Lower bounds by recursion theoretic arguments
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  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:
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 ).
No EXPTIME-complete set can be within a distance of 2nc to any set in P for some constant c>0.
The second result improves Yesha's by at least two exponentials.
KeywordsPolynomial Time SIAM Journal Recursive Function Random Oracle Recursion Theory
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- 1.J.L. Balcázar and U. Schöning, Bi-immune sets for complexity classes, Mathematical Systems Theory 18 (1985), 1–10.Google Scholar
- 2.C.H. Bennett and J. Gill, Relative to a random oracle A, pA≠NPA≠coNPA with probability 1, SIAM Journal on Computing 10 (1981), 96–113.Google Scholar
- 3.L. Berman and J. Hartmanis, On isomorphism and density of NP and other complete sets, SIAM Journal on Computing 6 (1977), 305–327.Google Scholar
- 4.H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.Google Scholar
- 5.A.C. Yao, Theory and applications of trapdoor functions, 23rd IEEE Symp. Foundations of Computer Science 1982, 80–91.Google Scholar
- 6.Y. Yesha, On certain polynomial-time truth-table reducibilities of complete sets to sparse sets, SIAM Journal on Computing 12 (1983), 411–425.Google Scholar
- 7.P. Young, Some structural properties of polynomial reducibilities and sets in NP, Proc. 15th Ann. ACM Symp. Theory of Computing, 1983, 392–401.Google Scholar