Does this regular expression equal this automata? I just came across an exercise which is to find a regular expression for the following automata, such that the regular expression and the automata generate the same language.



One solution presents the following expression:

$\qquad \displaystyle r_A = a^+b^+(c\mid ca^*b^+)^*$

However, can this be true? I think not, because the all words created from the regular expression will have at least one $b$ in it, whereas the automata accepts words without $b$, such as $aaa$.

What is your opinion?
In general, you can decide equivalence of regular expression and finite automaton using standard algorithms:


Construct an NFA from the regular expression,
Determinise and minimise both automata,
Check for equality (up to isomorphism).


These algorithms and proof that the minimal DFA of any regular language is unique (up to isomorphism) can be found in introductory textbooks.

If the underlying task is to "read off" a regular expression from an automaton, don't guess &amp; proof but rather build the expression algorithmically to save time.