Check for Regular Language

Question

Check whether given language is Regular or not?

WXW^R / W,X∈(0,1)⁺

Comments

a (a+b)⁺ a and b (a+b)⁺ b is regular expression So yes Regular.

Check here 

Answer 1

Yes this is a regular language. Actually $X$ plays a very important role here. Since $X \in {(0 + 1)}^+$ , it consumes all the middle string as $X$.

For example check this,

Let String is = $011 00 110$

Above it is clear that W = $011$ and X = $00$

But the question is why we are taking X = $00$ only. We can take all the middle symbol in $X$ string.

So Regular Expression will becomes $ 0.(0+1)^+ . 0 + 1 . (0+1)^+ . 1 $

Ultimately it becomes, A language which start and end with same symbol.

If you can write Regular expression then it's regular languages. Hence Regular language.

Answer 2

WXW^R is regular as X∈(0+1)+ ,middle sub string can be merged and RE will be a (a+b)⁺ a U b (a+b)⁺ b

NB: If X ∉ (0+1)+ then WXW^R    is not regular. we need a PDA to recognize it. hence Context free