Formal languages and automata theory

Li Fan

Pumping Lemma

• Let L be a regular set. Then there is a constant n such that if z is any word in L, and |z|>=n, we may write z=uvw in such a way that |uv|<=n, |v|>=1, and for all i>=0, u(v^i)w is in L. Furthermore, n is no greater than the number of states of the smallest FA accepting L.

Exercise

• The set L={0^(i^2) | i is an integer, i>=1}, which consists of all strings of 0’s whose length is a perfect square, is not regular.

• Prove that L = {(a^n)(b^k): n > k and n>=0} is not regular.

Which of the following languages are regular sets?

• {0^(2n) | n>=1}

• {(0^m)(1^n)(0^(m+n)) | m>=1 and n>=1}

• {0^n | n is a prime}

• The set of all strings that do not have three consecutive 0’s

• The set of all strings with an equal number of 0’s and 1’s

• {x | x in (0+1)*, and x=x^R} x^R is x written backward; for example, (011)^R=110.

• {xwx^R | x,w in (0+1)+}.

Thank you