Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Computer Science Logic

CSL 2000: Computer Science Logic pp 172–186Cite as

The Descriptive Complexity of the Fixed-Points of Bounded Formulas

  • Conference paper
  • First Online:

  • 403 Accesses

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1862))

Abstract

We investigate the complexity of the fixed-points of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a built-in BIT predicate, or equivalently, with a built-in membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomial-time. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by bounded-depth, polynomial-size circuits, and polylogarithmic-time uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmic-time hierarchy with an oracle to deterministic polynomial-time. Finally, we discuss the connection of this result with the well-studied algorithms for integer division.

Supported by the CUR, Generalitat de Catalunya, through grant 1999FI 00532, and partially supported by ALCOM-FT, IST-99-14186.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Allender and V. Gore. Rudimentary reductions revisited. Information Processing Letters, 40:89–95, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  2. E. Allender and V. Gore. A uniform circuit lower bound for the permanent. SIAM Journal of Computing, 23(5):1026–1049, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Atserias and Ph. G. Kolaitis. First-order logic vs. fixed-point logic in finite set theory. In 14th IEEE Symposium on Logic in Computer Science, pages 275–284, 1999.

    Google Scholar 

  4. D. M. Barrington and N. Immerman. Time, hardware, and uniformity. In Complexity Theory Retrospective II, pages 1–22. Springer-Verlag, 1997.

    Google Scholar 

  5. D.M. Barrington, N. Immerman, and H. Straubing. On uniformity within NC1. Journal of Computer and System Sciences, 41(3):274–306, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Barwise. Admissible Sets and Structures. Springer-Verlag, 1975.

    Google Scholar 

  7. P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal of Computing, 15(4):994–1003, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. H. Bennett. On Spectra. PhD thesis, Princeton University, 1962.

    Google Scholar 

  9. S. R. Buss. The boolean function value problem is in ALOGTIME. In 28th Annual IEEE Symposium on Foundations of Computer Science, pages 123–131, 1987.

    Google Scholar 

  10. A. K. Chandra, D. C. Kozen, and L. J. Stockmeyer. Alternation. Journal of the ACM, 28:114–133, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  11. A. Dawar, K. Doets, S. Lindell, and S. Weinstein. Elementary properties of finite ranks. Mathematical Logic Quarterly, 44:349–353, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Dawar and L. Hella. The expressive power of finitely many generalized quantifiers. Information and Computation, 123:172–184, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Dawar, S. Lindell, and S. Weinstein. First order logic, fixed point logic and linear order. In Computer Science Logic’ 95, volume 1092 of Lecture Notes in Computer Science, pages 161–177. Springer-Verlag, 1996.

    Google Scholar 

  14. L. Fortnow. Time-space tradeoffs for satisfiability. In 12th IEEE Conference in Computational Complexity, pages 52–60, 1997. To appear in Journal of Computer and System Sciences.

    Google Scholar 

  15. Y. Gurevich, N. Immerman, and S. Shelah. McColm’s conjecture. In 9th IEEE Symposium on Logic in Computer Science, pages 10–19, 1994.

    Google Scholar 

  16. Y. Gurevich and S. Shelah. Fixed-point extensions of first-order logic. Annals of Pure and Applied Logic, 32(3):265–280, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  17. N. Immerman. Relational queries computable in polynomial time. Information and Computation, 68:86–104, 1986.

    MATH  MathSciNet  Google Scholar 

  18. N. Immerman. Expressibility and parallel complexity. SIAM Journal of Computing, 18:625–638, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  19. N. Immerman and S. Landau. The complexity of iterated multiplication. Information and Computation, 116(1):103–116, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  20. N. Jones. Context-free languages and rudimentary attributes. Mathematical Systems Theory, 3:102–109, 1969.

    Article  MATH  MathSciNet  Google Scholar 

  21. N. D. Jones. Space-bounded reducibility among combinatorial problems. Journal of Computer and System Sciences, 11:68–85, 1975. Corrigendum: Journal of Computer and System Sciences 15:241, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  22. Ph. G. Kolaitis and M. Y. Vardi. Fixpoint logic vs. infinitary logic in finite-model theory. In 7th IEEE Symposium on Logic in Computer Science, pages 46–57, 1992.

    Google Scholar 

  23. S. Lindell. A purely logical characterization of circuit uniformity. In 7th IEEE Structure in Complexity Theory, pages 185–192, 1992.

    Google Scholar 

  24. R. J. Lipton and A. Viglas. On the complexity of SAT. In 40th Annual IEEE Symposium on Foundations of Computer Science, pages 459–464, 1999.

    Google Scholar 

  25. J. A. Makowsky. Invariant definability and P/poly. To appear in Lecture Notes in Computer Science, Proceedings of Computer Science Logic 1998, 1999.

    Google Scholar 

  26. Y. N. Moschovakis. Elementary Induction on Abstract Structures. North-Holland, 1974.

    Google Scholar 

  27. V. A. Nepomnjascii. Rudimentary predicates and Turing calculations. Soviet Math. Dokl., 11:1462–1465, 1970.

    Google Scholar 

  28. J. H. Reif. On threshold circuits and polynomial computation. In 2nd IEEE Structure in Complexity Theory, pages 118–123, 1987.

    Google Scholar 

  29. W. L. Ruzzo. On uniform circuit complexity. Journal of Computer and System Sciences, 22:365–383, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  30. V. Y. Sazonov. On bounded set theory. In Logic and Scientific Methods, pages 85–103. Kluwer Academic Publishers, 1997.

    Google Scholar 

  31. M. Sipser. Borel sets and circuit complexity. In 15th Annual ACM Symposium on the Theory of Computing, pages 61–69, 1983.

    Google Scholar 

  32. R. Smullyan. Theory of formal systems. In Annals of Mathematics Studies, volume 47. Princeton University Press, 1961.

    Google Scholar 

  33. L. Stockmeyer. The polynomial-time hierarchy. Theoretical Computer Science, 3:1–22, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  34. I. Wegener. The Complexity of Boolean Functions, pages 243–247. John Wiley & Sons, 1987.

    Google Scholar 

  35. C. Wrathall. Rudimentary predicates and relative computation. SIAM Journal of Computing, 7(2):194–209, 1978.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, c/ Jordi Girona Salgado, 1-3, Edif. C6., 08034, Barcelona, Spain

    Albert Atserias

Authors
  1. Albert Atserias
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science and Department of Biology, Boston College, Fulton Hall 410B, Chestnut Hill, MA, 02467, USA

    Peter G. Clote

  2. Mathematisches Institut, Ludwig-Maximilian-Universität München, Theresienstr. 39, 80333, München, Germany

    Helmut Schwichtenberg

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Atserias, A. (2000). The Descriptive Complexity of the Fixed-Points of Bounded Formulas. In: Clote, P.G., Schwichtenberg, H. (eds) Computer Science Logic. CSL 2000. Lecture Notes in Computer Science, vol 1862. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44622-2_11

Download citation

  • DOI: https://doi.org/10.1007/3-540-44622-2_11

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67895-3

  • Online ISBN: 978-3-540-44622-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation