For each problem, choose a partner, and then solve the problem by using the pummping lemma.

Write down the proof for the problem on a sheet of paper.

We will discuss solutions for each problem, before moving on to the next problem.

You should use a different partner for each problem.

1. Show that L1 = {0^m1^m | m ≥ 0} is not regular. {"",01,0011,000111, ...}
2. Show that L2 = {xx | x ∈ {0, 1}^*} is not regular. {"",00,11,0000,0101,1010,1111, ...}
3. Show that L3 = {1ⁿ² | n ≥ 0} is not regular. {"",1,1111,111111111, ...}
4. Show that L5 = {xx^R y | x,y in (0+1)⁺} is not regular.
5. Show that L6 = {O^m1ⁿO^m+n | m,n ≥ 0 } is not regular.

• For every regular language L
• There exists a constant n
• For every string w in L such that |w| ≥ n,
• There exists a way to break up w into three strings w = xyz such that
  1. |y| > 0
  2. |xy| ≤ n
  3. For every k ≥ 0 , the string xy^kz is also in L.