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Im struggling to understand how to transform Nondeterministic finite automaton (NFA) of the following form:

enter image description here

To a regular expression equivalent. What I have tried was using arden's rule. However I just cant figure out how to simplify and return the appropriate regular expression corresponding to that NFA.

First I have created the initial equation corresponding to those states:

$1: q3 = q_1 0 + q_1 1$

$2: q1 = q_0 0 + q_1 1$

$3: q0 = q_0 0 + q_0 1 + \epsilon$

Which I have tried to simplify:

$1: q3 = (q_0 0 + q_1 1)0 + (q_0 0 + q_1)1$

$1: q3 = q_0 00 + q_1 100 + q_0 01 + q_1 11$

$1: q3 = q_0(0+1) + q_1(0+1)$

$2: q1 = q_0 00 + q_0 10 + \epsilon 0 + q_0 01 + q_1 11$

$2: q1 = q_0(0+0+1)+ \epsilon 0 + q_1 11$

$3: q0 = q_0 0 + q_0 1 + \epsilon$

$3: q0 = q_0(0+1) + \epsilon$

Here I just lost. Maybe there is a different approach suitable in this context.

Appreciate any help!

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  • $\begingroup$ Did you draw the transition table and try subset construction ? $\endgroup$
    – user93
    Commented Jan 22, 2018 at 5:21
  • $\begingroup$ Hi yes, I think your example is not correct since if you convert that NFA to DFA it will have multiple final states. $\endgroup$
    – Googme
    Commented Jan 22, 2018 at 5:42
  • $\begingroup$ I used your nfa diagram to draw the transition table and use subset construction to get the dfa. nfa to dfa does not always need to have multiple final states $\endgroup$
    – user93
    Commented Jan 22, 2018 at 5:49
  • $\begingroup$ Nevertheless, it does not illustrate how to approach the problem I have. $\endgroup$
    – Googme
    Commented Jan 22, 2018 at 6:11
  • $\begingroup$ The video provided in the link explains it in details $\endgroup$
    – user93
    Commented Jan 22, 2018 at 6:34

1 Answer 1

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The Arden's rule, as it is usually stated, is easier to use if you consider equations on $(L_q)_{q\in Q}$ depicting the language $L_q$ the automaton accepts from state $q$. Doing this, you obtain the following equations. Check that you understand this properly :

  • $L_0 = 0L_0 + 1L_0 + 0L_1$
  • $L_1 = 0L_3 + 1L_3$
  • $L_3 = \varepsilon$

(I use $L$ instead of $q$ as it looks less misleading to me)

Once you have those equations, you can solve this as follows

  • $L_1 = 0 + 1$ (I replaced $L_3$ with its value)
  • $L_0 = (0+1)L_0 + 0(0+1)$ (Replaced $L_1$ and factorized. This is ready for Arden's rule)
  • Since $\epsilon\not\in L_0$, from Arden's rule : $L_0 = (0+1)^*0(0+1)$

The language accepted by the automaton is always the union of the $L_{q_i}$, where the $q_i$ are the initial states. So here, $L = L_0 = (0+1)^*0(0+1)$

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  • $\begingroup$ Hi wazdra, is there a way you could show me how to solve them since thats where Im struggling at $\endgroup$
    – Googme
    Commented Jan 22, 2018 at 7:38
  • $\begingroup$ Did you understand how I got the equations ? I solve them right underneath. If you are talking about your equations, those are really not suited for Arden's rule $\endgroup$
    – wazdra
    Commented Jan 22, 2018 at 7:41
  • $\begingroup$ Hmm Im not 100% sure, you are saying that the regular expression corresponding to that NFA in my example is (0+1)*0(0+1)? $\endgroup$
    – Googme
    Commented Jan 22, 2018 at 7:50
  • $\begingroup$ Yes. Solving is easier than creating the equations at start. $\endgroup$
    – wazdra
    Commented Jan 22, 2018 at 7:59
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    $\begingroup$ Yes, $L_3$ corresponds to the language this automaton would accept if you started from $q_3$. Since $q_3$ is an accepting state, you would accept e. Since there is no transition out of $q_3$, this is all you can accept. So $L_3 = e$. $\endgroup$
    – wazdra
    Commented Jan 22, 2018 at 8:20

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