Sythesis and complexity of cellular circuitry asymptotically optimal by reliability

Background. The work relates to one of the most important divisions of mathematical cybernetics - the theory of synthesis, reliability and complexity of control systems. Such models of discrete functions computing as functional elements circuits are well-known. These circuits, consisting of absolutely reliable or unreliable elements, have been studied for a long time; researchers have obtained multiple results for them. However in real circuits it is necessary to take into account not just functional elements, but also circuit's geometry. Thereby, the authors suggested a model of cellular circuits consisting of functional elements, where the model is represented in the form of a rectangle, divided into cells, which contain circuit elements having certain sizes and occupying certain area. Cellular elements may be both functional, i.e. realizing some function from their inputs, and commutational, which transmit signals to a next element with possible alteration of direction. The study suggests that commutational elements are absolutely reliable, and inverse malfunctions occur independently on any of two functional element outputs with the same probability. Such circuits are a natural mathematical model of integral circuits and, therefore, have multiple applications in various fields of science and technology, and also appear to be relevant for research. The aim of the work is to build cellular circuits, asymptotically optimal by reliability, and to estimate complexity thereof. Materials and methods. In order to build cellular circuits, asymptorically optimal by reliability, the authors used a method based on the method of synthesizing functional element circuits, asymptotically optimal by reliability. For this method the researchers built cellular circuits, proved the corresponding theorems on upper and lower reliability estimates and estimated the built circuits. Results. The authors suggested a method of synthesis of cellular circuits, asymptotically optimal by reliability, and obtained upper and lower reliability estimates of the said circuits. For the first time the authors proved the estimate of complexity of cellular circuits, asymptotically optimal by reliability. Conclusions. In order to build cellular circuits, asymptotically optimal by reliability, it is possible to use the methods of synthesis of functional element circuits, asymptotically optimal by reliability.