Complicated complementations

Kolmogorov complexity has proven to be a very useful tool in simplifying and improving proofs that use complicated combinatorial arguments. Using Kolmogorov complexity for oracle construction, we obtain separation results that are much stronger than separations obtained previously even with the use of very complicated combinatorial arguments. Moreover the use of Kolmogorov arguments almost trivializes the construction itself: In particular we construct relativized worlds where: 1. NP/spl cap/CoNP/spl isin/P/poly. 2. NP has a set that is both simple and NP/spl cap/CoNP-immune. 3. CoNP has a set that is both simple and NP/spl cap/CoNP-immune. 4. /spl Pi//sub 2//sup p/ has a set that is both simple and /spl Pi//sub 2//sup p//spl cap//spl Sigma//sup 2p/-immune.

[1]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

[2]  Chee-Keng Yap,et al.  Some Consequences of Non-Uniform Conditions on Uniform Classes , 1983, Theor. Comput. Sci..

[3]  Lance Fortnow,et al.  Circuit Lower Bounds à la Kolmogorov , 1995, Inf. Comput..

[4]  Nikolai K. Vereshchagin.  NP-sets are Co-NP-immune relative to a random oracle , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[5]  Riccardo Silvestri,et al.  Easily Checked Generalized Self-Reducibility , 1995, SIAM J. Comput..

[6]  José L. Balcázar.  Simplicity, Relativizations and Nondeterminism , 1985, SIAM J. Comput..

[7]  Nikolai K. Vereshchagin,et al.  Relationships between NP-sets, Co-NP-sets, and P-sets relative to random oracles , 1993, SCT.

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

[9]  Theodore P. Baker,et al.  A second step toward the polynomial hierarchy , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

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

[11]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[12]  A. Yao.  Separating the polynomial-time hierarchy by oracles , 1985 .

[13]  Sarah Mocas,et al.  Nonuniform Lower Bounds for Exponential Time Classes , 1995, MFCS.

[14]  Danilo Bruschi.  Strong Separations of the Polynomial Hierarchy with Oracles: Constructive Separations by Immune and Simple Sets , 1992, Theor. Comput. Sci..

[15]  Ker-I Ko,et al.  A note on separating the relativized polynomial time hierarchy by immune sets , 1990, RAIRO Theor. Informatics Appl..

[16]  Peter van Emde Boas,et al.  Simplicity, Immunity, Relativizations and Nondeterminism , 1989, Inf. Comput..

[17]  Lance Fortnow,et al.  Nonrelativizing separations , 1998, Proceedings. Thirteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat. No.98CB36247).

[18]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[19]  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.

[20]  Juris Hartmanis,et al.  Robust Machines Accept Easy Sets , 1990, Theor. Comput. Sci..

[21]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[22]  Lance Fortnow,et al.  The Role of Relativization in Complexity Theory , 1994, Bull. EATCS.

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