Sparse Selfreducible Sets and Polynomial Size Circuit Lower Bounds
 Harry Buhrman,
 Leen Torenvliet,
 Falk Unger
 … show all 3 hide
Abstract
It is wellknown that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXP^{NP}, or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are selfreducible? It follows from earlier work of Lozano and Toran [10] that EXP^{NP} does not have sparse selfreducible hard sets. We define a natural version of selfreduction, treeselfreducibility, and show that NEXP does not have sparse treeselfreducible hard sets. We also show that this result is optimal with respect to relativizing proofs, by exhibiting an oracle relative to which all of EXP is reducible to a sparse treeselfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as superpolynomial circuit lower bounds for NEXP.
 Agrawal, M., Arvind, V. (1996) Quasilinear truthtable reductions to pselective sets. Theoretical Computer Science 158: pp. 361370 CrossRef
 Beigel, R., Kummer, M., Stephan, F. (1995) Approximable sets. Information and Computation 120: pp. 304314 CrossRef
 Berman, L., Hartmanis, H. (1977) On isomorphisms and density of NP and other complete sets. SIAM J. Comput. 6: pp. 305322 CrossRef
 Buhrman, H., Fortnow, L., Thierauf, T.: Nonrelativizing separations. In: IEEE Conference on Computational Complexity, pp. 8–12. IEE Computer Society Press (1998)
 Buhrman, H., Torenvliet, L. (1996) Pselective selfreducible sets: A new characterization of P. J. Computer and System Sciences 53: pp. 210217 CrossRef
 Fortnow, L., Klivans, A. (2005) NP with small advice. Proceedings of the 20th IEEE Conference on Computationa Complexity. IEEE Computer Society Press, Los Alamitos
 Hemaspaandra, L., Torenvliet, L. (2002) Theory of SemiFeasible Algorithms. Monographs in Theoretical Computer Science. Springer, Heidelberg
 Ko, K.I. (1983) On selfreducibility and weak Pselectivity. J. Comput. System Sci. 26: pp. 209211 CrossRef
 Ko, K., Schöning, U. (1985) On circuitsize and the low hierarchy in NP. SIAM J. Comput. 14: pp. 4151 CrossRef
 Lozano, A., Torán, J.: Selfreducible sets of small density. Mathematical Systems Theory (1991)
 Meyer, A.: oral communication. cited in [3] (1977)
 Mocas, S.: Separating Exponential Time Classes from Polynomial Time Classes. PhD thesis, Northeastern University (1993)
 Ogihara, M. (1995) Polynomialtime membership comparable sets. SIAM Journal on Computing 24: pp. 11681181 CrossRef
 Ogiwara, M., Watanabe, O. (1991) On polynomial time bounded truthtable reducibility of NP sets to sparse sets. SIAM J. Comput. 20: pp. 471483 CrossRef
 Papadimitriou, C. (1994) Computational Complexity. AddisonWesley, Reading
 Selman, A. (1979) Pselective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Math. Systems Theory 13: pp. 5565 CrossRef
 Selman, A. (1982) Analogues of semicursive sets and effective reducibilities to the study of NP complexity. Information and Control 52: pp. 3651 CrossRef
 Shaltiel, R., Umans, C.: Pseudorandomness for approximate counting and sampling. Technical Report TR04086, ECCC (2004)
 Wagner, K. (1988) Bounded query computations. Proc. 3rd Structure in Complexity in Conference. IEEE Computer Society Press, Los Alamitos, pp. 260278
 Wilson, C. (1985) Relativized circuit complexity. J. Comput. System Sci. 31: pp. 169181 CrossRef
 Title
 Sparse Selfreducible Sets and Polynomial Size Circuit Lower Bounds
 Book Title
 STACS 2006
 Book Subtitle
 23rd Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 2325, 2006. Proceedings
 Pages
 pp 455468
 Copyright
 2006
 DOI
 10.1007/11672142_37
 Print ISBN
 9783540323013
 Online ISBN
 9783540322887
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 3884
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 Springer Berlin Heidelberg
 Additional Links
 Topics
 Keywords

 Computational Complexity
 Sparseness
 Selfreducibility
 Industry Sectors
 eBook Packages
 Editors

 Bruno Durand ^{(16)}
 Wolfgang Thomas ^{(17)}
 Editor Affiliations

 16. LIF, CNRS & Univ. de Provence,
 17. RWTH Aachen University,
 Authors

 Harry Buhrman ^{(18)} ^{(19)}
 Leen Torenvliet ^{(19)}
 Falk Unger ^{(18)}
 Author Affiliations

 18. CWI Amsterdam,
 19. Universiteit van Amsterdam,
Continue reading...
To view the rest of this content please follow the download PDF link above.