Honors Theory, Spring 2002, Yap Homework 5

This is due on Monday April 1. Please try to rewrite your solutions to make them more concise (without being more obscure). Always introduce

your paragraphs with an introduction (e.g., what method of proof will be used, etc).

1. (10 Points) Question 3 of the midterm is a basic type of argument that you need to master in computability theory.

Fix any r.e. set A. You were asked to construct a total recursive function t : N → N such that for all x ∈ N, x ∈ A iff t(x) ∈ HALT. Now, you should realize that t(x) is the index of a suitable STM (isn’t that all that HALT knows about?). In our solution, we described this STM and called it M_{x}. Then we simply said “the function t(x) which gives the index of M_{x} is clearly total computable”. Most of the time, this claim is sufficient. But for the present question, we ask you to provide precisely this detail.

2. (10 Points) Construct a non-recursive bijection f : A → N where A is an infinite r.e. set. NOTE: This possibility seems to have been overlooked by several students that I decided you should try to construct one for yourself.

3. (5 Points) Show that NP = co-NP iff there is an NP-complete language that is in co- − NP. There is an implicit notion of reducibility (which I will simply denote as “≤”) in these terminology. You have to tell us what property is needed of ≤ to make this result true.

4. (15 Points) Show that if A is PSPACE-complete under Karp-reducibility, then P^{A} = NP^{A}. NOTE: P^{A} means the class of language accepted in polynomial time by a deterministic oracle machine with A as oracle. The class NP^{A} is similarly defined, but in the nondeterministic mode.

5. (15 Points) Let s be a space-constructible function (see Chapter 2 for definition). Assume s(n) < ∞ for all n. Prove a (DSPACE(s), RE) separation result.

6. (20 Points) A Simple Alternating Machine is like an alternating machine, except that it has 1 worktape and no input tapes. The input is initially placed on this worktape. Show that such a machine can accept the palindrome language in linear time, using only a constant number of alternations. HINT: guess the positions (in binary) of about log n equally spaced input symbols, writing them directly under the corresponding symbols. Call these the “marker positions”. The positions of the other symbols can be determined relative to these marker positions.