A Note on Universal Measures for Weak Implicit Computational Complexity

  • Arnold Beckmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2514)


This note is a case study for finding universal measures for weak implicit computational complexity. We will instantiate “universal measures” by “dynamic ordinals”, and “weak implicit computational complexity” by “bounded arithmetic”. Concretely, we will describe the connection between dynamic ordinals and witness oracle Turing machines for bounded arithmetic theories.


Bounded arithmetic Dynamic ordinals Witness oracle Turing machines Weak implicit computational complexity 


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  1. 1.
    Toshiyasu Arai. Some results on cut-elimination, provable well-orderings, induction and reflection. Ann. Pure Appl. Logic, 95:93–184, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Theodore Baker, John Gill, and Robert Solovay. Relativizations of the P question. SIAM J. Comput., 4:431–442, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Arnold Beckmann. Seperating fragments of bounded predicative arithmetic. PhD thesis, Westf. Wilhelms-Univ., Münster, 1996.Google Scholar
  4. 4.
    Arnold Beckmann. Dynamic ordinal analysis. Arch. Math. Logic, 2001. accepted for publication.Google Scholar
  5. 5.
    Samuel R. Buss. Bounded arithmetic, volume 3 of Stud. Proof Theory, Lect. Notes. Bibliopolis, Naples, 1986.Google Scholar
  6. 6.
    Samuel R. Buss. Relating the bounded arithmetic and the polynomial time hierarchies. Ann. Pure Appl. Logic, 75:67–77, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Samuel R. Buss and Jan Krajíček. An application of boolean complexity to separation problems in bounded arithmetic. Proc. London Math. Soc., 69:1–21, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Johan Håstad. Computational Limitations of Small Depth Circuits. MIT Press, Cambridge, MA, 1987.Google Scholar
  9. 9.
    Jan Johannsen. A note on sharply bounded arithmetic. Arch. Math. Logik Grundlag., 33:159–165, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jan Krajíček. Fragments of bounded arithmetic and bounded query classes. Trans. Amer. Math. Soc., 338:587–98, 1993.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Jan Krajíček. Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press, Heidelberg/New York, 1995.zbMATHGoogle Scholar
  12. 12.
    Jan Krajíček, Pavel Pudlák, and Gaisi Takeuti. Bounded arithmetic and the polynomial hierarchy. Ann. Pure Appl. Logic, 52:143–153, 1991.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Daniel Leivant. Substructural termination proofs and feasibility certification. In Proceedings of the 3rd Workshop on Implicit Computational Complexity (Aarhus), pages 75–91, 2001.Google Scholar
  14. 14.
    Rohit J. Parikh. Existence and feasibility in arithmetic. J. Symbolic Logic, 36:494–508, 1971.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Wolfram Pohlers. Proof Theory. An Introduction. Number 1407 in Lect. Notes Math. Springer, Berlin/Heidelberg/New York, 1989.zbMATHGoogle Scholar
  16. 16.
    Chris Pollett. Structure and definability in general bounded arithmetic theories. Ann. Pure Appl. Logic, 100:189–245, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gaisi Takeuti. RSUV isomorphism. In Peter Clote and Jan Krajíček, editors, Arithmetic, proof theory, and computational complexity, Oxford Logic Guides, pages 364–86. Oxford University Press, New York, 1993.Google Scholar
  18. 18.
    Andrew C. Yao. Separating the polynomial-time hierarchy by oracles. Proc. 26th Ann. IEEE Symp. on Foundations of Computer Science, pages 1–10, 1985.Google Scholar
  19. 19.
    Domenico Zambella. Notes on polynomially bounded arithmetic. J. Symbolic Logic, 61:942–966, 1996.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Arnold Beckmann
    • 1
  1. 1.Institute of Algebra and Computational MathematicsVienna University of TechnologyViennaAustria

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