0

Let

  1. $L$ be a language over $\Sigma$. And $\Sigma = \{0,1\}$ is a set of input alphabets.
  2. $L = \{ w \mid w \in \Sigma^*, \text{ where $w$ is a string with numbers of 0s divisible by 3 and number of 1s divisible by 5}\}$.

What will be the minimum numbers of states in the DFA accepting $L$?

My approach is to create a grid like this:

enter image description here

I am getting 5 x 3 = 15 states.

How can I apply Myhill Nerode theorem to solve this kind of problems? I don't understand this theorem at all.

rsonx
  • 281
  • 1
  • 11
  • 3
    Please read the wikipedia article on DFA minimization or relevant chapters of some textbooks for general algorithms that minimize the number of states of a DFA. – John L. Apr 10 '19 at 17:31
  • The empty string satisfies the condition but is not recognised by your DFA. Remember that 0 is a multiple of 3 (and of 5); the empty string is not the only valid string not recognised. Fixing that will produce a smaller automaton. – rici Apr 10 '19 at 17:51
  • @rici yeah I missed that. The start state will also be an accepting state. What are the other valid strings it does not recognises. – rsonx Apr 10 '19 at 18:28
  • Any string with no 0s or no 1s – rici Apr 10 '19 at 19:04
  • Wow I did not saw that. Lets number the states with its rows and cols. Then I can fix it by making with transitions $\delta$(00,0) = 51 and $\delta$(00,1) = 13 – rsonx Apr 10 '19 at 19:46
  • The minimal number of states for your language is 15. See if you can find a DFA with that number of states. – Yuval Filmus Apr 10 '19 at 20:41
  • Hi, @YuvalFilmus thanks for the edit. Yes, I am able to create a DFA with 15 states. The answer of D.W link talks about Myhill Nerode theorem. I don't understand the approach. Can anyone please make me understand? – rsonx Apr 11 '19 at 09:25
  • If you're familiar with Myhill-Nerode theory, you can come up with the proof yourself. Otherwise, get yourself familiar with Myhill-Nerode theory. – Yuval Filmus Apr 11 '19 at 11:01

0 Answers0