Considering this FSM: finite state machine

Can someone explain me why the grammar is not:

  1. $S \rightarrow aA$
  2. $A \rightarrow aA \mid bB \mid \varepsilon$
  3. $B \rightarrow bB \mid \varepsilon$

Why does A not have a transition to $\varepsilon$?

  • 1
    $\begingroup$ The grammar given by the FSM and the FSM don't agree. A grammar from the FSM picture would be $S \to \epsilon \mid a A ; A \to \epsilon \mid a A \mid b B ; B \to \epsilon \mid b B$. There are productions to $\epsilon$ for all final states (all in the picture). $\endgroup$ – vonbrand Apr 9 '13 at 19:48
  • $\begingroup$ There is an error in my manual then. Thank you $\endgroup$ – Pier-Alexandre Bouchard Apr 10 '13 at 3:41

Both grammars are wrong: The first does allow String $b$, which is not accepted by the NFA, the second does not allow $\epsilon$, which is accepted by the NFA.

Note that some definitions of regular grammars don't allow renaming of variables, which makes the first grammar irregular. In the same way $\varepsilon$-transitions don't extend the expressive power of NFAs, renaming variables does not extend the expressive power of regular grammars.

So if the automaton is what you want (i.e. ensure that a word which contains a $b$, contains an $a$, too), just add $S \rightarrow \varepsilon$ to your second grammar. If the first grammar is what you want ($\{a\}^*\{b\}^*$) add $S{b \atop \rightarrow }B$ to your automaton and $S \rightarrow \varepsilon\mid bB$ to the second grammar.

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  • $\begingroup$ The first one is copy paste of my AI manual.. $\endgroup$ – Pier-Alexandre Bouchard Apr 9 '13 at 18:33
  • $\begingroup$ The first one allows b, because B is double rounded $\endgroup$ – Pier-Alexandre Bouchard Apr 9 '13 at 18:34
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    $\begingroup$ No: There has to be a path from $S$ only using $b$. $\endgroup$ – frafl Apr 9 '13 at 18:36
  • $\begingroup$ You are right, I think S and A from the automata should not be double rounded, in the first grammar. $\endgroup$ – Pier-Alexandre Bouchard Apr 9 '13 at 18:50
  • $\begingroup$ No, that's fine, since $S\Rightarrow A \Rightarrow B \Rightarrow \varepsilon$. You should an an arc $S{\rightarrow \atop b}B$ or make either $A$ or $B$ an initial state. $\endgroup$ – frafl Apr 9 '13 at 19:19

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