## without context – The language \$ {ww | w in {0,1 } ^ {*} } \$ n is not a CFL

We proved that the language $$L = { Omega Omega | omega in {0,1 } ^ {*} }$$ It's not a CFL, and we did it using the pumping lemma. And the proof is clear to me. But I thought about the following CFG:

$$G = ( {S, S_ {1} }, {0,1 }, R, S)$$ where R has the following rules:

$$S rightarrow S_ {1} S_ {1} | epsilon$$

$$S_ {1} rightarrow 0S_ {1} | 1S_ {1} | epsilon$$

It seems that the language of this CFG should be the language $$L$$ that I've defined above since each substitution adds the same letter on both sides. But it can not be so since we can use the pumping lemma on the word $$0 ^ {l} 1 ^ {l} 0 ^ {l} 1 ^ {l}$$ (or $$l$$ is the pumping length). So, either I do not do the substitution incorrectly, or the CFG language contains $$L$$ and has more words than I currently do not see …

Can any one help me and point out where is my mistake?

## Conversion of CFG to CFL [on hold]

Is it possible to convert
(1) Each CFL CFL
(2) Each CFL at CFL

If we take a string where the number of a and the number of b equals the number of c
then we get a CFG that does not have CFL.
So, can CFG at CFL be possible or not?

## CFL and LED mixed

I shoot a video on a green screen. I need to turn on the screen as well as the subject. I have a bunch of 6000K CFL and I want to get some LEDs. If both types have the same temperature, 6000 K, can I mix them in a video or will that be a problem?

## Prove that each CFL is decidable in a space O (n)

This question was raised while a group of students from my school were studying for our qualifying exams. The question on an old exam was:

Prove that each language without context $$A$$ is in $$mathrm {SPACE} (n)$$. You can assume that $$A$$ is given in CNF (normal form of Chomsky)

I've seen the CYK algorithm but its spatial complexity is $$O (n ^ 2)$$.