Lower bounds by recursion theoretic arguments

Abstract

Using methods and notions stemming from recursion theory, new lower bounds on the "distance" between certain intractable sets (like NP-complete or EXPTIME-complete sets) and the sets in P are obtained. Here, the distance of two sets A and B is a function on natural numbers that, for each n, gives the number of strings of size n on which A and B differ. Yesha [6] has shown that each NP-complete set has a distance of at least O(log log n) from each set in P, assuming P ≠ NP. Similarly, whithout an additional assumption, each EXPTIME-complete set has a distance of at least O(log log n) from each set in P.

In this paper the following will be shown:

• 1. Assuming P ≠ NP, no NP-complete set that is a (weak) p-cylinder can be within a distance of q(n) to any set in P where q is any polynomial. (Note that all "naturally known" NP-complete sets have been shown to be p-cylinders [3]).

• 2. No EXPTIME-complete set can be within a distance of 2^{n c} to any set in P for some constant c>0.

The second result improves Yesha's by at least two exponentials.

(Extended Abstract)