System Informatics, 2025, # 27

The article is devoted to the author's results of the analysis of the genetically inherited ability of living bodies to intellectual activity (for example, the ability to echolocate in dolphins), which resulted in the emergence of algebraic formalisms called tensor-unitary transformations. Genetic intelligence is understood as that part of the intellectual potential of living organisms that allows, on the basis of genetic information in DNA and RNA molecules, to build, for example, from one fertilized cell an organism with trillions of cells so that the parental traits are reproduced in it in a multichannel noise-resistant manner, despite strong noise and constantly changing conditions of nutrition and external influences during life. In this case, we are talking about the systematic growth in the course of ontogenesis of the number of parameters and degrees of freedom of the body with a corresponding increase in the dimensionality of its configuration space of states. With such growth, the organism at successive stages of its development, acquiring new degrees of freedom and knowledge, somehow retains the memory of the skills and knowledge that it possessed at previous stages of life. The author develops the algebraic foundations for modeling this fundamental feature of the development of living bodies in the tensor-matrix language of systems of multidimensional vector configuration spaces. Tensor-unitary transformations are operators that preserve the lengths of vectors during their tensor transformation into vectors of a space of increased dimension (in contrast to conventional unitary transformations that transform vectors into a space of the same dimension). They are operators of expansion of stochastic-deterministic memory with preservation of all previous memory. Possible applications of tensor-unitary transformations for the development of AI, genetic algorithms, etc. are discussed.

The theory of categorical systems developed by the author allows to naturally model traditional artificial neural networks of arbitrary topology, networks of living neurons, which in addition to spike communication have several dozen other types of cellular communication, as well as network structures similar to higher categories. The mathematical apparatus of categorical systems is the theory of categorical splices, this work is devoted to the following issues of this theory. Directed binary categorical splices are introduced and studied, which are a generalization of ordinary categories, in the theory of which, as is known, the concept of duality of categories and the principle of duality based on duality play a large role. In the theory of categorical splices, in addition to duality similar to duality generated by replacing the direction of category arrows, there is a new type of duality associated with replacing the names of arrows with the names of convolutions generalizing the usual operation of composition. The construction of dual in both senses categorical splices of the studied type is carried out, theorems corresponding to the principles of duality for the specified two types of duality are proved. The theorems are given within the framework of the theory of proofs, for the special case of ordinary categories giving new proofs of the principle of duality. Generalization of the approach to convolutional analogs of multicategories find applications in neural networks, in particular, for the well-known formulas of S. Osovsky in the method of backpropagation of errors.

The paper studies the approach of LLM adaptation for queries in the mathematical subject area. The subject area is presented as an ontology of a semantic library LibMeta, where data navigation is carried out using KG MathSemanticLib. The descriptions of the mathematical subject area are based on mathematical encyclopedias of the Soviet and Russian mathematical schools, and the filling of the LibMeta subject area library is carried out by integrating subject areas of specialized mathematical journals. A procedure for integrating LLM and KG MathSemanticLib is proposed. It is shown that as a result of this approach, LLM does not go beyond the subject area, which allows us to state a more relevant answer to the query.

This paper describes a method for constructing an instruction dataset for fine-tuning a large language model (LLM) to serve as a generator within a retrieval-augmented generation (RAG) pipeline. The practical implementation of this method is demonstrated through the construction of a dataset tailored for fine-tuning the generator of a corporate intelligent assistant based on the RAG architecture.

The paper explores the problem of building trustworthy artificial intelligence based on large language models and p-computable checkers. For this purpose we present a concept of framework for reliable verification of answers obtained by large language models (LLMs). We focus on the application of this framework to digital twin systems, particularly for smart cities, where LLMs are not yet widely used due to their resource intensity and potential for hallucination. Taking into account the fact that solution verification from a suitable set of tasks is p-computable and in most cases less complex than computing and implementing the whole task, we present a methodology that uses checkers to assess the validity of LLM-generated solutions. These checkers are implemented within the methodology of polynomial-time programming in Turing-complete languages, and guarantee a polynomial-time complexity. Our system was tested on the 2-SAT problem. This framework offers a scalable way to implement trustworthy AI systems with guaranteed polynomial complexity, ensuring error detection and preventing system hangups.