An important feature of modern scientific and technological revolution is the rapid growth of collecting and processing information in almost all areas of science and technology. This gives rise to a large number of difficult tasks (problems) for a variety of reasons that people are not able to solve on their own because of their huge information capacity and complexity. It should be noted that in recent years there has been a steady trend of increasing the role of computers in the task, which is associated with problems of processing and presentation of data not only in quantitative terms but also in qualitative form - in the form of relationships, text description, obtained in economics, medicine, biology etc. The creation and distribution of software for data processing, which would allow users-nonprogrammers, working in various fields of science and technology, to solve problems on the basis of a computer, is also extremely relevant. Improving computer element base occurs with great predominance increase the degree of integration in comparison with other parameters. Therefore, the primary means of increasing productivity is to increase computer equipment - the number of gates used in it. In this paper we study the problem of implementing any Boolean function in the class of formulas and - circuits of functional elements in standard and Zhegalkin bases. An efficient method for the synthesis of formulas and schemes on the basis of the recurrence relations (functional equations), followed by obtaining advance analytically upper estimates of various indicators of complexity (the number of letters, number subformulas; on the number of functional elements over the depth of the formula and the depth of the circuit), including and schemes for the minimum of difficulty. If necessary to clarify the upper bounds for the complexity, the computational algorithm is being proposed.