COMP 494: Problem Set 2 --- Due Tuesday, July 18, 2000

1. Minimal-size DFA (postponed from problem set 1)
   1. Show that there is a DFA with n+1 states that recognizes the language (1^n)*. The alphabet is Sigma={0,1}
   2. Show that there does not exist a smaller DFA for this language. (smaller = fewer states).
   
   
2. Suppose that L (a language over Sigma) is DFA-acceptable. Show that the following language is DFA-acceptable, too:
   MAX(L) = {x in L: there is no y in Sigma+ s.t. xy in L}.
   // A+ means AA*
   
   
3. Same instruction as the last problem, for
   ECHO(L) = {a_1 a_1 a_2 a_2 ... a_n a_n: a_1 a_2 ... a_n in L}.
   
   
4. Page 85 of your book, Exercise 1.12.
   
   
5. Construct a regular expression for each of the languages from Problem Set 1, problem 1:
   1. The set of all strings with 010 as a substring
   2. The set of all strings which do not have 010 as a substring
   3. The set of all strings which have an even number of 0's or an even number of 1's
   4. The complement of {1,10}*
   5. The binary encodings of numbers divisible by 3: {0}* {e, 11, 110, 1001, 1100, 1111, ...}