On the complexity of the sequential realization of partial Boolean functions by schemes

A pair $(f,g)$ of partial Boolean functions is characterized by a tuple of parameters $l_{\alpha\beta}$ that is the number of tuples $\tilde x$ such that $(f(\tilde x),g(\tilde x))=(\alpha,\beta)$, where $\alpha$ and $\beta$ take the values 0, 1, and an undefined value. The sequential computation of $(f,g)$ is considered when a circuit $S_f$ for $f$ is constructed first, and, next, it is completed by the construction up to a circuit $S_{f,g}$. It is shown that if the domain $D(f)$ includes $D(g)$ then it is possible to compute sequentially $f$ and $g$ in such a way that $S_f$ and $S_{f,g}$ are asymptotically minimal simultaneously (i.e., they satisfy the asymptotically best bounds on the complexity for corresponding classes); and, in general, these functions cannot be sequentially computed in the order $g$, $f$ so that $S_g$ and $S_{f,g}$ are asymptotically minimal. An attainable lower bound is obtained on the size of the circuit $S_{f,g}$ for the sequential computation. The information properties of partially defined data play an essential role whose study in the previous papers of the author is continued here.