Distributionally-Hard Languages

Define a set L to be distributionally-hard to recognize if for every polynomial-time computable distribution µ with infinite support, L is not recognizable in polynomial time on the µ-average. Cai and Selman [5] defined a modification of Levin's notion of average polynomial time and proved that every P-bi-immune language is distributionally-hard. Pavan and Selman [23] proved that there exist distributionally-hard sets that are not P-bi-immune if and only P contains P-printable-immune sets. We extend this characterizion to include assertions about several traditional questions about immunity, about finding witnesses for NP-machines, and about existence of one-way functions. Similarly, we address the question of whether NP contains sets that are distributionally hard. Several of our results are implications for which we cannot prove whether or not their converse holds. In nearly all such cases we provide oracles relative to which the converse fails. We use the techniques of Kolmogorov complexity to describe our oracles and to simplify the technical arguments.

[1]  José L. Balcázar,et al.  The polynomial-time hierarchy and sparse oracles , 1986, JACM.

[2]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[3]  Lane A. Hemaspaandra,et al.  Defying Upward and Downward Separation , 1993, Inf. Comput..

[4]  Aduri Pavan,et al.  Complete distributional problems, hard languages, and resource-bounded measure , 2000, Theor. Comput. Sci..

[5]  Eric Allender,et al.  P-Printable Sets , 1988, SIAM J. Comput..

[6]  D. Juedes,et al.  Weak completeness in E and E 2 , 1995 .

[7]  Jack H. Lutz,et al.  Cook Versus Karp-Levin: Separating Completeness Notions if NP is not Small , 1996, Theor. Comput. Sci..

[8]  Yuri Gurevich,et al.  Average Case Completeness , 1991, J. Comput. Syst. Sci..

[9]  Jörg Rothe,et al.  Easy sets and hard certificate schemes , 1995, Acta Informatica.

[10]  Timothy J. Long,et al.  Relativizing complexity classes with sparse oracles , 1986, JACM.

[11]  Johan Håstad,et al.  Almost optimal lower bounds for small depth circuits , 1986, STOC '86.

[12]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[13]  Andrew Chi-Chih Yao,et al.  Separating the Polynomial-Time Hierarchy by Oracles (Preliminary Version) , 1985, FOCS.

[14]  Rainer Schuler,et al.  Sets Computable in Polynomial Time on Average , 1995, COCOON.

[15]  Jack H. Lutz,et al.  The quantitative structure of exponential time , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[16]  J. Hartmanis,et al.  Computation Times of NP Sets of Different Densities , 1984, Theor. Comput. Sci..

[17]  Ker-I Ko,et al.  Computational Complexity of Real Functions , 1982, Theor. Comput. Sci..

[18]  Russell Impagliazzo,et al.  Decision versus search problems in super-polynomial time , 1989, 30th Annual Symposium on Foundations of Computer Science.

[19]  Jack H. Lutz.  Almost Everywhere High Nonuniform Complexity , 1992, J. Comput. Syst. Sci..

[20]  Lane A. Hemaspaandra,et al.  Defying Upward and Downward Separation , 1993, STACS.

[21]  Ker-I Ko.  On Self-Reducibility and Weak P-Selectivity , 1983, J. Comput. Syst. Sci..

[22]  Peter Bro Miltersen,et al.  Relative to a Random Oracle, NP Is Not Small , 1996, J. Comput. Syst. Sci..

[23]  Jie Wang,et al.  Average-case computational complexity theory , 1998 .

[24]  Elvira Mayordomo.  Almost Every Set in Exponential Time is P-bi-Immune , 1994, Theor. Comput. Sci..

[25]  Neil Immerman,et al.  Sparse Sets in NP-P: EXPTIME versus NEXPTIME , 1985, Inf. Control..

[26]  Lance Fortnow.  Relativized Worlds with an Infinite Hierarchy , 1999, Inf. Process. Lett..

[27]  Ronald V. Book,et al.  Tally Languages and Complexity Classes , 1974, Inf. Control..

[28]  Jin-Yi Cai,et al.  Fine Separation of Average-Time Complexity Classes , 1999, SIAM J. Comput..

[29]  Leonid A. Levin,et al.  Average Case Complete Problems , 1986, SIAM J. Comput..

[30]  Russell Impagliazzo,et al.  A personal view of average-case complexity , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[31]  Jack H. Lutz,et al.  Cook Versus Karp-Levin: Separating Completeness Notions if NP Is not Small (Extended Abstract) , 1994, STACS.

[32]  Oded Goldreich,et al.  On the Theory of Average Case Complexity , 1992, J. Comput. Syst. Sci..

[33]  Sebastiaan Terwijn,et al.  Resource Bounded Randomness and Weakly Complete Problems , 1997, Theor. Comput. Sci..