My notes — Section 2.3

A new pumping lemma

If A is a context-free language
  There exists a number p
    Such that
      If s ∈ A and |s|≥p
        There exists strings u, v, x, y, z
          Such that
            s = uvxyz
            |v| > 0 or |y| > 0
            |vxy| ≤ p
          And
            For all i ≥ 0
              uvixyiz ∈ A

Proof:
    Ponder Figure 2.35

Example 2.36: {anbncn | n ≥ 0}
    The trick: Use apbpcp and pump up
    The idea:
        aaaaaaaa...aaaaaabbbbbbbb....bbbbbbcccccccc....cccccc
                     vvxxxyyy
        !!! v and y consist of one kind of letter (one could be empty)
        !!! pumping gives too many a's or b's
        aaaaaaaa...aaaaaabbbbbbbb....bbbbbbcccccccc....cccccc
                   vvxxxyyy
        !!! v or y consist of more than one kind of letter
        !!! pumping gives alternating sub-sequence .. such abbabbabb
	      
Example 2.37: {aibjck | 0≤i≤j≤k}
    The trick: Use apbpcp and pump down
        !!! Check the book... 

Example 2.38: {ww | w ∈ {0,1}*}
    The trick: Use 0p1p0p1p
        !!! Check the book...