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Theorem. Show that if $L$ is regular, then so is $$ \varphi(L)=\left\{w \in \Sigma^{*} \mid \text {there exists an } \alpha \in \Sigma^{*} \text { with }|\alpha|=|w| \text { and } \alpha w \in L\right\} $$ Proof. Let $L$ be a regular language. Because $L$ is regular, there exists a DFA $M$ that accepts it. We construct out of $M$ a $\lambda$NFA $M'$ whose language is $\varphi(L)$.

We can think of a computation of $M$ as moving a token across the states of $M$. The machine $M'$ will reuse the states of $M$, but will use three tokens: white, red and blue. Initially, the blue token is put on $q_0$ (the initial state of $M$), and the white and red tokens are put on a nondeterministically guessed state (both tokens on the same state).

  1. Describe the transitions of $M'$.
  2. Describe the acceptance condition of $M'$.

Could someone please help me with that? I research regular expressions, DFAs, NFAs, and conversions between all of theses, but I still don't know how to solve this question.

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  • $\begingroup$ We're happy to help you understand the concepts but just solving exercises for you is unlikely to achieve that. You might find this page helpful in improving your question. $\endgroup$ – D.W. Nov 6 '19 at 21:22
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    $\begingroup$ This question is still oddly formatted. Why did you put everything in \$\$ + \text?! This ist still an exercise question. What have you tried so far to solve it? $\endgroup$ – ttnick Nov 6 '19 at 21:46
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The white token is a placeholder, which just remembers the original state of the red token (of course, the roles of these two tokens can be switched); this is the state at which $M$ would be after reading $\alpha$. The red token corresponds to the $w$ part of the word, and the blue token to its $\alpha$ part.

When reading a symbol, the red token advances according to the rules of $M$, the white token stays put, and the blue token guesses a symbol and advances according to the rules of $M$ (in effect, it guesses one symbol of $\alpha$).

The machine accepts if the blue and white tokens are at the same state (so we guessed correctly the state at which $M$ is after reading $\alpha$), and the red token is at an accepting state of $M$ (so $\alpha w \in L$).

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