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Weak Completeness Notions for Exponential Time

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Lutz (SIAM J. Comput. 24(6), 1170–1189, 1995) proposed the following generalization of hardness: While a problem A is hard for a complexity class C if all problems in C can be reduced to A, Lutz calls a problem weakly hard if a nonnegligible part of the problems in C can be reduced to A. For the linear exponential time class E = DTIME(2lin), Lutz formalized these ideas by introducing a resource-bounded (pseudo) measure on this class and by saying that a subclass of E is negligible if it has measure 0 in E. In this paper we introduce two new weak hardness notions for E – E-nontriviality and strongly E-nontriviality. They generalize Lutz’s weak hardness notion for E, but are much simpler conceptually. Namely, a set A is E-nontrivial if, for any k ≥ 1, there is a set Bk ∈E which can be reduced to A (by a polynomial time many-one reduction) and which cannot be computed in time O(2kn), and a set A is strongly E-nontrivial if the set Bk can be chosen to be almost everywhereO(2kn)-complex, i.e. if Bk can be chosen such that any algorithm that computes Bk runs for more than 2k|x| steps on all but finitely many inputs x.

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References

  1. Ambos-Spies, K.: Randomness relativizations, and polynomial reducibilities. Structure in complexity theory (Berkeley, Calif., 1986), 23–34, Lecture Notes in Comput Sci., vol. 223. Springer, Berlin (1986)

    Google Scholar 

  2. Ambos-Spies, K.: Minimal pairs for polynomial time reducibilities. Computation theory and logic, 1–13, Lecture Notes in Comput Sci., vol. 270. Springer, Berlin (1987)

    Google Scholar 

  3. Ambos-Spies, K.: Honest polynomial time reducibilities and the P =? NP problem. J. Comput. Syst. Sci. 39(3), 250–281 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ambos-Spies, K.: Resource-bounded genericity. Computability, enumerability, unsolvability, 1–59, London Math. Soc. Lecture Note Ser., vol. 224. Cambridge Univ. Press, Cambridge (1996)

    Google Scholar 

  5. Ambos-Spies, K.: Polynomial time reducibilities and degrees. Handbook of computability theory, 683–705, Stud. Logic Found. Math., 140, North-Holland, Amsterdam (1999)

  6. Ambos-Spies, K, Bakibayev, T.: Weak completeness notions for exponential time. Automata, languages and programming. Part I, 503–514, Lecture Notes in Comput Sci., vol. 6198. Springer, Berlin (2010)

    MATH  Google Scholar 

  7. Ambos-Spies, K., Bakibayev, T.: Nontriviality for exponential time w.r.t. weak reducibilities. Theoret. Comput. Sci. 494, 2–12 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ambos-Spies, K., Bakibayev, T.: Comparing nontriviality for. E and EXP. Theory Comput. Syst. 51(1), 106–122 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ambos-Spies, K., Mayordomo, E., Zheng, X.: A comparison of weak completeness notions. In: Proceedings of the 11th annual IEEE conference on computational complexity, 171–178 (1996)

  10. Ambos-Spies, K., Terwijn, S.A., Zheng, X.: Resource bounded randomness and weakly complete problems. Theoret. Comput. Sci. 172(1-2), 195–207 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Balcázar, J., Díaz, J., Gabarró J.: Structural complexity. I. Second edition. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (1995)

    Google Scholar 

  12. Balcázar, J.L., Díaz, J., Gabarró, J.: Structural complexity. II. EATCS monographs on theoretical computer science, vol. 22. Springer, Berlin (1990)

    Google Scholar 

  13. Buhrman, H., Mayordomo, E.: An excursion to the Kolmogorov random strings. Tenth Annual Conference on Structure in Complexity Theory (Minneapolis, MN. J. Comput. Syst. Sci. 54(3), 393–399 (1997)

    Article  MATH  Google Scholar 

  14. Geske, J.G., Huỳnh, D.T., Selman, A.L.: A hierarchy theorem for almost everywhere complex sets with application to polynomial complexity degrees. STACS 87 (Passau, 1987), 125–135, Lecture Notes in Comput Sci., 247. Springer, Berlin (1987)

    Google Scholar 

  15. Juedes, D.W., Lutz, J.H.: Weak completeness in E and E2. Theoret. Comput. Sci. 143(1), 149–158 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ladner, R.E.: On the structure of polynomial time reducibility. J. Assoc. Comput. Mach. 22, 155–171 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, M., Vitányi, P.: An introduction to Kolmogorov complexity and its applications Graduate texts in computer science. 2nd edn. Springer-Verlag, New York (1997)

  18. Lutz, J.H.: Weakly hard problems. SIAM J. Comput. 24(6), 1170–1189 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lutz, J.H., Mayordomo, E.: Measure stochasticity, and the density of hard languages. SIAM J. Comput. 23(4), 762–779 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Watanabe, O.: A comparison of polynomial time completeness notions. Theoret. Comput. Sci. 54(2-3), 249–265 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank the referees for their very helpful comments and suggestions.

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Authors and Affiliations

  1. Institut für Informatik, University of Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Klaus Ambos-Spies

  2. Almaty Management University, 227 Rozybakiyev Street, Almaty, 050050, Kazakhstan

    Timur Bakibayev

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  1. Klaus Ambos-Spies
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  2. Timur Bakibayev
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Correspondence to Timur Bakibayev.

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The authors were supported in part by the Ministry of Education and Science of Kazakhstan research grant number AP05131579. Some results of this paper have been presented at the ICALP Conference, Bordeaux, in 2010 [6].

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Ambos-Spies, K., Bakibayev, T. Weak Completeness Notions for Exponential Time. Theory Comput Syst 63, 1388–1412 (2019). https://doi.org/10.1007/s00224-019-09920-4

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