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 $aaa$.

• You already gave the answer, the DFA accepts a, but a is not contained in the language described by the RE. Jul 21 '12 at 13:56
• It's pretty clear that the two are a mismatch, for the reason you give. Is there a reason why you doubt yourself? Jul 21 '12 at 14:10
• It's pretty simple indeed. I doubted because it was the officially provided solution in a teaching book. I am going to send the author an e-mail or check the errata if I find them. Thanks!
– Erik
Jul 21 '12 at 14:26
• I think it should be a+b*(c|ca+b*)*
– Erik
Jul 21 '12 at 17:56
• Note that this question can be seen as too localized; you should try to formulate a more general question in most cases. Also note that we can use LaTeX here for typesetting maths.
– Raphael
Jul 23 '12 at 7:41

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 & proof but rather build the expression algorithmically to save time.

• You are absolutely right. But sometimes thinking saves time and improves your understanding before doing it the mechanical way ;)
– Erik
Jul 23 '12 at 14:19
• @Erik Maybe I am no longer creative enough for such exercises, but what kind of thinking do you do? Trying inputs at random, hoping to refute correctness? Not very educational. Trying to find an intuitively correct mapping between expression and automaton? If that even exists for moderately complex cases, I doubt it will be worth the time. YMMV.
– Raphael
Jul 23 '12 at 15:32