Asymptotically best method for synthesis of Boolean recursive circuits

Abstract

Models of multi-output and scalar recursive Boolean circuits of bounded depth in an arbitrary basis are considered. Methods for lower and upper estimates for the Shannon function for the complexity of circuits of these classes are provided. Based on these methods, an asymptotic formula for the Shannon function is put forward. Moreover, in the above classes of recursive circuits, upper estimates for the complexity of implementation of some functions and systems of functions used in applications are obtained.

Asymptotically Best Synthesis Methods for Reflexive-Recursive Circuits

“…The concepts that are presented below without explicit definitions can be found, e.g., in [17]. Before introducing the class of reflexive-recursive circuits, we define many-output recursive circuits and scalar recursive circuits, as well as some auxiliary concepts from [16].…”

“…Using this notation, the results of [13][14][15] mentioned in the beginning can be represented as follows.…”

“…The following asymptotic behavior of the Shannon function has been established in [15] for the class of manyoutput recursive circuits of bounded depth r in the standard basis:…”

“…The results of [15] have been generalized in [16], where asymptotic expressions have been obtained for the Shannon function for both many-output and scalar recursive circuits of bounded depth r in an arbitrary complete finite basis B:…”

“…Many-output recursive circuits of bounded depth in a standard basis have been considered in [15], where r r p 2 n /n asymptotic growth has been obtained for the Shannon function for circuits of this class. Asymptotically exact bounds of the Shannon function for both many-output and scalar recursive circuits in an arbitrary finite complete basis have been derived in [16].…”