Usual distances on DFAs (Deterministic Finite Automata)? I've been searching in the literature for examples of distances defined on the set of the DFAs (or on the set of minimal DFAs) that are defined on a given alphabet sigma.

Since the languages they describe (regular languages) can potentially have an infinite size, defining a distance is not a trivial matter.  

Nevertheless, having a distance on these objects can be useful, in order to fit these in metric spaces, which allows for a range of things (in my case to assess the performance of an algorithm).  

My only consistent idea so far is to create a distance similar to the edit-distance in labeled graphs on the minimized DFAs.

Does someone have ever heard of other distances ?
There are many possible distance metrics, and without any criteria, we have no basis to choose.  Here are two plausible ones.

Let $L_1,L_2$ be the languages of the two DFAs.  Let $L$ be the symmetric difference of those languages (i.e., $(L_1 \setminus L_2) \cup (L_2 \setminus L_1)$).

One dissimilarity measure is $d(L_1,L_2) = 2^{-n}$ where $n=\min\{|x| : x \in L\}$, i.e., $n$ is the length of the shortest word in $L$.  This is not a distance metric, because it does not satisfy the triangle inequality.

Another possibility is to use the density of $L$, i.e.,

$$d(L_1,L_2) = \sum_{n=0}^\infty {|L \cap \Sigma^n| \over 2^n |\Sigma|^n}.$$

This one is a distance metric.

These can be computed given DFAs for $L_1,L_2$.  You can easily compute a DFA for $L$.  Then, it is easy to find the length of the shortest word in $L$ using breadth-first search.

Computing the density is a bit trickier, but can be done efficiently using a little more math.  Let $A$ denote the transition matrix of the DFA for $L$, i.e., $A_{ij}=1$ if there is a transition (on any input symbol) from state $i$ to state $j$, or $A_{ij}=0$ if not.  Let $s$ denote the one-hot vector indicating the start state, i.e., $s_i=1$ iff $i$ is the start state, and $f$ denote the one-hot vector indicating the final states, i.e., $f_i=1$ iff $i$ is an accepting state.  Now $|L \cap \Sigma^n| = s^\top A^n f$, so the distance is given by

$$d(L_1,L_2) = \sum_{n=0}^\infty {s^\top A^n f \over 2^n |\Sigma|^n}
= s^\top C f$$

where

$$C = \sum_{n=0}^\infty B^n = {1 \over I - B} = (I - B)^{-1}$$

where $B={1 \over 2|\Sigma} A$ and $I$ represents the identity matrix.  This can be computed in time proportional to the cube of the number of states in the DFA for $L$ using standard algorithms for matrix inversion.