Competing provers yield improved Karp-Lipton collapse results

Information and Computation

Volumn 198, Issue 1, 2005, Pages 1-23

  Cai, Jin Yi ^a,c   Chakaravarthy, Venkatesan T ^a   Hemaspaandra, Lane A ^b   Ogihara, Mitsunori ^b  

^a University of Wisconsin Madison   (United States)

^b University of Rochester   (United States)

^c TSINGHUA UNIVERSITY   (China)

Author keywords

Competing provers; K mper AFK theorem; Karp Lipton theorem; Lowness; Nonuniform complexity; Structural complexity; Symmetric alternation; Yap's theorem

Indexed keywords

APPROXIMATION THEORY; COMPUTER PROGRAMMING LANGUAGES; POLYNOMIALS; PROBLEM SOLVING; SET THEORY; THEOREM PROVING;

COMPETING PROVERS; KAMPER-AFK THEOREM; KARP-LIPTON THEOREM; LOWNESS; NONUNIFORM COMPLEXITY; STRUCTURAL COMPLEXITY; SYMMETRIC ALTERATION; YAP'S THEOREM;

COMPUTER SCIENCE;

EID: 15844376276     PISSN: 08905401     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ic.2005.01.002     Document Type: Article

References (54)

1. M. Abadi, J. Feigenbaum, and J. Kilian On hiding information from an oracle Journal of Computer and System Sciences 39 1989 21 50
2. M. Agrawal, and S. Biswas Universal relations Proceedings of the 7th structure in complexity theory conference 1992 IEEE Computer Society Press Silver Spring, MD 207 220
3. V. Arvind, Y. Han, L. Hemachandra, J. Köbler, A. Lozano, M. Mundhenk, M. Ogiwara, U. Schöning, R. Silvestri, and T. Thierauf Reductions to sets of low information content K. Ambos-Spies S. Homer U. Schöning Complexity theory 1993 Cambridge University Press Cambridge 1 45
4. V. Arvind, J. Köbler, U. Schöning, and R. Schuler If NP has polynomial-size circuits, then MA=AM Theoretical Computer Science 137 2 1995 279 282
5. L. Babai, and L. Fortnow Arithmetization: a new method in structural complexity theory Computational Complexity 1 1 1991 41 66
6. J. Balcázar Self-reducibility Journal of Computer and System Sciences 41 3 1990 367 388
7. J. Balcázar, R. Book, T. Long, U. Schöning, A. Selman, Sparse oracles and uniform complexity classes, in: Proceedings of the 25th IEEE Symposium on Foundations of Computer Science, 1984, pp. 308-313
8. J. Balcázar, R. Book, and U. Schöning The polynomial-time hierarchy and sparse oracles Journal of the ACM 33 3 1986 603 617
9. J. Balcázar, J. Díaz, J. Gabarró, in: Structural Complexity I, second ed., EATCS Texts in Theoretical Computer Science, Springer-Verlag, Berlin, 1995
10. J. Balcázar, and U. Schöning Logarithmic advice classes Theoretical Computer Science 99 2 1992 279 290
11. N. Bshouty, R. Cleve, R. Gavaldà, S. Kannan, and C. Tamon Oracles and queries that are sufficient for exact learning Journal of Computer and System Sciences 52 3 1996 421 433
12. H. Buhrman, J. Kadin, and T. Thierauf Functions computable with nonadaptive queries to NP Theory of Computing Systems 31 1 1998 77 92
13. J. Cai S 2 p ⊆ ZPP NP Proceedings of the 42nd IEEE symposium on foundations of computer science 2001 IEEE Computer Society Press Silver Spring, MD 620 629
14. J. Cai, V. Chakaravarthy, L. Hemaspaandra, M. Ogihara, Competing provers yield improved Karp-Lipton collapse results, in: Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, vol. 2607, Springer-Verlag, Berlin, 2003, pp. 535-546
15. J. Cai, L. Hemaspaandra, and G. Wechsung Robust reductions Theory of Computing Systems 32 6 1999 625 647
16. R. Canetti More on BPP and the polynomial-time hierarchy Information Processing Letters 57 5 1996 237 241
17. R. Chang Bounded queries, approximations, and the boolean hierarchy Information and Computation 169 2 2001 129 159
18. R. Gavaldà, and J. Balcázar Strong and robustly strong polynomial time reducibilities to sparse sets Theoretical Computer Science 88 1 1991 1 14
19. Y. Gurevich Algebras of feasible functions Proceedings of the 24th IEEE symposium on foundations of computer science 1983 IEEE Computer Society Press Silver Spring, MD 210 214
20. J. Hartmanis, and L. Hemachandra Complexity classes without machines: On complete languages for UP Theoretical Computer Science 58 1-3 1988 129 142
21. J. Hartmanis, N. Immerman, On complete problems for NP∩coNP, in: Proceedings of the 12th International Colloquium on Automata, Languages, and Programming, Lecture Notes in Computer Science, vol. 194, Springer-Verlag, Berlin, 1985, pp. 250-259
22. L. Hemaspaandra, A. Hoene, A. Naik, M. Ogiwara, A. Selman, T. Thierauf, and J. Wang Nondeterministically selective sets International Journal of Foundations of Computer Science 6 4 1995 403 416
23. L. Hemaspaandra, S. Jain, and N. Vereshchagin Banishing robust Turing completeness International Journal of Foundations of Computer Science 4 3 1993 245 265
24. L. Hemaspaandra, A. Naik, M. Ogihara, and A. Selman Computing solutions uniquely collapses the polynomial hierarchy SIAM Journal on Computing 25 4 1996 697 708
25. L. Hemaspaandra, and M. Ogihara The complexity theory companion 2002 Springer-Verlag Berlin
26. J. Hopcroft, Recent directions in algorithmic research, in: Proceedings 5th GI Conference on Theoretical Computer Science, Lecture Notes in Computer Science, vol. 104, Springer-Verlag, Berlin, 1981, pp. 123-134
27. N. Immerman, and S. Mahaney Relativizing relativized computations Theoetical Computer Science 68 3 1989 267 276
28. J. Kämper Non-uniform proof systems: A new framework to describe non-uniform and probabilistic complexity classes Theoretical Computer Science 85 2 1991 305 331
29. R. Karp, and R. Lipton Some connections between nonuniform and uniform complexity classes Proceedings of the 12th ACM symposium on theory of computing 1980 ACM Press New York 302 309 an extended version has also appeared as: Turing machines that take advice, L'Enseignement Mathématique, second series, vol. 28, 1982, pp. 191-209
30. K. Ko On self-reducibility and weak P-selectivity Journal of Computer and System Sciences 26 1983 209 221
31. J. Köbler Locating P/poly optimally in the extended low hierarchy Theoretical Computer Science 134 2 1994 263 285
32. J. Köbler On the structure of low sets Proceedings of the 10th structure in complexity theory conference 1995 IEEE Computer Society Press Silver Spring, MD 246 261
33. J. Köbler, and O. Watanabe New collapse consequences of NP having small circuits SIAM Journal on Computing 28 1 1998 311 324
34. T. Long Strong nondeterministic polynomial-time reducibilities Theoretical Computer Science 21 1982 1 25
35. T. Long, and A. Selman Relativizing complexity classes with sparse oracles Journal of the ACM 33 3 1986 618 627
36. C. Lund, L. Fortnow, H. Karloff, and N. Nisan Algebraic methods for interactive proof systems Journal of the ACM 39 4 1992 859 868
37. A. Meyer, M. Paterson, With what frequency are apparently intractable problems difficult? Tech. Rep. MIT/LCS/TM-126, Laboratory for Computer Science, MIT, Cambridge, MA, 1979
38. C. Papadimitriou Computational complexity 1994 Addison-Wesley Reading, MA
39. D. Ranjan, S. Chari, and P. Rohatgi Improving known solutions is hard Computational Complexity 3 2 1993 168 185
40. A. Russell, and R. Sundaram Symmetric alternation captures BPP Computational Complexity 7 2 1998 152 162
41. U. Schöning A low and a high hierarchy within NP Journal of Computer and System Sciences 27 1 1983 14 28
42. U. Schöning, in: Complexity and Structure, Lecture Notes in Computer Science, vol. 211, Springer-Verlag, Berlin, 1986
43. A. Selman Polynomial time enumeration reducibility SIAM Journal on Computing 7 4 1978 440 457
44. A. Selman A taxonomy of complexity classes of functions Journal of Computer and System Sciences 48 2 1994 357 381
45. J. Simon, On some central problems in computational complexity, Ph.D. thesis, Cornell University, Ithaca, NY, available as Cornell Department of Computer Science Technical Report TR75-224 (1975)
46. S. Toda PP is as hard as the polynomial hierarchy SIAM Journal on Computing 20 5 1991 865 877
47. S. Toda, and M. Ogihara Counting classes are at least as hard as the polynomial-time hierarchy SIAM Journal on Computing 21 2 1992 316 328
48. J. Torán Complexity classes defined by counting quantifiers Journal of the ACM 38 3 1991 753 774
49. V. Variyam, A note on NP∩coNP/poly (note: the author uses NP∩coNP/poly to denote what in the present paper is denoted (NP∩coNP)/poly), Tech. Rep. RS-00-19, BRICS, Aarhus, Denmark, 2000
50. exp⊈(NP∩coNP)/poly Information Processing Letters 89 1 2004 43 47
51. K. Wagner The complexity of combinatorial problems with succinct input representations Acta Informatica 23 3 1986 325 356
52. K. Wagner Bounded query computation Proceedings of the 3rd structure in complexity theory conference 1988 IEEE Computer Society Press Silver Spring, MD 260 277
53. C. Yap Some consequences of non-uniform conditions on uniform classes Theoretical Computer Science 26 3 1983 287 300
54. S. Zachos Robustness of probabilistic complexity classes under definitional perturbations Information and Computation 54 3 1982 143 154