Skip to main content
SpringerLink
Log in

Algebraic and logical characterizations of deterministic linear time classes

  • Complexity Theory III
  • Conference paper
  • First Online:
STACS 97 (STACS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1200))

Included in the following conference series:

Abstract

In this paper an algebraic characterization of the class DLIN of functions that can be computed in linear time by a deterministic RAM using only numbers of linear size is given. This class was introduced by Grandjean, who showed that it is robust and contains most computational problems that are usually considered to be solvable in deterministic linear time.

The characterization is in terms of a recursion scheme for unary functions. A variation of this recursion scheme characterizes DLINEAR, the class which allows polynomially large numbers. A second variation defines a class that still contains DTIME(n), the class of functions that are computable in linear time on a Turing machine.

From these algebraic characterizations, logical characterizations of DLIN and DLINEAR as well as complete problems (under DTIME(n) reductions) are derived.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Bellantoni. Predicative recursion and the polytime hierarchy. In P. Clote and J.B. Remmel, editors, Feasible Mathematics II, pages 15–29. Birkhäuser, 1995.

    Google Scholar 

  2. S. Bellantoni and S. Cook. A new recursion-theoretic characterization of the polytime functions. Comp. Complexity, 2:97–110, 1992.

    Article  Google Scholar 

  3. S.A. Bloch, J.F. Buss, and J. Goldsmith. Sharply bounded alternation within P. Technical Report TR96-011, Electronic Colloquium on Computational Complexity, 1996. http://ftp.eccc.uni-trier.de/pub/eccc/reports/1996/TR96-011/index.html.

    Google Scholar 

  4. A. Cobham. The intrinsic computational complexity of functions. In Y. Bar-Hillel, editor, Proceedings of: Logic, Methodology, and the Philosophy of Science, pages 24–30. North-Holland, 1964.

    Google Scholar 

  5. K. J. Compton and C. Laflamme. An algebra and a logic for NC1. Information and Control, 87:241–263, 1990.

    Google Scholar 

  6. H.-D. Ebbinghaus and J. Flum. Finite Model Theory. Springer-Verlag, 1995.

    Google Scholar 

  7. E. Grädel. On the notion of linear time computability. Int. J. Found. Comput. Sci., 1:295–307, 1990.

    Article  Google Scholar 

  8. E. Grädel. The expressive power of second order horn logic. In Proc. 8th Symposium on Theoretical Aspects of Computer Science STACS 91, volume 480 of LNCS, pages 466–477. Springer-Verlag, 1991.

    Google Scholar 

  9. E. Grädel and Y. Gurevich. Tailoring recursion for complexity. Journal of Symbolic Logic, 60:952–969, 1995.

    Google Scholar 

  10. E. Grandjean. A natural NP-complete problem with a nontrivial lower bound. SIAM Journal of Computing, 17:786–809, 1988.

    Article  Google Scholar 

  11. E. Grandjean. A nontrivial lower bound for an NP problem on automata. SIAM Journal of Computing, 19:438–451, 1990.

    Article  Google Scholar 

  12. E. Grandjean. Invariance properties of RAMs and linear time. Computational Complexity, 4:62–106, 1994.

    Article  Google Scholar 

  13. E. Grandjean. Linear time algorithms and NP-complete problems. SIAM Journal of Computing, 23:573–597, 1994.

    Article  Google Scholar 

  14. E. Grandjean. Sorting, linear time and the satisfiability problem. Annals of Mathematics and Artificial Intelligence, 16:183–236, 1996.

    Article  Google Scholar 

  15. E. Grandjean and F. Olive. Monadic logical definability of NP-complete problems. submitted, 1996.

    Google Scholar 

  16. Y. Gurevich. Algebras of feasible functions. In Proc. 24th IEEE Symp. on Foundations of Computer Science, pages 210–214, 1983.

    Google Scholar 

  17. Y. Gurevich and S. Shelah. Nearly linear time. In A. Meyer and M. Taitslin, editors, Logic at Botik '89, Lecture Notes in Computer Science 363, pages 108–118. Springer-Verlag, 1989.

    Google Scholar 

  18. D. Leivant. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In P. Clote and J.B. Remmel, editors, Feasible Mathematics II, pages 320–343. Birkhäuser, 1995.

    Google Scholar 

  19. K. Regan. Machine models and linear time complexity. SIGACT News, 24:4, Fall 1993.

    Article  Google Scholar 

  20. C. P. Schnorr. Satisfiability is quasilinear complete in NQL. Journal of the ACM, 25:136–145, 1978.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Johannes-Gutenberg Universität Mainz, Germany

    Thomas Schwentick

Authors
  1. Thomas Schwentick
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Rüdiger Reischuk Michel Morvan

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Schwentick, T. (1997). Algebraic and logical characterizations of deterministic linear time classes. In: Reischuk, R., Morvan, M. (eds) STACS 97. STACS 1997. Lecture Notes in Computer Science, vol 1200. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023481

Download citation

  • DOI: https://doi.org/10.1007/BFb0023481

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-62616-9

  • Online ISBN: 978-3-540-68342-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation