The Arden's lemma states that there exists a solution to the equation between regular expressions r = sr + t, with r unknown, and it is s*t.
I went through some other topics on the forum and I always saw it being applied, for example, on grammars that have productions like
so that L(S) = L(A)L(S) + b and the solution is L(A)*b.
However, on some student notes from a classmate I found this example:
and the Arden's rule is applied in this way without any further manipulation of the grammar:
L(S) = L(A)L(S)L(A) + L(A) and it concludes by saying that L(S) = L(A)*
This seems wrong to me, but I want to check first whether there is some further hypothesis that can be applied here to make the statement valid. For example, I wonder if it is decidable (and of course valid) whether the productions S->ASA|A generate the same language of the productions S->AAS|A.
Can somebody please help me?