Sparse hard sets for P: Resolution of a conjecture of Hartmanis

Journal of Computer and System Sciences

Volumn 58, Issue 2, 1999, Pages 280-296

  Cai, Jin Yi ^a   Sivakumar, D ^b  

^a State University of New York   (United States)

^b University of Houston   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL COMPLEXITY; POLYNOMIALS;

SPARSE HARD SETS;

SET THEORY;

EID: 0032624451     PISSN: 00220000     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcss.1998.1615     Document Type: Article

References (30)

1. [AGHP90] N. Alon, O. Goldreich, J. Hastad, and R. Peralta, Simple constructions of almost k-wise independent random variables, in "Proc. 31st Annual IEEE Symposium on Foundations of Computer Science, 1990," pp. 544-553.
2. +92] S. Buss, S. Cook, A. Gupta, and V. Ramachandran, An optimal parallel algorithm for formula evaluation, SIAM J. Comput. 21 (1992), 755-780.
3. [BH77] L. Berman and J. Hartmanis, On isomorphisms and density of NP and other complete sets, SIAM J. Comput. 6 (1977), 305-321. [A preliminary version appeared in STOC 1976]
4. [Bus87] S. Buss, The boolean formula value problem is in ALOGTIME, in "Proc. 19th Annual ACM Symposium on the Theory of Computing, 1987," pp. 123-131.
5. [BvzGH82 ] A. Borodin, J. von zur Gathen, and J. Hopcroft, Fast parallel matrix and GCD computations, Inform. and Control 52 (1982), 241-256.
6. [Chi85] A. L. Chistov, Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic, in "Proc. 5th International Conference on Fundamental of Computation Theory," Lecture Notes in Computer Science, pp. 63-69, Springer-Verlag, New York/Berlin, 1985.
7. [CNS95] J. Cai, A. Naik, and D. Sivakumar, On the existence of hard sparse sets under weak reductions, in "Proc. Symposium on Theoretical Aspects of Computer Science," Lecture Notes in Computer Science, pp. 307-318, Springer-Verlag, 1996.
8. [CS95] J. Cai and D. Sivakumar, Resolution of Hartmanis' conjecture for NL-hard sparse sets, in "Proc. Third Annual International Computing and Combinatorics Conference (COCOON)," Lecture Notes in Computer Science, Vol. 1276, pp. 62-71, Springer-Verlag, New York/Berlin, 1997.
9. [CVS84] A. Chandra, L. Stockmeyer, and U. Vishkin, Constant-depth reducibility, SIAM J. Comput. 13 (1984), 423-439.
10. [Coo85] S. Cook, A taxonomy of problems with fast parallel algorithms, Information and Control 64 (1985), 2-22.
11. [For79] S. Fortune, A note on sparse complete sets, SIAM J. Comput. 8, No. 3 (1979), 431-433.
12. [GJ79] M. R. Garey and D. S. Johnson, "Computers and Intractability: A Guide to the Theory of NP-Completeness," Freeman, New York, 1979.
13. [GL89] O. Goldreich and L. Levin, A hard-core predicate for all one-way functions, in "Proc. 21st Annual ACM Symposium on the Theory of Computing, 1989," pp. 25-32.
14. [Har78] J. Hartmanis, On log-tape isomorphisms of complete sets, Theoret. Comput. Sci. 7, No. 3 (1978), 273-286.
15. [HOT94] L. Hemachandra, M. Ogiwara, and S. Toda, Space-efficient recognition of sparse self-reducible languages, Comput. Complexity 4 (1994), 262-296.
16. [HOW92] L. Hemachandra, M. Ogiwara, and O. Watanabe, How hard are sparse sets, in "Proc. 7th Annual IEEE Conference on Structure in Complexity Theory, 1992," pp. 222-238.
17. [Imm88] N. Immerman, Nondeterministic space is closed under complementation, SIAM J. Comput. 17 (1988), 935-938.
18. [KL82] R. Karp and R. Lipton, Turing machines that take advice, Enseig. Math. 28, No. 2 (1982), 191-209. [A prelimary version appeared in STOC 1980].
19. [Lad75] R. Ladner, The circuit value problem in log space complete for P, SIGACT News 7, No. 1 (1975), 18-20.
20. [Mah82] S. Mahaney, Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis, J. Comput. System Sci. 25, No. 2 (1982), 130-143. [A preliminary version appeared in FOCS 1980.]
21. [Mel96] D. Van Melkebeek, Reducing P to a sparse set using a constant number of queries collapses P to L, in "Proc. 11th IEEE Conference on Computational Complexity Theory, 1996," pp. 88-96.
22. [Mul87] K. Mulmuley, A fast parallel algorithm to compute the rank of a matrix over an arbitrary field, Combinatorica 7, No. 1 (1987), 101-104.
23. [NN90] J. Naor and M. Naor, Small-bias probability spaces: efficient constructions and applications, in "Proc. 22nd Annual ACM Symposium on the Theory of Computing, 1990," pp. 213-223.
24. [Ogi95] M. Ogihara, Sparse hard sets for P yield space-efficient algorithms, in "Proc. 36th Annual IEEE Symposium on Foundations of Computer Science, 1995," pp. 354-361.
25. [OW91] M. Ogiwara and O. Watanabe, On polynomial-time bounded truth-table reducibility of NP sets to sparse sets, SIAM J. Comput. 20, No. 3 (1991), 471-483. [A preliminary version appeared in STOC 1990].
26. [Pip79] N. Pippenger, On simultaneous resource bounds, in "Proc. 20th Annual IEEE Symposium on Foundations of Computer Science, 1979," pp. 307-311.
27. [Sze87] R. Szelepcsényi, The method of forcing for nondeterministic automata, Bull. EATCS 33 (1987), 96-100.
28. [vL91] J. H. van Lint, "Introduction to Coding Theory," Springer-Verlag, 1991.
29. [You92a] P. Young, How reductions to sparse sets collapse the polynomial-time hierarchy: A primer (Part I), SIGACT News 23, No. 3 (1992), 107-117.
30. [You92b] P. Young, How reductions to sparse sets collapse the polynomial-time hierarchy: A primer (Part II), SIGACT News 23, No. 4 (1992), 83-94.