The linked edges should be used simultaneously. If a vertex is incident to a multiple edge, it can be also incident to other multiple edges and it can be the common end of k linked edges of some multi-edge. If a vertex is the common end of some multi-edge, it cannot be the common end of another multi-edge. A multiple tree is a connected multiple graph with no cycles. Unlike ordinary trees, the number of edges in a multiple tree is not fixed. The problem of finding the spanning tree can be set for a multiple graph. Complete spanning trees form a special class of spanning trees of a multiple graph. Their peculiarity is that a multiple path joining any two selected vertices exists in the tree if and only if such a path exists in the initial graph. If the multiple graph is weighted, the minimum spanning tree problem and the minimum complete spanning tree problem can be set. Also we can formulate the problems of recognition of the spanning tree and complete spanning tree of the limited weight. The main result of this article is the proof of NPcompleteness of such recognition problems for arbitrary multiple graphs as well as for divisible multiple graphs in the case when multiplicity k ≥ 3. The corresponding optimization problems are NP-hard.

Obfuscation is used to protect programs from analysis and reverse engineering. There are theoretically effective and resistant obfuscation methods, but most of them are not implemented in practice yet. The main reasons are large overhead for the execution of obfuscated code and the limitation of application only to a specific class of programs. On the other hand, a large number of obfuscation methods have been developed that are applied in practice. The existing approaches to the assessment of such obfuscation methods are based mainly on the static characteristics of programs. Therefore, the comprehensive (taking into account the dynamic characteristics of programs) justification of their effectiveness and resistance is a relevant task. It seems that such a justification can be made using machine learning methods, based on feature vectors that describe both static and dynamic characteristics of programs. In this paper, it is proposed to build such a vector on the basis of characteristics of two compared programs: the original and obfuscated, original and deobfuscated, obfuscated and deobfuscated. In order to obtain the dynamic characteristics of the program, a scheme based on a symbolic execution is constructed and presented in this paper. The choice of the symbolic execution is justified by the fact that such characteristics can describe the difficulty of comprehension of the program in dynamic analysis. The paper proposes two implementations of the scheme: extended and simplified. The extended scheme is closer to the process of analyzing a program by an analyst, since it includes the steps of disassembly and translation into intermediate code, while in the simplified scheme these steps are excluded. In order to identify the characteristics of symbolic execution that are suitable for assessing the effectiveness and resistance of obfuscation based on machine learning methods, experiments with the developed schemes were carried out. Based on the obtained results, a set of suitable characteristics is determined.

Self-adaptation of complex systems is a very active domain of research with numerous application domains. Component systems are designed as sets of components that may reconfigure themselves according to adaptation policies, which describe needs for reconfiguration. In this context, an adaptation policy is designed as a set of rules that indicate, for a given set of configurations, which reconfiguration operations can be triggered, with fuzzy values representing their utility. The adaptation policy has to be faithfully implemented by the system, especially w.r.t. the utility occurring in the rules, which are generally specified for optimizing some extra-functional properties (e.g. minimizing resource consumption). In order to validate adaptive systems’ behaviour, this paper presents a model-based testing approach, which aims to generate large test suites in order to measure the occurrences of reconfigurations and compare them to their utility values specified in the adaptation rules. This process is based on a usage model of the system used to stimulate the system and provoke reconfigurations. As the system may reconfigure dynamically, this online test generator observes the system responses and evolution in order to decide the next appropriate test step to perform. As a result, the relative frequencies of the reconfigurations can be measured in order to determine whether the adaptation policy is faithfully implemented. To illustrate the approach the paper reports on experiments on the case study of platoons of autonomous vehicles.

To ensure traffic safety of railway transport, non-destructive tests of rails are regularly carried out by using various approaches and methods, including eddy-current flaw detection methods. An automatic analysis of large data sets (defectograms) that come from the corresponding equipment is an actual problem. The analysis means a process of determining the presence of defective sections along with identifying structural elements of railway tracks in defectograms. This article continues the cycle of works devoted to the problem of automatic recognizing images of defects and structural elements of rails in eddy-current defectograms. In the process of forming these images, only useful signals are taken into account, the threshold levels of amplitudes of which are determined automatically from eddy-current data. The previously used algorithm for finding threshold levels was focused on situations in which the vast majority of signals coming from the flaw detector is a rail noise. A signal is considered useful and is subject to further analysis if its amplitude is twice the corresponding noise threshold. The article is devoted to the problem of correcting threshold levels, taking into account the need to identify extensive surface defects of rails. An algorithm is proposed for finding the values of threshold levels of rail noise amplitudes with their subsequent correction in the case of a large number of useful signals from extensive defects. Examples of the algorithm’s operation on real eddy-current data are given.

The issues of building an automated learning system “Sets” which will allow students to master one of the important topics of the discipline “Discrete Mathematics” and to develop logical and mathematical thinking in this direction are studied. The corresponding topic of the 1st part of the project includes materials related to the concept of a set, operations on sets, algebra of sets, proofs of statements for sets, and the derivation of formulas for the number of set elements. The system is based on a construction of the statements proof editor for a set and of the formulas derivation editor for the number of set elements, both editors are to be used for teaching. The first of these allows students to split the original statement into a number of simpler statements, taken together equivalent to the original statement, to choose a method of proving each simple statement and to conduct their step-by-step proof. The second editor allows (using the inclusion-exclusion principle and the formula of the number of complement elements) to derive a step-by-step formula for the number of set elements through the specified numbers of elements for sets from which the resulting set is constructed. An important part of the system is to monitor the correctness of all actions of students, and on this basis the entire learning system is developed. The logical supervision over the correctness of the selected action in the first editor is performed by a Boolean function created by the system and corresponding to this action and by checking it for identical truth. In the second editor, invariants such as characteristic strings of the set and of its number of elements are used for verification. The rest of the system is related to learning of set algebra and to preparation to editors usage. The main focus here is on the learning strategy in which testing the understanding of the learned material is rather rigorous and eliminating the random choice of answers. The division of the material into sections with verification of the success of teaching not only by tests, but also by exercises and tasks, allows students to master the complex logical and mathematical techniques of proving statements for sets and derivation of formulas for the number of set elements.

The article is written in support of the educational discipline “Non-classical logics”. Within the framework of this discipline, the objects of study are the basic principles and constructive elements, with the help of which the formal construction of various non-classical propositional logics takes place. Despite the abstractness of the theory of non-classical logics, in which the main attention is paid to the strict mathematical formalization of logical reasoning, there are real practical areas of application of theoretical results. In particular, languages of temporal modal logics are widely used for modeling, specification, and verification (correctness analysis) of logic control program systems. This article demonstrates, using the linear temporal logic LTL as an example, how abstract concepts of non-classical logics can be reƒected in practice in the field of information technology and programming. We show the possibility of representing the behavior of a software system in the form of a set of LTL-formulas and using this representation to verify the satisfiability of program system properties through the procedure of proving the validity of logical inferences, expressed in terms of the linear temporal logic LTL. As program systems, for the specification of the behavior of which the LTL logic will be applied, Minsky counter machines are considered. Minsky counter machines are one of the ways to formalize the intuitive concept of an algorithm. They have the same computing power as Turing machines. A counter machine has the form of a computer program written in a high-level language, since it contains variables called counters, and conditional and unconditional jump operators that allow to build loop constructions. It is known that any algorithm (hypothetically) can be implemented in the form of a Minsky three-counter machine.