Prove/ Disprove: If $L$ is a CFL then $A(L)$ is a CFL too

Question

Consider the operation $A(L)$:

$$A(L) = \{ w: w\in L \land w_R \notin L \}$$

where $w_R$ is the reverse of $w$.

Prove/ Disprove: if $L$ is a CFL language so does $A(L)$.

I am almost certain there's a counter-example but I couldn't find a proper one. I'd be glad for help!

Thanks

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score 3 · Accepted Answer · answered May 18 '15 at 16:09

3

A simple first step.

Consider any language $K \subseteq \{a,b\}^*$, and let $1,2$ be two new symbols. Let $L = 1\cdot \{a,b\}^* \cdot 2 \cup 2 \cdot K \cdot 1$.

What if $1x2\in A(L)$?

answered May 18 '15 at 16:09

Hendrik Jan

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score 1 · Answer 2 · answered May 18 '15 at 18:12

Hint: First, try the following. Prove/disprove: if $L$ is regular, then so is $A(L)$.

Another hint: That should help you narrow down your search for a counterexample. Next, try to come up with a language that you know is context-free but not regular. Try tweaking it a little bit so that it has a non-trivial intersection with its reversal. What do you come up with? I was able to very quickly solve this problem using this approach -- the first example language I came up with, and the first modification I tried, worked.

Prove/ Disprove: If $L$ is a CFL then $A(L)$ is a CFL too

2 Answers2

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