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I'm doing a problem where I need to find the $≡_A$ equivalence classes of the language $$A = \{ 0^{n}x \mid n \in \mathbb Z^+, x \in \{0, 1\}^*, \text{ and } \#_0(x) ≥ n \}. $$

The best way I've learned to find the equivalence classes from a formal language is to create an automation and minimize it. Is there a conventional way of finding the equivalence classes of a language more quickly and intuitively? Because I do not know the DFA for this problem. Thanks for the help!

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Your language is $01^*0(0+1)^*$, i.e., all words starting with 0 and containing at least one more 0.

Indeed, a word of the form $0y0z$ is in $A$ since we can take $n=1$ and $x = y0z$. Conversely, if $w = 0^nx \in A$ then $w$ contains at least $2n \geq 2$ many 0's, and starts with a 0.

Therefore the Myhill–Nerode equivalence classes of your language are: $$ \epsilon, 1(0+1)^*, 01^*, 01^*0(0+1)^*. $$

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  • $\begingroup$ Since there are finite number of equivalence classes, does that make the language regular? Also what would change of the equivalence classes if #0(x)<n instead of >n $\endgroup$ – James Pekon Feb 21 at 22:59
  • $\begingroup$ Yes, a language is regular iff there are finitely equivalence classes. $\endgroup$ – Yuval Filmus Feb 22 at 3:31

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