Question

Let L be the language of binary strings of odd length such that the middle bit is 0.

Show that L is not regular, and show that L is context-free.

Answer 1

Let L be the language pf binary strings odd length such that the middle bit it 0.

Let w = (0+1)ⁿ 0 (0+1)ⁿ

We note that |w| ≥ n.

Let x,y,z ∈ Σ^*  w = xyz such that

x = (0+1)^n-2

y = (0+1)

z = (0+1) 0 (0+1)ⁿ

Checking the conditions of Pumping Lemma for Regular expressions,

1. |xy| ≤ n

2. y ≠ ∈ (or y ≥ 1)

3. for all k ≥ 0, the string xy^kz ∈ L.

Let k =0

xy⁰z = (0+1)^n-2 (0+1)⁰ (0+1) 0 (0+1)ⁿ

xy⁰z = (0+1)^n-2 (0+1) 0 (0+1)

xy⁰z = (0+1)^n-1 0 (0+1)ⁿ   ∉ L

Therefore, L is not regular.

We have 2 condition to be checked here ​​​​​​:

(a) The String is of odd length

(b) The middle element is 0

If we can find mid-point in the expression even in a non-deterministic way, then it is context free language

Therefore, L is a Context Free Language.