Boolean equations with many unknowns
The author rediscovers the well-known method of successive elimination of variables for solving a Boolean equation in several unknowns. Further, he considers the partial derivatives \(F_x(y)=F(1,y)+F(0,y)\) and \(F_y(x)=F(x,1)+ F(x,0)\) of a Boolean function \(F\), where \(+\) denotes symmetric difference. It is proved that the system of functional equations \(F_x(y)=f_1(y)\), \(F_y(x)=f_2(x)\), is consistent if and only if \(f_1(1)+ f_1(0)= f_2(1)+f_2(0)\), in which case the solutions are functions of the form \(F(x,y)= xf_1(y)+ \overline yf_2(0)+a\), where \(a\) is a constant. Reviewer's remarks. The author seems to be unaware that the field ob Boolean equations over an arbitrary algebra has been extensively studied [see, e.g., the reviewer, Boolean functions and equations, North-Holland, Amsterdam (1974; Zbl 0321.06013)]. In particular the author's solution of a specific synthesis problem using an RS flip-flop can be obtained from the reviewer's solution of the general problem (op. cit., Table 16.3) by taking \(d_0=x_1\overline x_2\) and \(d_1=x_1\). An explicit (i.e., not recursive) form of the solution of a Boolean ring equation with unique solution was also obtained by the reviewer [Discr. Math. 122, No. 1-3, 381-383 (1993; Zbl 0788.06014)].
scientific article; zbMATH DE number 1499050