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Conference on Computability in Europe

CiE 2006: Logical Approaches to Computational Barriers pp 279–288Cite as

Complexity-Theoretic Hierarchies

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3988))

Abstract

We introduce two hierarchies of unknown ordinal height. The hierarchies are induced by natural fragments of a calculus based on finite types and Gödel’s T, and all the classes in the hierarchies are uniformly defined without referring to explicit bounds. Deterministic complexity classes like Logspace, p, pspace, linspace and exp are captured by the hierarchies. Typical subrecursive classes are also captured, e.g. the small relational Grzegorczyk classes \({\mathcal{E}}^0_*\), \({\mathcal{E}}^1_*\) and \({\mathcal{E}}^2_*\).

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References

  1. Avigad, J., Feferman, S.: Gödel’s functional interpretation. In: Buss, S. (ed.) Handbook of Proof Theory. Elsevier, Amsterdam (1998)

    Google Scholar 

  2. Bel’tyukov, A.: A machine description and the hierarchy of initial Grzegorczyk classes. J. Soviet Math (1982), Zap. Naucn. Sem. Leninigrad. Otdel. May. Inst. Steklov (LOMI) 88, 30–46 (1982)

    MATH  Google Scholar 

  3. Clote, P.: Computation models and function algebra. In: Griffor, E. (ed.) Handbook of Computability Theory. Elsevier, Amsterdam (1996)

    Google Scholar 

  4. Ehrenfeucht, A.: Polynomial functions with exponentiation are well ordered. Algebra Universalis 3, 261–262 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Esbelin, M.A., More, M.: Rudimentary relations and primitive recursion: A toolbox. Theoretical Computer Science 193, 129–148 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gandy, R.: Some relations between classes of low computational complexity. Bulletin of London Mathematical Society, pp. 127–134 (1984)

    Google Scholar 

  7. Goerdt, A., Seidl, H.: Characterizing complexity classes by higher type primitive recursive definitions, part II. In: Dassow, J., Kelemen, J. (eds.) IMYCS 1990. LNCS, vol. 464, pp. 148–158. Springer, Heidelberg (1990)

    Chapter  Google Scholar 

  8. Goerdt, A.: Characterizing complexity classes by higher type primitive recursive definitions. Theoretical Computer Science 100(1), 45–66 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grzegorczyk, A.: Some classes of recursive functions. In: Rozprawy Matematyczne, Warszawa, vol. IV (1953)

    Google Scholar 

  10. Jones, N.: The expressive power of higher-order types or, life without CONS. Journal of Functional Programming 11, 55–94 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kristiansen, L.: Appendix to the present paper (2006), Available from the author’s home page: http://www.iu.hio.no/~larskri

  12. Kristiansen, L.: Neat function algebraic characterizations of LOGSPACE and LINSPACE. Computational Complexity 14(1), 72–88 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kristiansen, L., Barra, G.: The small Grzegorczyk classes and the typed λ-calculus. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 252–262. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  14. Kristiansen, L., Voda, P.: Complexity classes and fragments of C. Information Processing Letters 88, 213–218 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kristiansen, L., Voda, P.: The surprising power of restricted programs and gödel’s functionals. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 345–358. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  16. Kristiansen, L., Voda, P.: Programming languages capturing complexity classes. Nordic Journal of Computing 12, 1–27 (2005); Special issue for NWPT 2004

    MathSciNet  MATH  Google Scholar 

  17. Kristiansen, L., Voda, P.: The trade-off theorem and fragments of gödel’s t. In: Cai, J.-Y., Cooper, S.B., Li, A. (eds.) TAMC 2006. LNCS, vol. 3959, pp. 654–674. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  18. Kutylowski, M.: Small Grzegorczyk classes. Journal of the London Mathematical Society 36(2), 193–210 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Levitz, H.: An ordinal bound for the set of polynomial functions with exponentiation. Algebra Universalis 8, 233–244 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  20. Odifreddi, P.: Classical recursion theory, vol. II. North-Holland Publishing Co., Amsterdam (1999)

    MATH  Google Scholar 

  21. Paris, J., Wilkie, A.: Counting problems in bounded arithmetic. In: Methods in mathematical logic. Proceedings, Caracas 1983. Lecture Notes in Mathematics, vol. 1130, pp. 317–340. Springer, Heidelberg (1985)

    Chapter  Google Scholar 

  22. Ritchie, R.W.: Classes of predictably computable functions. Transactions of the American Mathematical Society 106, 139–173 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rose, H.: Subrecursion. In: Functions and hierarchies. Clarendon Press (1984)

    Google Scholar 

  24. Schwichtenberg, H.: Classifying recursive functions. In: Griffor, E. (ed.) Handbook of computability theory, pp. 533–586. Elsevier, Amsterdam (1996)

    Google Scholar 

  25. Skolem, T.: An ordered set of arithmetic functions representing the least ε-number. Det Kongelige Norske Videnskabers Selskabs Forhandlinger 29(12), 54–59 (1956)

    MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Faculty of Engineering, Oslo University College, PO Box 4, St. Olavs plass, NO-0130, Oslo, Norway

    Lars Kristiansen

  2. Department of Mathematics, University of Oslo, Norway

    Lars Kristiansen

Authors
  1. Lars Kristiansen
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Swansea University, Singleton Park, SA2 8PP, Swansea, United Kingdom

    Arnold Beckmann

  2. Department of Computer Science, University of Wales Swansea, Singleton Park, SA2 8PP, Swansea, UK

    Ulrich Berger

  3. Universiteit van Amsterdam, Plantage Muidergracht 24, 1018, Amsterdam, TV, The Netherlands

    Benedikt Löwe

  4. School of Physical Sciences, Swansea University, Singleton Park, SA2 8PP, Swansea, Wales, United Kingdom

    John V. Tucker

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© 2006 Springer-Verlag Berlin Heidelberg

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Kristiansen, L. (2006). Complexity-Theoretic Hierarchies. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds) Logical Approaches to Computational Barriers. CiE 2006. Lecture Notes in Computer Science, vol 3988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780342_30

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  • DOI: https://doi.org/10.1007/11780342_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35466-6

  • Online ISBN: 978-3-540-35468-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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