Gathering files

The aim of this section is to decrease the size needed to store all the information contained by an element of $\Sigma$. We will make groups of files in $F$, using the structure induced by the filesytem. The fact that files stored in the same directory often share the same properties or same function make it easy to use this structure.

Let's call $\psi:F\rightarrow F$ the function that maps to the file $f$ its parent. For example :

$$\psi( ) = \psi( )$$

$$\psi( ) = \psi( )$$

$$\psi( ) = \psi( )$$

Let $G$ be a subset of $F$ so that "/"$\in G$.

Let's call $\alpha_G(x)=\min\{n \in \mathbb{N}~\vert~ \psi^n(x) \in G\}$. Let's call $\psi_G$ the function that map to a file $f$ the nearest parent in $G$ :

$$\psi_G~:~F \rightarrow G$$

$$x \mapsto \psi^{\alpha_G(x)}(x)$$

Let $\mathcal{R}_G$ be the binary relation so that

$$x \mathcal{R}_G y \Leftrightarrow \psi_G(x)=\psi_G(y)$$

We can easily verify that $\mathcal{R}_G$ is an equivalence relation^3.1.

If $x \in F$, we write $\bar{x}=\{y \in F ~\vert~ y \mathcal{R}_G x\} \in \frac {F}{\mathcal{R}_G}$ the equivalence class of $x$. $\frac {F}{\mathcal{R}_G}$ is the quotient set, i.e. the set of all the equivalence classes.

Figure: From $G$ to the equivalence classes of $\frac {F}{\mathcal{R}_G}$.

to 1cm to 2cm $\longrightarrow$

We can now reduce the quantity of informations used in $\sigma$ or $\mu$ by using functions whose starting set is $G$. We have to chose correctly the set $G$ so that we can factorize $\sigma$ or $\mu$ and can write $\sigma=\sigma'\circ\psi_G$ and $\mu=\mu'\circ\psi_G$.

This amounts to the same thing as to group files that are given the same rights and to choose $G$ so that the equivalence classes match the groups.