I have this problem but my knowledge of the methods of linear algebra is totally useless if it exists at all (I know what is a matrix at least... maybe).
I have a three collections of objects ${\bf a}\in A$, ${\bf w}\in W$ and ${\bf b} \in B$ and I have a process that takes a triple and automatically gives me a vector of natural number $F:A\times W \times B\to\Bbb N^3$.
$$ F({\bf a},{\bf w},{\bf b}) = \begin{pmatrix} \sigma_1\\ \sigma_2\\ \sigma_3 \end{pmatrix} $$
- I know that each element of $A$, $W$ and $B$ have to be vectors so that we have $A\subseteq X^3$, $W\subseteq X^3$ and $B\subseteq X^3$ for some set of numbers $X\subseteq \Bbb Q$
$${\bf a}=\begin{pmatrix} a_1\\ a_2\\ a_3 \end{pmatrix} ; {\bf w}=\begin{pmatrix} w_1\\ w_2\\ w_3 \end{pmatrix} {\text{ and }} {\bf b}=\begin{pmatrix} b_1\\ b_2\\ b_3 \end{pmatrix} $$
I also know that the values of $\sigma_1,\sigma_2$ and $\sigma_3$ belong to the set $5:=\{0,1,2,3,4\}$
and I also know that $F$ is evaluated using an unknown $3$-ary function $f:\Bbb Q^3\to \Bbb N$ so that
$${\rm F}\begin{pmatrix} a_1 & w_1 & b_1 \\ a_2 & w_2 & b_2 \\ a_3 & w_3 & b_3 \end{pmatrix} = \begin{pmatrix} f(a_1,w_1,b_1)=\sigma_1\\ f(a_2,w_2,b_2)=\sigma_2\\ f(a_3,w_3,b_3)=\sigma_3 \end{pmatrix} $$
The problem is that the values of every vector ${\bf a}\in A$, ${\bf w}\in W$ and ${\bf b} \in B$ are "hidden" (I can't see7know them) and I can only see the values $\sigma_1,\sigma_2$ and $\sigma_3$, the outcome of $ F({\bf a},{\bf w},{\bf b})$.
Questions:
Given only the behaviour of $\sigma_1,\sigma_2$ and $\sigma_3$ (varying the arguments of $F$) how much it is possible to understand about the values of the vectors ${\bf a}\in A$, ${\bf w}\in W$ and ${\bf b} \in B$ and of $f$?
What it's needed in order to solve the functional equation (find $f$) in an unique way?