Given any language L that does not include Λ, let us define its cousin language |L| as follows:

Question

Given any language $L$ that does not include $\Lambda$, let us define its cousin language $|L|$ as follows: For any string of $a$ 's and $b$ 's, if the word formed by concatenating the second, fourth, sixth, . . . letters of this string is a word in $L$, then the whole string is a word in $|L|$. For instance, if $b b b$ is a word in $L$, then $a b a b b b b$ and $b b a b a b a$ are both words in $|L|$. (i) Show that if there is some PDA that accepts $L$, then there is some PDA that accepts $|L|$. (ii) If $L$ is regular, is $|L|$ necessarily regular too?

   Given any language $L$ that does not include $\Lambda$, let us define its cousin language $|L|$ as follows: For any string of $a$ 's and $b$ 's, if the word formed by concatenating the second, fourth, sixth, . . . letters of this string is a word in $L$, then the whole string is a word in $|L|$. For instance, if $b b b$ is a word in $L$, then $a b a b b b b$ and $b b a b a b a$ are both words in $|L|$.
(i) Show that if there is some PDA that accepts $L$, then there is some PDA that accepts $|L|$.
(ii) If $L$ is regular, is $|L|$ necessarily regular too?

Introduction to computer theory

Daniel I. A. Cohen 2nd Edition

Chapter 14, Problem 19

Instant Answer

Step 1

Let $M$ be a PDA that accepts $L$. We can modify $M$ to accept $|L|$ by adding an extra state and transitions. First, we add a new start state $q_0'$ and an epsilon transition from $q_0'$ to the original start state $q_0$ of $M$. This allows us to start in Show more…

Show all steps

Thanks for your feedback!

Given any language $L$ that does not include $\Lambda$, let us define its cousin language $|L|$ as follows: For any string of $a$ 's and $b$ 's, if the word formed by concatenating the second, fourth, sixth, . . . letters of this string is a word in $L$, then the whole string is a word in $|L|$. For instance, if $b b b$ is a word in $L$, then $a b a b b b b$ and $b b a b a b a$ are both words in $|L|$. (i) Show that if there is some PDA that accepts $L$, then there is some PDA that accepts $|L|$. (ii) If $L$ is regular, is $|L|$ necessarily regular too?

Play audio

Feedback

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback