May 7, 1998
1. Find an FA that recognizes the same language as the following
NFA:
The initial state is 0 and the accepting state is 2.
2. Let L be a language over {a,b} defined recursively as follows:
Basis Clause: 
, and 
.
Inductive clause: For any string w, if 
, then 
and 
.
Extremal Clause: Nothing is in L unless it is obtained from the above two clauses.
Answer the following questions:
(a) List all strings of L of length 4 or less.
(b) Prove by structural induction that for any w in L, 
holds, where 
is the reversal of w.
3. Prove that the language { 
} is non-regular.
4. For the language 
} over the alphabet {a,b}, design the following Turing machines:
(a) A Turing machine that decides L.
(b) A Turing machine that accepts but does not decide L.
5. Let L be a language. Suppose that L and 
are accepted by Turing machines.
Is L, then, decidable ? Justify your answer.
Hint: You may use computers in place of Turing machines in your argument.