Abstract
We study properties of non-uniform reductions and related completeness notions. We strengthen several results of Hitchcock and Pavan (ICALP (1), Lecture Notes in Computer Science, vol. 4051, pp. 465–476, Springer, 2006) and give a trade-off between the amount of advice needed for a reduction and its honesty on NEXP. We construct an oracle relative to which this trade-off is optimal. We show, in a more systematic study of non-uniform reductions, among other things that non-uniformity can be removed at the cost of more queries. In line with Post’s program for complexity theory (Buhrman and Torenvliet in Bulletin of the EATCS 85, pp. 41–51, 2005) we connect such ‘uniformization’ properties to the separation of complexity classes.
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References
Agrawal, M.: Pseudo-random generators and structure of complete degrees. In IEEE Conference on Computational Complexity, pp. 139–147 (2002)
Agrawal, M., Biswas, S.: Polynomial isomorphism of 1-L complete sets. In: Proc. Structure in Complexity Theory 7th Annual Conference, San Diego, California, pp. 75–80. IEEE Computer Society, Los Alamitos (1993)
Allender, E.: Isomorphisms and 1-L reductions. J. Comput. Syst. Sci. 36(6), 336–350 (1988)
Allender, E., Buhrman, H., Koucký, M., van Melkebeek, D., Ronneburger, D.: Power from random strings. In: FOCS, pp. 669–678. IEEE Computer Society, Los Alamitos (2002)
Ambos-Spies, K.: p-mitotic sets. In: Börger, E., Hasenjäger, G., Roding, D. (eds.) Logic and Machines. Lecture Notes in Computer Science, vol. 177, pp. 1–23. Springer, Berlin (1984)
Balcázar, J., Díaz, J., Gabarró, J.: Structural Complexity I. Springer, Berlin (1988)
Berman, L., Hartmanis, H.: On isomorphisms and density of NP and other complete sets. SIAM J. Comput. 6, 305–322 (1977)
Buhrman, H., Mayordomo, E.: An excursion to the Kolmogorov random strings. In: Proceedings Structure in Complexity Theory, 10th Annual Conference (STRUCTURES95), Minneapolis, pp. 197–205. IEEE Computer Society, Los Alamitos (1995)
Buhrman, H., Torenvliet, L.: Complicated complementations. In: Proceedings 14th IEE Conference on Computational Complexity, pp. 227–236. IEEE Computer Society, Los Alamitos (1999)
Buhrman, H., Torenvliet, L.: Separating complexity classes using structural properties. In: Proceedings 19th IEE Conference on Computational Complexity, pp. 130–138. IEEE Computer Society, Los Alamitos (2004)
Buhrman, H., Torenvliet, L.: A Post’s program for complexity theory. In Bulletin of the EATCS 85, pp. 41–51 (2005)
Buhrman, H., Homer, S., Torenvliet, L.: On complete sets for nondeterministic classes. Math. Syst. Theory 24, 179–200 (1991)
Buhrman, H., Spaan, E., Torenvliet, L.: Bounded reductions. In: Ambos-Spies, K., Homer, S., Schöning, U. (eds.) Complexity Theory, pp. 83–99. Cambridge University Press, Cambridge (1993)
Buhrman, H., Spaan, E., Torenvliet, L.: The relative power of logspace and polynomial time reductions. Comput. Complexity 3(3), 231–244 (1993)
Buhrman, H., van Melkebeek, D., Fortnow, L., Torenvliet, L.: Using autoreducibility to separate complexity classes. SIAM J. Comput. 29(5), 1497–1520 (2000)
Fenner, S., Fortnow, L., Kurtz, S.A.: The isomorphism conjecture holds relative to an oracle. In: Proc. 33rd IEEE Symposium Foundations of Computer Science, pp. 30–39 (1992)
Ganesan, K., Homer, S.: Complete problems and strong polynomial reducibilities. In: Proc. Symposium on Theoretical Aspects of Computer Science. Springer Lecture Notes in Computer Science, vol. 349, pp. 240–250. Springer, Berlin (1988)
Glaßer, C., Selman, A.L., Travers, S.D., Zhang, L.: Non-mitotic sets. In: Arvind, V., Prasad, S. (eds.) STTCS. Lecture Notes in Computer Science, vol. 4855, pp. 146–157. Springer, Berlin (2007)
Hartmanis, J., Hemachandra, L.: One-way functions and the non-isomorphism of NP-complete sets. Theor. Comput. Sci. 81(1), 155–163 (1991)
Homer, S., Kurtz, S., Royer, J.: A note on many-one and 1-truth table complete sets. Theor. Comput. Sci. 115(2), 383–389 (1993)
Hitchcock, J.M., Pavan, A.: Hardness hypotheses, derandomization, and circuit complexity. In: 24th Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 336–347. Springer, Berlin (2004)
Hitchcock, J.M., Pavan, A.: Comparing reductions to NP-complete sets. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP (1). Lecture Notes in Computer Science, vol. 4051, pp. 465–476. Springer, Berlin (2006)
Homer, S., Selman, A.L.: Oracles for structural properties: the isomorphism problem and public-key cryptography. J. Comput. Syst. Sci. 44(2), 287–301 (1992)
Homer, S., Selman, A.L.: Computability and Complexity Theory. Springer, New York (2001)
Kurtz, S., Mahaney, S., Royer, J.: The isomorphism conjecture fails relative to a random oracle. In Proc. 21nd Annual ACM Symposium on Theory of Computing, pp. 157–166 (1989)
Krentel, M.: The complexity of optimization problem. J. Comput. Syst. Sci. 36, 490–509 (1988)
Ladner, R., Lynch, N., Selman, A.: A comparison of polynomial time reducibilities. Theor. Comput. Sci. 1, 103–123 (1975)
Mayordomo, E.: Almost every set in exponential time is p-bi-immune. Theor. Comput. Sci. 136(2), 487–506 (1994)
Ronneburger, D.: Kolmogorov complexity and derandomization. PhD thesis, Rutgers University, New Brunswick, NJ, October 2004
Watanabe, O.: A comparison of polynomial time completeness notions. Theor. Comput. Sci. 54, 249–265 (1987)
Young, P.: Juris Hartmanis: Fundamental contributions to the isomorphism problems. In: Selman, A.L. (ed.) Complexity Theory Retrospective, pp. 108–146. Springer, Berlin (1990)
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Buhrman, H., Hescott, B., Homer, S. et al. Non-Uniform Reductions. Theory Comput Syst 47, 317–341 (2010). https://doi.org/10.1007/s00224-008-9163-5
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DOI: https://doi.org/10.1007/s00224-008-9163-5