1. CS 461 – Sept. 16 Review Pumping lemma Applications of FA:
How do we show language non-regular?
Applications of FA:
Scanning (part of compilation)
Searching for text, using regular expression

2. Proving non-regularity
It’s like a 2-player game…
Adversary picks secret number p.
We select any string we want, in terms of p (e.g. 0p1p)
Adversary will break up s into xyz subject to constraints.
The place to pump has length at least 1.
The place to pump appears in the first p positions.
Be ready to show that xyiz won’t be in language for some i.

3. Example languages Bit strings that are palindromes
Bit strings with equal # of 0s and 1s.
More 0s than 1s.
{ 0n : n is a perfect square }
{ 0n : n is prime }
{ 03n + 2 : n  0 }
{ 0i 1j : i is even or i  j }
Notice that regular sets can’t handle counting or nonlinear patterns.

4. Consider complement
Show L = { 0i 1j : i is even or i  j } is not regular.
Let’s use some set theory…
L regular iff L’ regular
What is L’ ?
Hint: Need to consider (0 + 1)* - 0*1*.
Language L’ has “and” instead of “or”, so easier to produce a word not in language. 

5. continued L = { 0i 1j : i is even or i  j }
Then L’ is all words with:
The definition turned around.
Let A = { i is odd and i < j }.
Plus all words not of the form 0*1*.
Let B = (0 + 1)* - 0*1* = 0*11*0(0 + 1)*.
Then, L’ = A union B.
Let s = 02p+1 12p+2 and i = 2 and see how it works in pumping lemma…