Advertisement

A Higher-Order Characterization of Probabilistic Polynomial Time

  • Ugo Dal Lago
  • Paolo Parisen Toldin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7177)

Abstract

We present RSLR, an implicit higher-order characterization of the class PP of those problems which can be decided in probabilistic polynomial time with error probability smaller than \(\frac{1}{2}\). Analogously, a (less implicit) characterization of the class BPP can be obtained. RSLR is an extension of Hofmann’s SLR with a probabilistic primitive, which enjoys basic properties such as subject reduction and confluence. Polynomial time soundness of RSLR is obtained by syntactical means, as opposed to the standard literature on SLR-derived systems, which use semantics in an essential way.

Keywords

Normal Form Polynomial Time Error Probability Inference Rule 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arora, S., Barak, B.: Computational Complexity, A Modern Approach. Cambridge University Press (2009)Google Scholar
  2. 2.
    Bellantoni, S.J., Niggl, K.H., Schwichtenberg, H.: Higher type recursion, ramification and polynomial time. Annals of Pure and Applied Logic 104(1-3), 17–30 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bellantoni, S.: Predicative recursion and the polytime hierarchy. In: Clote, P., Remmel, J.B. (eds.) Feasible Mathematics II, pp. 15–29. Birkhäuser (1995)Google Scholar
  4. 4.
    Bellantoni, S., Cook, S.A.: A new recursion-theoretic characterization of the polytime functions. Computational Complexity 2, 97–110 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bonfante, G., Kahle, R., Marion, J.-Y., Oitavem, I.: Recursion Schemata for \({\mathit NC}^k\). In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 49–63. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Dal Lago, U., Martini, S., Sangiorgi, D.: Light logics and higher-order processes. In: Fröschle, S.B., Valencia, F.D. (eds.) Proceedings of the 17th International Workshop Expressiveness in Concurrency. EPTCS, vol. 41 (2010)Google Scholar
  7. 7.
    Dal Lago, U., Masini, A., Zorzi, M.: Quantum implicit computational complexity. Theoretical Computer Science 411(2), 377–409 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dal Lago, U., Parisen Toldin, P.: An higher-order characterization of probabilistic polynomial time (long version), http://arxiv.org/abs/1202.3317
  9. 9.
    Hofmann, M.: A Mixed Modal/Linear Lambda Calculus with Applications to Bellantoni-Cook Safe Recursion. In: Nielsen, M., Thomas, W. (eds.) CSL 1997. LNCS, vol. 1414, pp. 275–294. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Mitchell, J.C., Mitchell, M., Scedrov, A.: A linguistic characterization of bounded oracle computation and probabilistic polynomial time. In: Proceedings of the 39th Annual Symposium Foundations of Computer Science, pp. 725–733. IEEE Computer Society (1998)Google Scholar
  11. 11.
    Jones, N.D.: Logspace and ptime characterized by programming languages. Theoretical Computer Science 228, 151–174 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Leivant, D.: Stratified functional programs and computational complexity. In: Proceedings of the 20th International Symposium Principles of Programming Languages, pp. 325–333. ACM (1993)Google Scholar
  13. 13.
    Leivant, D., Marion, J.-Y.: Ramified Recurrence and Computational Complexity II: Substitution and Poly-Space. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 486–500. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  14. 14.
    Schwichtenberg, H., Bellantoni, S.: Feasible computation with higher types. In: Proof and System-Reliability, pp. 399–415. Kluwer Academic Publishers (2001)Google Scholar
  15. 15.
    Zhang, Y.: The computational SLR: a logic for reasoning about computational indistinguishability. Mathematical Structures in Computer Science 20(5), 951–975 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ugo Dal Lago
    • 1
  • Paolo Parisen Toldin
    • 1
  1. 1.Dipartimento di Scienze dell’InformazioneUniversità di Bologna, Équipe FOCUS, INRIA Sophia AntipolisBolognaItaly

Personalised recommendations