Directions of Development of Computability Theory in the Era of Artificial Intelligence

Abstract

This article provides a systematic analysis of the connection between computability theory and modern artificial intelligence (AI) systems. Based on a synthesis of existing research, it is shown that the fundamental principles laid down by the Turing machine determine not only the capabilities but also the fundamental limitations for all existing AI architectures. The study explores the relationship between the classical Turing machine and modern AI paradigms from the perspective of computability theory and its boundaries. Through a formal classification of computational models, they are compared with practical AI architectures, such as symbolic, neural network, probabilistic, and hybrid ones. The work consistently examines Neural Turing Machines, convolutional networks, reinforcement learning, and transformers from the standpoint of the Church-Turing thesis and the unsolvable halting problem. Particular attention is paid to the theoretical limits of AI verification and safety, as well as methodological issues of applying computability theory to machine learning systems. An analysis of the limitations arising from undecidability theorems and resource-bounded complexity is conducted, and how heuristics and approximations allow bypassing classical obstacles in applied problems is considered. Special focus is given to the formalization of aspects of "intelligence" within the Turing machine framework, the study of the boundaries of theoretical reproducibility of learning systems, and the possibility of implementing a practical form of hypercomputation. The work presents a review of existing results and a critical analysis of practical implications for the development of reliable and verifiable AI systems. Conclusions are drawn about where computability theory finds its application, and directions for further research are proposed.