Formulas of the General Solutions of Boolean Equations

Dragić Banković

We explicitly give various formulas of the general solutions of Boolean equations in $n$ unknows. The method presented in the paper is based on a Prešić's idea of the solving function from [4], but we have it here in more general form. We build the cycle using the sequence $i_1,i_2,\ldots,i_{\nu}$ ($\nu = 2^n$) where $\{i_1,i_2,\ldots,i_{\nu}\} = \{0,1,2,\ldots,\nu-1\}$. We can chose the sequence so that we obtain the formulas of the general solution in the triangular form. Specially, when $i_1 = 2^n-1$, we have the reproductive solutions. This paper enables one to make the program (we wrote it in FORTRAN IV) for digital computer which gives the formulas of the general solutions of Boolean equations, where the number of unknowns can be large. The limitation results only from the number of the elements of the sequence $i_1,i_2,\ldots,i_{\nu}$ i.e. of the memory of the computer.