Boolean circuit

Abstract: In an earlier paperf we have developed an abstract theory of Boolean algebras and their representations by algebras of classes. We now relate this theory to the study of general topology. The first part of our discussion is devoted to showing that the theory of Boolean rings is mathematically equivalent to the theory of locally-bicompact totally-disconnected topological spaces. In R we have already prepared the way for a topological treatment of the perfect representation of an arbitrary Boolean ring. Continuing in this way, we find that the perfect representation is converted by the introduction of a suitable topology into a space of the indicated type. We have no difficulty in inverting this result, proving that every locally-bicompact totally-disconnected topological space arises by the same procedure from a suitable Boolean ring.' It is thus convenient to call the spaces corresponding in this manner to Boolean rings, Boolean spaces. The algebraic properties of Boolean rings can, of course, be correlated in detail with the topological properties of the corresponding Boolean spaces. A simple instance of the correlation is the theorem that the Boolean rings with unit are characterized as those for which the corresponding Boolean spaces are bicompact. A familiar example of a bicompact Boolean space is the Cantor discontinuum or ternary set, which we discuss at the close of Chapter I. Having established this direct connection between Boolean rings and topology, we proceed in the second part of the discussion to considerations of a yet more general nature. We propose the problem of representing an arbitrary TVspace by means of maps in bicompact Boolean spaces. Our solution of this problem embodies an explicit construction of such maps, which we shall now describe briefly. In a given TVspace dt, the open sets and the nowhere dense sets generate a Boolean ring, with 9Î as unit, which characterizes the topological structure of 9Î. Those subrings which contain 9Î and which are so large that the interiors of their member sets constitute bases for 9î, also char-

Abstract: A systematic procedure is presented for writing a Boolean function as a minimum sum of products This procedure is a simplification and extension of the method presented by W V Quine Specific attention is given to terms which can be included in the function solely for the designer's convenience

Abstract: A Boolean network is a logical dynamic system, which has been used to describe cellular networks. Using a new matrix product, called semi-tensor product of matrices, a logical function can be expressed as an algebraic function. This expression can covert the Boolean networks into discrete-time linear dynamic systems. Similarly, the Boolean control networks can also be converted into discrete time bilinear dynamic systems. Under these forms the standard matrix analysis can be used to consider the structure and the control problems of Boolean (control) networks. After the detailed description of this new approach, the controllability of Boolean control networks is considered in the paper as an application.

Abstract: We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fan-in circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(O(n1/2k)) gates to calculate MODr functions for any r ≠ pm. This statement contains as special cases Yao's PARITY result [ Ya 85 ] and Razborov's new MAJORITY result [Ra 86] (MODm gate is an oracle which outputs zero, if the number of ones is divisible by m).