Synthesis of formulas whose complexity and depth do not exceed the asymptotically best estimates of high degree of accuracy

We study problems concerning optimal realizations of arbitrary Boolean functions by formulas in the standard basis {&;, V, ¬} in the presence of two optimality criteria: the depth and the complexity. Both the complexity and depth of Boolean functions are investigated from the point of view of so-called asymptotically best estimates of high degree of accuracy for the corresponding Shannon functions. Such estimates produce an asymptotics not only for the Shannon function, but also for the first remainder term of its standard asymptotic expansion. For any Boolean function depending on n variables, we prove that it is possible to construct a realizing formula in the basis {&;, V, ¬} so that its depth and complexity do not exceed values of the corresponding Shannon functions for the argument equal to n in the sense of asymptotic estimates of high degree of accuracy.