Homework 3: Honors Theory, Spring'99, Yap

Honors Theory of Computation, Spring'99

Homework 3

-- DATE DUE: Feb 24, 1999
-- PROBLEMS:

Do question 4.1 in Chapter 4: definition of NP-completeness.

Do question 4.5 in Chapter 4: closure of DLOG under Karp reducibility.
NOTE: I erroneously said question 4.4 earlier.

Do question 4.7 in Chapter 4: Show NP = co-NP iff there is a NP-complete language in NP. You must say what restrictions we need on the underlying reducibility ``<=''.
NOTE: I erroneously said question 4.6 earlier.

Recall that for any class K defined by complexity bounds, and for any language A, we can define the ``relativized class'' K^A which are accepted by oracle machines with the same complexity bounds as K, and with A as oracle. E.g., P^A is the class of languages accepted in deterministic polynomial time by oracle machines using A as oracle.
Let A be complete for PSPACE under Karp reducibility. Show that P^A= NP^A.

For any language A over \Sigma, let len(A) denote the language {w\in\Sigma^* : There exists x\in A such that |x|=|w| }. Construct a langauge A such that len(A) is not in P^A. HINT: assume Q={Q₀, Q₁, Q₂, ... } is an efficient universal oracle machine that runs in deterministic polynomial time. You need to construct A to satisfy the infinite list {C₀, C₁, C₂,...} of conditions: C_i : len(A) is not equal to L(Q_i^(A)) Read the related sections in chapter 4.