A New Approach to Solving the Problem of Generating Sets of Complex Structural Objects Based on a Quasi-Equivalent Transformation of a Labeling Scheme
Authors: Tkachov Igor 
Published: April 2021

The paper presents the results of a theoretical study related to the development of methods for constructing generating structures based on labeling schemes for generating sets of complex structural objects. In a theoretical aspect, generated objects are mappings of sets of objects into a set of labels, and in practical terms, they can be, in particular, visual images. The scientific and practical interest in generative constructions is that they can be used to determine whether objects belong to a certain class, that is, to solve the problem of pattern recognition.

The problem of constructing generating labeling scheme belongs to a wide section of modern applied informatics that embraces Constraint Satisfaction Problem and related themes [1–4]. But this problem has not been posed before and there are still no regular methods for solving it.

The analysis of the above methods is based on the formalism of the consistent labeling problem [6, 10, 11], which is, on the one hand, a generalization of many statements of discrete problems of Constraint Satisfaction, and, on the other hand, a transparent theoretical construction with a well-developed mathematical foundation.

The problem of constructing a relational scheme (in this case, labeling scheme) that generates a given set of mappings, by analogy with linguistic models, may be named “the problem of grammar restoration” [12–14].

In previous studies it was shown that to solve this problem it makes sense to use equivalent transformations of the labeling scheme [11]. This is because the source table listing all the complex objects that should be generated by the target scheme is itself a trivial variant of the scheme with a given set of consistent labelings. This means that the source scheme and target scheme are equivalent. However, one of the equivalent operations – disunion of a column – cannot be used regularly, since it requires certain conditions to be met regarding the internal structure of the column.

In this case, to expand the capabilities of four known equivalent transformations of the labeling scheme – deleting and appending nonexistent labeling, as well as joining of columns and column disunion – a non-equivalent transformation was added – "coloring the column labelings".

The purpose of the paper is to introduce and investigate operation of "coloring the column labelings" that leads to a non-equivalent transformation of a labeling scheme. Show the advisability of using the known equivalent and the introduced quasi-equivalent transformations of the labeling scheme to solve the problem of constructing generating structures based on labeling schemes.

Results. The transformation of the labeling scheme, called "coloring the labelings of the scheme column", has been introduced. It is shown that its implementation leads to a quasi-equivalent labeling scheme, by solving which it is possible to uniquely restore the solution of the original problem. A method is proposed for using the newly introduced operation to transform the labeling scheme into a quasi-equivalent labeling scheme, in which it becomes possible to regularly perform the column decoupling operation. This ability of the operation of "coloring the column labelings" opens the way to the creation of a method for solving the problem of restoring a labeling scheme that generates a given set of consistent labelings.

Keywords: relational scheme, consistent labeling scheme, equivalent labeling scheme transformations, constraint satisfaction problem.