High Accuracy Asymptotic Bounds for Predicate Circuits over a Class of Specific Predicate Bases

Synthesis problem for a specific class of control systems called predicate circuits is considered. This class generalizes most of well-known control-system classes (such as circuit of functional elements, contact circuits etc.). Predicate circuits are constructed with the use or predicate elements and therefore usually have no predefined direction of signal distribution.
Asymptotic behavior of the Shannon's function $\mathcal L_\mathfrak B(n)$ is investigated for complexity of $n$-variable predicate implementation with the use of predicate circuits over basis $\mathfrak B$ of specific structure. The following high accuracy asymptotic bounds are acquired
$$ \mathcal L_\mathfrak B(n)=\rho_\mathfrak B\frac{2^n}n\Biggl(1+\frac{\bigl(2+\frac1{k_\mathfrak B-1}\bigr)\log_2n\pm O(1)}n\Biggr), $$
where $\rho_\mathfrak B$ and $k_\mathfrak B$ are basis-dependent constants.