COMP 494: Problem Set 1 --- Due Thursday, July 6, 2000

1. Give DFAs for the following languages. Assume Sigma = {0,1}.
   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, ...}
   
   
2. State whether the following proposition are true or false, proving each answer.
   1. Every DFA-acceptable language can be accepted by a DFA with an even number of states.
   2. Every DFA-acceptable language can be accepted by a DFA whose start state is never visited twice.
   3. Every DFA-acceptable language can be accepted by a DFA no state of which is ever visited more than once.
   4. Every DFA-acceptable language can be accepted by a DFA with only a single final state.