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Is $L = \{ w : \#_a(w) = \#_b(w) \}$ regular? I do not think it is. I recently posted a question and from there I was thinking if this language is regular.

If we assume on the contrary, then there exists a pumping length $p$. In that case, the word looks like $xyz$ where $|xy| \leq p$. If $y$ does not have the same number of $a$'s and $b$'s then $xy^iz$ for $i>1$ will give us a contradiction. Otherwise $y$ has the same number of $a$'s and $b$'s. This does not lead me to anywhere. I tried creating cases where this would fail -- seems like there isn't any.

This link answers it, but I do not understand what they mean by "a regular language is one that uses finite memory". I have encountered similar reasoning before, I have not been able to figure it out.

This answer seems to check if a number satisfies the criteria using regex. Is that a different form of regex?

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    $\begingroup$ Recall that regular languages are closed under intersection, and that $a^*b^*$ is regular. Can you use that to prove that the language is not regular? Alternatively, recall that for the pumping lemma, you can choose the word $w$ you work with. Try words of the form $a^pb^p$. $\endgroup$ – Shaull Oct 15 '20 at 8:41
  • $\begingroup$ Makes sense. In $a^p b^p$, we have $xy = a^{p-k}$ for some $k \geq 0$. That does the job, since $y^{i}$ for $i>1$ adds more $a$'s. Even $i^0$ gives us a contradiction. $\endgroup$ – oldsailorpopoye Oct 15 '20 at 8:46
  • $\begingroup$ Perhaps you can answer your own question? $\endgroup$ – Yuval Filmus Oct 15 '20 at 10:48
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This can also be proved easily using Myhill-Nerode theorem.

Myhill-Nerode Theorem: Given a language $ L \subseteq \Sigma^* $, Suppose $$ \forall x,y \in S, (x \neq y) \wedge (\exists z \in \Sigma^* ,L(xz) \neq L(yz)) $$ where S is an infinite set. Then L is not a regular language.

(Here $L(w) = 1$ if $w \in L$ and $L(w) = 0$ if $w \notin L$.)

For the given problem, We have $L=\{w:\#_a(w)=\#_b(w)\}$.

Take $S = a^*$ (note: the set $S$ is infinite).

Now take any two distinct elements from set $S$, $x=a^i$ and $y=a^j$, and take $z=b^i$.

So $xz \in L$ and $yz \notin L$.

Hence, we will get an infinite number of distinct quotients as $S$ is an infinite distinctive set.

Thus, $L$ is not regular.

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Use the pumping lemma: If $L$ is regular, there is a constant $N \ge 1$ such that any string $\sigma \in L$ can be divided as $\sigma =\alpha \beta \gamma$ with $\lvert \alpha \beta \rvert \le N$, $\beta \ne \varepsilon$ so that for all $k$ the string $\alpha \beta^k \gamma \in L$.

Proof is by contradiction. Assume your $L$ is regular, let $N$ be the lemma's constant and pick $\sigma = a^N b^N \in L$, $\lvert \sigma \rvert = 2 N \ge N$. You see that $\alpha \beta$ is just 'a', so leaving out $\beta$ (i.e., $k = 0$) destroys the balance between 'a' and 'b', the result is not in $L$. Contradiction.

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