as i know that DCFL is not closed under kleen closure but here Language is given so we cant use closure property ...

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Let the language be $L1= \left \{ 0^{N}1^{N} | N\geq 0 \right \}$

For this language, we can design a context free grammar as

$A\rightarrow \epsilon /0A1$

Now, according to the question, $L= L1$*

We can write grammar for this as

$S\rightarrow \epsilon /AS$

$A\rightarrow \epsilon /0A1$

This grammar generates $\epsilon , A,AA,AAA,AAAA,AAAAA,.................................$

Now, Is this language DCFL or NCFL ?

**We can try to make a DPDA for it. If it exists,,then this language is DCFL.**

This language L is DCFL and as well as NCFL.

1

But the grammar which u have written is not linear and we know that every dcfg need to be a linear grammar that is having only one variable in the production.

But u r doing S --> AS which is not allowed in linear grammar.

Also u cannot write a grammar for the above language that is linear.

So the language should be ncfl only not dcfl

But u r doing S --> AS which is not allowed in linear grammar.

Also u cannot write a grammar for the above language that is linear.

So the language should be ncfl only not dcfl