Convex Continuation of a Boolean Function and Its Applications
Dostonjon Numonjonovich Barotov · Journal of Applied and Industrial Mathematics · 2024
A convex continuation of an arbitrary Boolean function to the set $$ [0,1]^n $$ is constructed. Moreover, it is proved that for any Boolean function $$ f(x_1,x_2,\dots ,x_n) $$ that has no neighboring points on the set $$ \mathrm{supp} f $$ , the constructed function $$ f_C(x_1,x_2, \dots ,x_n) $$ is the only totally maximally convex continuation to $$ [0,1]^n $$ . Based on this, in particular, it is constructively stated that the problem of solving an arbitrary system of Boolean equations can be reduced to the problem of minimizing a function any local minimum of which in the desired region is a global minimum, and thus for this problem the problem of local minima is completely resolved.