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Robust Simulations and Significant Separations

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

Abstract

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.

Keywords

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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References

  1. [Bar02]
    Barak, B.: A probabilistic-time hierarchy theorem for “Slightly Non-uniform” algorithms. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 194–208. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  2. [BFL91]
    Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity 1, 3–40 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [BFS09]
    Buhrman, H., Fortnow, L., Santhanam, R.: Unconditional lower bounds against advice. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 195–209. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. [BFT98]
    Buhrman, H., Fortnow, L., Thierauf, T.: Nonrelativizing separations. In: Proceedings of 13th Annual IEEE Conference on Computational Complexity, pp. 8–12 (1998)Google Scholar
  5. [Cai01]
    Cai, J.-Y.: S2 P ⊆ ZPPNP. In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, pp. 620–629 (2001)Google Scholar
  6. [Coo72]
    Cook, S.: A hierarchy for nondeterministic time complexity. In: Fourth Annual ACM Symposium on Theory of Computing, Conference Record, Denver, Colorado, May 1-3, pp. 187–192 (1972)Google Scholar
  7. [Coo88]
    Cook, S.: Short propositional formulas represent nondeterministic computations. Informations Processing Letters 26(5), 269–270 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [DF03]
    Downey, R., Fortnow, L.: Uniformly hard languages. Theoretical Computer Science 298(2), 303–315 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [FLvMV05]
    Fortnow, L., Lipton, R., van Melkebeek, D., Viglas, A.: Time-space lower bounds for satisfiability. Journal of the ACM 52(6), 833–865 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [For00]
    Fortnow, L.: Time-space tradeoffs for satisfiability. Journal of Computer and System Sciences 60(2), 337–353 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [FS04]
    Fortnow, L., Santhanam, R.: Hierarchy theorems for probabilistic polynomial time. In: Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pp. 316–324 (2004)Google Scholar
  12. [Hås86]
    Håstad, J.: Almost optimal lower bounds for small depth circuits. In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing, pp. 6–20 (1986)Google Scholar
  13. [IKW02]
    Impagliazzo, R., Kabanets, V., Wigderson, A.: In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences 65(4), 672–694 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [IW97]
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, pp. 220–229 (1997)Google Scholar
  15. [Kab01]
    Kabanets, V.: Easiness assumptions and hardness tests: Trading time for zero error. Journal of Computer and System Sciences 63(2), 236–252 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [Kan82]
    Kannan, R.: Circuit-size lower bounds and non-reducibility to sparse sets. Information and Control 55(1), 40–56 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [KL82]
    Karp, R., Lipton, R.: Turing machines that take advice. L’Enseignement Mathématique 28(2), 191–209 (1982)MathSciNetzbMATHGoogle Scholar
  18. [KvM99]
    Klivans, A., van Melkebeek, D.: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. In: Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pp. 659–667 (1999)Google Scholar
  19. [NW94]
    Nisan, N., Wigderson, A.: Hardness vs randomness. Journal of Computer and System Sciences 49(2), 149–167 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [Raz85]
    Razborov, A.: Lower bounds for the monotone complexity of some boolean functions. Soviet Mathematics Doklady 31, 354–357 (1985)zbMATHGoogle Scholar
  21. [RR97]
    Razborov, A., Rudich, S.: Natural proofs. Journal of Computer and System Sciences 55(1), 24–35 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [San07]
    Santhanam, R.: Circuit lower bounds for Merlin-Arthur classes. In: Proceedings of 39th Annual Symposium on Theory of Computing, pp. 275–283 (2007)Google Scholar
  23. [SFM78]
    Seiferas, J., Fischer, M., Meyer, A.: Separating nondeterministic time complexity classes. Journal of the ACM 25(1), 146–167 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [Vin05]
    Vinodchandran, V.: A note on the circuit complexity of PP. Theoretical Computer Science 347(1-2), 415–418 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [vMP06]
    van Melkebeek, D., Pervyshev, K.: A generic time hierarchy for semantic models with one bit of advice. In: Proceedings of 21st Annual IEEE Conference on Computational Complexity, pp. 129–144 (2006)Google Scholar
  26. [Wil10a]
    Williams, R.: Improving exhaustive search implies superpolynomial lower bounds. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing, pp. 231–240 (2010)Google Scholar
  27. [Wil10b]
    Williams, R.: Non-uniform ACC circuit lower bounds (2010) (manuscript)Google Scholar
  28. [Ž8́3]
    Žák, S.: A Turing machine time hierarchy. Theoretical Computer Science 26(3), 327–333 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

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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