Fine Separation of Average Time Complexity Classes

We extend Levin's theory of average polynomial time to arbitrary time-bounds in accordance with the following general principles: (1) It essentially agrees with Levin's notion when applied to polynomial time-bounds. (2) If a language L belongs to DTIME(T(n)), for some time-bound T(n), then every distributional problem (L,μ) is T on the μ-average. (3) If L does not belong to DTIME(T(n)) almost everywhere, then no distributional problem (L, μ) is T on the μ-average.

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

[2]  G. Hardy.  Properties of Logarithmico‐Exponential Functions , 2022 .

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

[4]  Stephen A. Cook,et al.  Time Bounded Random Access Machines , 1973, J. Comput. Syst. Sci..

[5]  Alan L. Selman,et al.  A Hierarchy Theorem for Almost Everywhere Complex Sets With Application to Polynomial Complexity Degrees , 1987, Symposium on Theoretical Aspects of Computer Science.

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

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

[8]  Jie Wang,et al.  Rankable Distributions Do Not Provide Harder Instances Than Uniform Distributions , 1995, COCOON.

[9]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

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

[11]  Ming Li,et al.  Average Case Complexity Under the Universal Distribution Equals Worst-Case Complexity , 1992, Inf. Process. Lett..

[12]  Joel I. Seiferas,et al.  A Note on Almost-Everywhere-Complex Sets and Separating Deterministic-Time-Complexity Classes , 1991, Inf. Comput..

[13]  Rüdiger Reischuk,et al.  Precise Average Case Complexity , 1993, STACS.

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