Towards an Implicit Characterization of NCk

  • G. Bonfante
  • R. Kahle
  • J. -Y. Marion
  • I. Oitavem
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4207)


We define a hierarchy of term systems T k by means of restrictions of the recursion schema. We essentially use a pointer technique together with tiering. We prove T k  ⊆ NC k  ⊆ T k + 1, for k ≥2. Special attention is put on the description of T 2 and T 3 and on the proof of T 2 ⊆ NC 2 ⊆ T 3. Such a hierarchy yields a characterization of NC.


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  1. [AJST01]
    Aehlig, K., Johannsen, J., Schwichtenberg, H., Terwijn, S.: Linear ramified higher type recursion and parallel complexity. In: Kahle, R., Schroeder-Heister, P., Stärk, R.F. (eds.) PTCS 2001. LNCS, vol. 2183, pp. 1–21. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  2. [BC92]
    Bellantoni, S., Cook, S.: A new recursion-theoretic characterization of the poly-time functions. Computational Complexity 2, 97–110 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [Blo94]
    Bloch, S.: Function-algebraic characterizations of log and polylog parallel time. Computational Complexity 4(2), 175–205 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BM04]
    Baillot, P., Mogbil, V.: Soft lambda-calculus: A language for polynomial time computation. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 27–41. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. [BO04]
    Bellantoni, S., Oitavem, I.: Separating NC along the δ axis. Theoretical Computer Science 318, 57–78 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Clo90]
    Clote, P.: Sequential, machine independent characterizations of the parallel complexity classes ALogTIME, AC k, NC k and NC. In: Buss, S., Scott, P. (eds.) Feasible Mathematics, pp. 49–69. Birkhäuser (1990)Google Scholar
  7. [Hof99]
    Hofmann, M.: Linear types and non-size-increasing polynomial time computation. In: Symposium on Logic in Computer Science (LICS 1999), pp. 464–473. IEEE, Los Alamitos (1999)Google Scholar
  8. [Hof02]
    Hofmann, M.: The strength of non-size increasing computation. In: POPL 2002: Proceedings of the 29th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, pp. 260–269. ACM Press, New York (2002)CrossRefGoogle Scholar
  9. [Lei95]
    Leivant, D.: Ramifed recurrence and computational complexity I: Word recurrence and polytime. In: Clote, P., Remmel, J.B. (eds.) Feasible Mathematics II, pp. 320–343. Birkhäuser (1995)Google Scholar
  10. [Lei98]
    Leivant, D.: A characterization of NC by tree recursion. In: FOCS 1998, pp. 716–724. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  11. [LM93]
    Leivant, D., Marion, J.-Y.: Lambda calculus characterizations of poly-time. Fundamenta Informaticae 19(1/2), 167–184 (1993)zbMATHMathSciNetGoogle Scholar
  12. [LM95]
    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
  13. [LM00]
    Leivant, D., Marion, J.-Y.: A characterization of alternating log time by ramified recurrence. Theoretical Computer Science 236(1–2), 192–208 (2000)MathSciNetGoogle Scholar
  14. [Nee04]
    Neergaard, P.M.: A functional language for logarithmic space. In: Chin, W.-N. (ed.) APLAS 2004. LNCS, vol. 3302, pp. 311–326. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. [Nig05]
    Niggl, K.-H.: Control structures in programs and computational complexity. Annals of Pure and Applied Logic 133(1-3), 247–273 (2005) (Festschrift on the occasion of Helmut Schwichtenberg’s 60th birthday)zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Oit04]
    Oitavem, I.: Characterizing NC with tier 0 pointers. Mathematical Logic Quarterly 50, 9–17 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Ruz81]
    Ruzzo, W.L.: On uniform circuit complexity. J. Comp. System Sci. 22, 365–383 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  18. [Sim88]
    Simmons, H.: The realm of primitive recursion. Archive for Mathematical Logic 27, 177–188 (1988)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • G. Bonfante
    • 1
  • R. Kahle
    • 2
  • J. -Y. Marion
    • 1
  • I. Oitavem
    • 3
    • 4
  1. 1.LoriaINPLVillers-lès-NancyFrance
  2. 2.Dept. MatemáticaUniversidade de Coimbra, Apartado 3008, 3001-454 Coimbra, Portugal, and CENTRIA, UNLCaparicaPortugal
  3. 3.CMAFUniversidade de LisboaLisboaPortugal
  4. 4.DM, UNLCaparicaPortugal

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