Robust Simulations and Significant Separations

  • Lance Fortnow
  • Rahul Santhanam
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


We define and study a new notion of “robust simulations” between complexity classes which is intermediate between the traditional notions of infinitely-often and almost-everywhere, as well as a corresponding notion of “significant separations”. A language L has a robust simulation in a complexity class C if there is a language in C which agrees with L on arbitrarily large polynomial stretches of input lengths. There is a significant separation of L from C if there is no robust simulation of L ∈ C.

The new notion of simulation is a cleaner and more natural notion of simulation than the infinitely-often notion. We show that various implications in complexity theory such as the collapse of PH if NP = P and the Karp-Lipton theorem have analogues for robust simulations. We then use these results to prove that most known separations in complexity theory, such as hierarchy theorems, fixed polynomial circuit lower bounds, time-space tradeoffs, and the recent theorem of Williams, can be strengthened to significant separations, though in each case, an almost everywhere separation is unknown.

Proving our results requires several new ideas, including a completely different proof of the hierarchy theorem for non-deterministic polynomial time than the ones previously known.


Turing Machine Complexity Class Input Length Promise Problem Nondeterministic Turing Machine 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Lance Fortnow
    • 1
  • Rahul Santhanam
    • 2
  1. 1.Northwestern UniversityUSA
  2. 2.University of EdinburghScotland, UK

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