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I need to find a DFA for this language: L = { $ \omega = xy \in (a,b)^*\mid |x|_a = |y|_b $ }

lluc22
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  • You may want to clarify what $|x|_y$ means: is this the same as $#_y(x)$, i.e. the count of character $y$ in the string $x$? – wvxvw Sep 29 '15 at 12:09
  • This is a trick question. The point is that DFA usually cannot compare lenghts. Here the language can be rewritten in a much simpler way. Well, see the answer. – Hendrik Jan Sep 29 '15 at 12:29
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1 Answers1

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Let $w=w_1\dots w_n$ be any word in $\{a,b\}^*$. For $i=0,\dots,n$ consider the values $d_i(w):=|w_1\dots w_i|_a - |w_{i+1}\dots w_n|_b$. We can prove

  • $d_0(w) = - |w|_b \le 0$,
  • $d_n(w) = |w|_a \ge 0$,
  • $d_{i+1}(w) = d_i(w) + 1$ for $i=0,\dots,n-1$.

This implies that there must be some $i$ with $d_i(w) = 0$, and therefore $w\in L$. So $L=\{a,b\}^*$.

Klaus Draeger
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