Given a 3D matrix of size $N \times N \times N$, let $\mathcal{S}$ be a set of points in the Matrix and $\mathcal{S}'$ be the complement of $\mathcal{S}$. Can we find a set of equations of the form:$F: (f_1 \land f_2 \land \ldots \land fn)$where $f_x$ is of the form$f_x \implies ai + bj + ck + dN + e \text{ } \{<,>,=,\neq\} \text{ } 0$$f_x$ is true if a point satisfies the aforementioned function and false otherwisewhere $i,j,k$ are the indices of the grid points in the Matrix (which vary from 1 to $N$) and $a,b,c,d,e$ are integers such that$F(\text{all points in }\mathcal{S}) = True$$F(\text{all points in }\mathcal{S}') = False$ For example:If $N = 4$, and $S = \{(1,1,1),(2,2,2),(3,3,3),(4,4,4)\}$, we can say that the functions $f_1 \implies i-k=0$ and $f_2 \implies j-k=0$ are one solution for this set.Ideally we are interested in a solution with the minimum number of equations, but this is not necessary.
Constructing a list of functions/formulas which describes a set of grid points in a 3D matrix
machine learning;combinatorics;matrices;statistics