Separability and One-Way Functions

We settle all relativized questions of the relationships between the following five propositions: P = NP. P = UP. P = NP $\cap$ coNP. All disjoint pairs of NP sets are P-separable. All disjoint pairs of coNP sets are P-separable. We make the first widespread use of variations of generic oracles to achieve the necessary relativized worlds.

[1]  Stuart A. Kurtz,et al.  The isomorphism conjecture holds relative to an oracle , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[2]  Stuart A. Kurtz,et al.  The isomorphism conjecture fails relative to a random oracle , 1995, JACM.

[3]  Juris Hartmanis,et al.  One-Way Functions and the Nonisomorphism of NP-Complete Sets , 1991, Theor. Comput. Sci..

[4]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[5]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[6]  N. K. Vereschchagin.  Relationships between NP-sets, Co-NP-sets, and P-sets relative to random oracles , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[7]  Alan J. Demers,et al.  Some Comments on Functional Self-Reducibility and the NP Hierarchy , 1976 .

[8]  Kurt Mehlhorn.  On the Size of Sets of Computable Functions , 1973, SWAT.

[9]  Stuart A. Kurtz,et al.  Gap-Definable Counting Classes , 1994, J. Comput. Syst. Sci..

[10]  Alan L. Selman,et al.  Complexity Measures for Public-Key Cryptosystems , 1988, SIAM J. Comput..

[11]  Russell Impagliazzo,et al.  Limits on the provable consequences of one-way permutations , 1988, STOC '89.

[12]  Joachim Grollmann,et al.  Relativizations of Unambiguous and Random Polynomial Time Classes , 1986, SIAM J. Comput..

[13]  Nikolai K. Vereshchagin,et al.  A General Method to Construct Oracles Realizing Given Relationships Between Complexity Classes , 1996, Theor. Comput. Sci..

[14]  Manuel Blum,et al.  Generic oracles and oracle classes , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[15]  Moni Naor,et al.  Decision trees and downward closures , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[16]  John Gill,et al.  Relative to a Random Oracle A, PA != NPA != co-NPA with Probability 1 , 1981, SIAM J. Comput..

[17]  Stuart A. Kurtz,et al.  On oracle builder's toolkit , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[18]  Steven Homer,et al.  Oracles for Structural Properties: The Isomorphism Problem and Public-Key Cryptography , 1992, J. Comput. Syst. Sci..