This is a question about notation, and clear exposition in a paper. (I haven't published a paper before).
I have a set of variables $\mathcal Y=\{Y_1,...,Y_n\}$, and a directed acyclic graph $D$ with the $Y_i$ as nodes. If we let $PA_D(Y_i)\subseteq \mathcal Y$ denote the "parents" of $Y_i$ in this graph (the set of variables from which there is a link to $Y_i$ in $D$), then I have for each variable $Y_i$, a function $f_i:\prod _{Y_j\in PA_D(Y_i)}Y_j\to Y_i$. In other words, this graph $D$ gives us a functional dependency relation between the variables.
Some variables will not have any parents. Call these the "root" nodes $\mathcal R=\{R_i\}$, and let $r_i\in R_i$ be some arbitrary value for them.
Then we can recursively define $y_i=f_i(pa(i))$, where $pa(i)$ is the restriction of $(y_1,...,y_n)$ to $PA_D(Y_i)$.
This means we can expand the function one step: $$f_i(pa(i))=f_i(f_{pa(i)_1}(pa(pa(i)_1),...,f_{pa(i)_n}(pa(pa(i)_n))$$
We can continue expanding all the way to the root variables, but the notation becomes messier and messier.
What I want to do is the following, which is even messier: I want to single out some variable $X=Y_j$ for some $j$, and expand the function $f_i(pa(i))$, while showing to the reader the occurances of $f_X$. e.g. the first step of the expansion would be:
$$f_i(pa(i))=f_i(f_{pa(i)_1}(pa(pa(i)_1),...,f_X(pa(X)),...,f_{pa(i)_n}(pa(pa(i)_n))$$
I am looking for a notation for this type of expansion of $f_i$ to illustrate to my reader an algorithm I'm doing:
I want to expand $f_i$ all the way to the root variables, so that we have an expanded function that is only a function of root variables.
I want to then replace every $f_X(...)$ with some constant value $x\in X$, and replace every function $f_j$ with a different function $f_j'$, that has the same type signature and is given same arguments. (i.e. in the string that you would get from expanding $f_i$, you simply replace every occurance of $f_j$ with $f_j'$ for every $j$.)
I want to do this for arbitrary variables $\mathcal Y$ and graph $D$ and functions $(f_1,...,f_n)$.
I am looking for a notation to illustrate to the reader that I'm doing this, and that doesn't look messy. Is there a good way to do this?