On sparse oracles separating feasible complexity classes
This note clarifies which oracles separate NP from P and which do not. In essence, we are changing our research paradigm from the study of which problems can be relativized in two conflicting ways to the study and characterization of the class of oracles achieving a specified relativization. Results of this type have the potential to yield deeper insights into the nature of relativization problems and focus our attention on new and interesting classes of languages.
A complete and transparent characterization of oracles that separate NP from P would resolve the long-standing P=?NP question. In this note, we settle a central case. We fully characterize the sparse oracles separating NP from P in worlds where P=NP. We display related results about coNP, E, NE, coNE, and PSPACE.
KeywordsComputation Path Kolmogorov Complexity Advice Function Oracle Query Polynomial Hierarchy
Unable to display preview. Download preview PDF.
- T. Baker, J. Gill, and R. Solovay, “Relativizations of the P=?NP Question,” SIAM Journal on Computing, 1975, pp. 431–442.Google Scholar
- J. Balcázar, R. Book, T. Long, U. Schöning, and A. Selman, “Sparse Oracles and Uniform Complexity Classes,” FOCS 1984, pp. 308–313.Google Scholar
- A. Chandra, D. Kozen, and L. Stockmeyer, “Alternation,” JACM, V. 26, #1, 1981.Google Scholar
- M. Furst, J. Saxe, and M. Sipser, “Parity, Circuits, and the Polynomial-Time Hierarchy,” FOCS 1981, pp. 260–270.Google Scholar
- W. Gasrarch, “Recursion Theoretic Techniques in Complexity Theory and Combinatorics,” Center for Research in Computing and Technology Report TR-09-85, Harvard University, May 1985.Google Scholar
- J. Hartmanis, "Generalized Kolmogorov Complexity and the Structure of Feasible Computations,” Cornell Department of Computer Science Technical Report TR 83-573, September 1983.Google Scholar
- J. Hartmanis, to appear in EATCS Bulletin.Google Scholar
- R. Karp and R. Lipton, “Some Connections Between Nonuniform and Uniform Complexity Classes,” STOC 1980, pp. 302–309.Google Scholar
- S. Mahaney “Sparse Complete Sets for NP: Solution of a Conjecture of Berman and Hartmanis,” FOCS 1980, pp. 54–60.Google Scholar
- M. Sipser, “A Complexity Theoretic Approach to Randomness,” STOC 1983, pp. 330–335.Google Scholar