Boolean Differential Calculus

After a brief outline of classical concepts relative to Boolean differential calculus, a theoretical study of various differential operators is undertaken. Application of these concepts to several important problems arising in switching practice is mentioned. General notations The following notations are used in this paper: Boolean disjunction + or ~ Boolean conjunction no symbol or IT Boolean negation Modulo two sum EB or E Boolean exponentiation: x(e) = x' if e = 0 x(e) = x if e = 1 x(C) = xo(eo) xt(el) ... xn_l(en-l) x" = 1 if e = 0 x" = x if e = 1 xe = xoeo xlet Xn_Ien-l Dx = Dxo, Doc., , DXn_1 f D is an operator Dxe = Dxoeo Xlel ... Xn_len-l (Dxye) = (Dxoyeo) (Dxl)(et) ... (Dxn_t)(en-l) (Dx)C = (DxoYo (DX1yl ... (Dxn_IYn-l xl=al orj(xi = al) is the value ofjfor Xl = al (x;') is the value of jfor Xl substituted by x,: where this last notation means the vector of complemented literals. . Introduetion The concept of a differential of a Boolean function was introduced in a revious paper 6). It was shown how the transient behaviour of a Boolean unction is completely characterized by the algebraic properties of its associated ifferential. This paper is mainly concerned with a further analysis of the variational roperties of Boolean functions. It will be shown how a slight modification of he definition of the differential allows to solve a much larger class of problems ithout modifying the algorithms and procedures with the earlier definition f the differential. A large variety of problems, such as hazard detection, fault