Skip to main content

A Generic Time Hierarchy with One Bit of Advice

Abstract.

We show that for any reasonable semantic model of computation and for any positive integer a and rationals 1 ≤ c < d, there exists a language computable in time n d with a bits of advice but not in time n c with a bits of advice. Our result implies the first such hierarchy theorem for randomized machines with zero-sided error, quantum machines with one- or zero-sided error, unambiguous machines, symmetric alternation, Arthur.Merlin games of any signature, etc. Our argument yields considerably simpler proofs of known hierarchy theorems with one bit of advice for randomized and quantum machines with two-sided error.

Our paradigm also allows us to derive stronger separation results in which the machine with the smaller running time can receive more advice than the one with the larger running time. We present a unified way to derive such results for randomized and quantum machines with two-sided error and for randomized machines with one-sided error.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dieter van Melkebeek.

Additional information

Manuscript received 12 September 2006

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

van Melkebeek, D., Pervyshev, K. A Generic Time Hierarchy with One Bit of Advice. comput. complex. 16, 139–179 (2007). https://doi.org/10.1007/s00037-007-0227-8

Download citation

Keywords.

  • Time hierarchy
  • semantic models
  • non-uniform models
  • probabilistic complexity classes
  • delayed diagonalization

Subject classification.

  • 68Q15
  • 68Q10