CS 321 HW 6
Due on Canvas Thursday 11/5 at 5pm.

All problems are highly relevant for Quiz 5.

0. Are the following languages regular? Prove your results. 
   (a) same number of 0s and 1s
   (b) same number of 01s and 10s
   (c) same number of 00s and 11s

1. Use the "pumping down" method to prove the following languages are non-regular:
   (a) a^n b^n
   (b) { w w | w \in {a,b}* }

2. When using the pumping lemma, for which of the following we can make our own choices (\exists),
   and for which of the following we have to consider *all* cases (\forall)?

   (a) pumping length p
   (b) string s
   (c) the decomposition s = xyz
   (d) the pumping factor i

3. Write context-free grammars for the following. If impossible, explain:
   (a) a^n b^2n
   (b) { w w | w \in {0,1}* }
   (c) a^2n b^n c^m d^3m
   (d) a^n b^{2m+3n} c^m
   (e) a^n b^n c^n
   (f) (0 (0|1)* 1) (00)+
   
4. Describe a (recursive) algorithm to convert any RE to CFG.

5. Consider the regular language L1 = a^n b^m and the context-free language L2 = a^n b^n.
   Clearly L1 is a superset of L2. But we all know that regular languages form a subset of 
   context-free languages. How do you explain this seemingly contradicting fact?