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On type-2 probabilistic quantifiers

  • Session 8: Complexity Theory
  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1099))

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Abstract

We define and examine several probabilistic operators ranging over sets (i.e., operators of type 2), among others the formerly studied ALMOST-operator. We compare their power and prove that they all coincide for a wide variety of classes. As a consequence, we characterize the ALMOST-operator which ranges over infinite objects (sets) by a bounded-error probabilistic operator which ranges over strings, i.e. finite objects. This leads to a number of consequences about complexity classes of current interest. As applications, we obtain (a) a criterion for measure 1 inclusions of complexity classes, (b) a criterion for inclusions of complexity classes relative to a random oracle, (c) a new upper time bound for ALMOST-PSPACE, and (d) a characterization of ALMOST-PSPACE in terms of checking stack automata. Finally, a connection between the power of ALMOST-PSPACE and that of probabilistic NC1 circuits is given.

Research supported by NSF Grant CCR-93-02057, DFG Grant Wa 847/1, and a Feodor-Lynen-Fellowship from the Alexander von Humboldt Foundation. Research performed while the second and third author were visiting the Department of Mathematics, University of California, Santa Barbara.

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References

  1. K. Ambos-Spies, Randomness, relativizations, and polynomial reducibilities, Proceedings of the 1st Structure in Complexity Theory Conference (1986), Springer Lecture Notes in Computer Science Vol. 223, pp. 200–207.

    Google Scholar 

  2. L. Babai, Trading group theory for randomness; Proceedings of the 17th Symposium on Foundations of Computer Science (1975), pp. 421–429.

    Google Scholar 

  3. J. L. Balcázar, J. Dí az, J. Gabarró, Structural Complexity I (Springer Verlag, Berlin-Heidelberg-New York, 1995).

    Google Scholar 

  4. D. Barrington, N. Immerman, H. Straubing, On uniformity within NC1; Journal of Computer and System Sciences41 (1990), pp. 274–306.

    Article  Google Scholar 

  5. C. Bennett, J. Gill, Relative to a random oracle PA≠ NPA ≠ co-NPA with probability 1; SIAM J. Comput.10 (1981), pp. 96–113.

    Article  Google Scholar 

  6. R. V. Book, Tally languages and complexity classes; Information and Control26 (1974), pp. 186–193.

    Article  Google Scholar 

  7. R. V. Book, On languages reducible to algorithmically random languages; SIAM J. Comput.23 (1994), pp. 1275–1282.

    Article  Google Scholar 

  8. R. V. Book, J. H. Lutz, K. W. Wagner, An observation on probability versus randomness with applications to complexity classes; Mathematical Systems Theory 27 (1994), pp. 201–209.

    Article  Google Scholar 

  9. D. P. Bovet, P. Crescenzi, R. Silvestri, A uniform approach to define complexity classes; Theoretical Computer Science 104 (1992), pp. 263–283.

    Article  Google Scholar 

  10. J. Y. Cai, With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy; Journal of Computer and System Sciences 38 (1989), pp. 68–85.

    Article  Google Scholar 

  11. S. A. Cook, A taxonomy of problems with fast parallel algorithms; Information and Control64 (1985), pp. 2–22.

    Article  Google Scholar 

  12. J. Gill, Computational complexity of probabilistic complexity classes; SIAM Journal on Computing 6 (1977), pp. 675–695.

    Article  Google Scholar 

  13. U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer, K.W. Wagner, On the power of polynomial time bit-reductions; Proceedings of the 8th Structure in Complexity Theory Conference (1993), pp. 200–207.

    Google Scholar 

  14. O. H. Ibarra, Characterizations of some tape and time complexity classes of Turing machines in terms of multihead and auxiliary stack automata; Journal of Computer and System Sciences 5 (1971), pp. 88–117.

    Google Scholar 

  15. S. Kautz, Degrees of random sets; Ph. D. dissertation, Cornell University, 1991.

    Google Scholar 

  16. C. Lautemann, BPP and the polynomial hierarchy; Information Processing Letters 117 (1983), pp. 215–217.

    Article  Google Scholar 

  17. J. Lutz, personal communication, 1995.

    Google Scholar 

  18. W. Merkle, Y. Wang, Separations by random oracles and “Almost” classes for generalized reducibilities; Proceedings of the 20th International Symposium on Mathematical Foundations of Computer Science (1995), Springer Lecture Notes in Computer Science Vol. 969, pp. 179–190.

    Google Scholar 

  19. N. Nisan, A. Wigderson, Hardness vs. Randomness; Journal of Computer and System Sciences49 (1994), pp. 149–167.

    Google Scholar 

  20. N. Nisan, Pseudorandom generators for space-bounded computation; Journal of Combinatorica12 (1992), pp. 449–461.

    Article  Google Scholar 

  21. N. Nisan, On read-once vs. multiple access to randomness in logspace; Theoretical Computer Sience107 (1993), pp. 135–144.

    Article  Google Scholar 

  22. P. Orponen, Complexity classes of alternating machines with oracles; Proceedings of the 10th International Colloquium on Automata, Languages and Programming (1983), Springer Lecture Notes in Computer Science Vol. 154, pp. 573–584.

    Google Scholar 

  23. C. H. Papadimitriou, S. K. Zachos, Two remarks on the power of counting; Proceedings of the 6th GI-Conference on Theoretical Computer Science (1983), Springer Lecture Notes in Computer Science Vol. 145, pp. 269–275.

    Google Scholar 

  24. K. W. Regan, J. S. Royer, On closure properties of bounded two-sided error complexity classes; Mathematical Systems Theory28 (1995), pp. 229–243.

    Article  Google Scholar 

  25. H. Rogers, Theory of Recursive Functions and Effective Computability (McGraw Hill, New York, NY, 1967).

    Google Scholar 

  26. U. Schöning, Probabilistic complexity classes and lowness; Journal of Computer and System Sciences39 (1989), pp. 84–100.

    Google Scholar 

  27. J. Simon, On Some Central Problems in Computational Complexity; Dissertation, Cornell University (1975).

    Google Scholar 

  28. M. Sipser, A complexity theoretic approach to randomness; Proceedings of the 15th Symposium on Theory of Computing (1983), pp. 330–335.

    Google Scholar 

  29. S. Toda, PP is as hard as the polynomial time hierarchy; SIAM Journal on Computing20 (1991) 865–877.

    Article  Google Scholar 

  30. K. W. Wagner, Some observations on the connection between counting and recursion; Theoretical Computer Science 47 (1986), pp. 131–147.

    Article  Google Scholar 

  31. K. W. Wagner, High-order operators in complexity theory; manuscript.

    Google Scholar 

  32. K. W. Wagner, G. Wechsung, Computational Complexity (Deutscher Verlag der Wissenschaften, Berlin, 1986).

    Google Scholar 

  33. C. Wrathall, Complete sets and the polynomial-time hierarchy; Theoretical Computer Science3 (1977), pp. 23–33.

    Article  Google Scholar 

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Friedhelm Meyer Burkhard Monien

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© 1996 Springer-Verlag Berlin Heidelberg

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Book, R.V., Vollmer, H., Wagner, K.W. (1996). On type-2 probabilistic quantifiers. In: Meyer, F., Monien, B. (eds) Automata, Languages and Programming. ICALP 1996. Lecture Notes in Computer Science, vol 1099. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61440-0_143

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  • DOI: https://doi.org/10.1007/3-540-61440-0_143

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-61440-1

  • Online ISBN: 978-3-540-68580-7

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