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Let $\Sigma=\{0, 1, \#\}$.

Is the following language regular; context-free but not regular; or not context-free?

Justify your answer $$L=\{x\#y :\ x, y \in\{0, 1\}^∗\text{ and }\operatorname{bin}(x) + 1 = \operatorname{bin}(\operatorname{rev}(y))\}$$

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    $\begingroup$ What are $\operatorname{bin}(x)$ and $\operatorname{rev}(y)$? I could guess the first is the number which binary representation is $x$ and the second is the reversed word. Is that the case? If so, if you take a word in $L$, for which $x$ is longer than the pumping length, then when you pump the word it will eventually fall out of $L$. Observe that the operation of adding $1$ can at most increate the length by $1$, but the pumping can make the part before the $\#$ arbitrarily longer than the part after the $\#$. $\endgroup$
    – plop
    Mar 18 at 12:28
  • $\begingroup$ This site has a convenient search option: Context free grammar for bin($n$)bin$(n+1)^R$ $\endgroup$ Mar 22 at 23:57
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Denote your language by $L$. Notice that $$ L \cap 1^*\#0^*1 = \{ 1^n \# 0^n 1 : n \geq 0 \}, $$ which should suffice for showing that $L$ isn't regular.

On the other hand, $$ L = \{ 0^a1^n \# 0^n 10^b : n,a,b \geq 0 \} \cup \{ 0^ax01^n \# 0^n1x^R0^b : n,a,b \geq 0, x \in \{0,1\}^* \}, $$ which should suffice for showing that $L$ is context-free.

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  • $\begingroup$ That’s part of your exercise. $\endgroup$ Mar 19 at 5:35
  • $\begingroup$ Thanks. I get that the first part shows L is not regular as there is not counting in regular languages and so there cannot be 1^n and 0^n together. May you elaborate on why the second equation is true? L={0^a 1^n#0^n 10^b ∶n,a,b ≥0}∪{0^a x01^n#0^n 1x^R 0^b:n,a,b≥0,x∈{0,1}^*} $\endgroup$
    – JYCT
    Mar 21 at 5:59
  • $\begingroup$ The second part does how to decompose words in the language in a way which allows generating them using a context-free grammar. $\endgroup$ Mar 21 at 6:53

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