### Limitations of FAs

Problem: Is there any set not regular ?

ans: yes!

example: B = {anbn | n 0 } = {e,ab,aabb,aaabbb,…}

Intuition: Any machine accepting B must be able to remember the number of a’s it has scanned before encountering the first b, but this requires infinite amount of memory (states) and is beyond the capability of any FA , which has only a finite amount of memory (states).

Pf: Assume B is regular = L(M) for some DFA M.

Let k = |Q|. consider the string anbn with n >> k.

Let s a s1 a s2 a …. s n-1 a sn b s n+1 b …. b s2n be any path witnessing the acceptance of anbn.

now consider the sequence of states s0,s1,…,sn, by pigeonhole principle there are 0 < i < j < n s.t. si = sj.

=> D(si, a n-i bn) = D(sj, a n-j bn) = s2n F, and D(s, ai) = D(s,aj) = si = sj. => D(s, a n-(j-i)bn) = D(s, ai an-jbn) = D(D(s,ai), an-jbn) = D(sj, an-jbn) = s2n F => an-(j-I)bn B, a contradiction.