Skip to main content
SpringerLink
Log in

Elementary Algebraic Specifications of the Rational Function Field

  • Conference paper
Logical Approaches to Computational Barriers (CiE 2006)

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

Included in the following conference series:

Abstract

The elementary algebraic specifications form a small subset of the range of techniques available for algebraic specifications and are based on equational specifications with hidden functions and sorts and initial algebra semantics. General methods exist to show that all semicomputable and computable algebras can be characterised up to isomorphism by such specifications. Here we consider these specification methods for specific computable rational number arithmetics. In particular, we give an elementary equational specification of the 0-totalised rational function field ℚ0(X) with its degree operator as an auxiliary function.

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

Access this chapter

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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adamek, J., Hebert, M., Rosicky, J.: On abstract data types presented by multiequations. Theoretical Computer Science 275, 427–462 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bergstra, J.A., Tucker, J.V.: The completeness of the algebraic specification methods for data types. Information and Control 54, 186–200 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bergstra, J.A., Tucker, J.V.: Initial and final algebra semantics for data type specifications: two characterisation theorems. SIAM Journal on Computing 12, 366–387 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bergstra, J.A., Tucker, J.V.: Algebraic specifications of computable and semicomputable data types. Theoretical Computer Science 50, 137–181 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bergstra, J.A., Tucker, J.V.: Equational specifications, complete term rewriting systems, and computable and semicomputable algebras. Journal of ACM 42, 1194–1230 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bergstra, J.A., Tucker, J.V.: The data type variety of stack algebras. Annals of Pure and Applied Logic 73, 11–36 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bergstra, J.A., Tucker, J.V.: The rational numbers as an abstract data type (submitted)

    Google Scholar 

  8. Bergstra, J.A., Tucker, J.V.: Elementary algebraic specifications of the rational complex numbers (submitted)

    Google Scholar 

  9. Calkin, N., Wilf, H.S.: Recounting the rationals. American Mathematical Monthly 107, 360–363 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Contejean, E., Marche, C., Rabehasaina, L.: Rewrite systems for natural, integral, and rational arithmetic. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 98–112. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  11. Goguen, J.A.: Memories of ADJ. Bulletin of the European Association for Theoretical Computer Science 36, 96–102 (1989)

    MATH  Google Scholar 

  12. Goguen, J.A.: Tossing algebraic flowers down the great divide. In: Calude, C.S. (ed.) People and ideas in theoretical computer science, pp. 93–129. Springer, Singapore (1999)

    Google Scholar 

  13. Meseguer, J., Goguen, J.A.: Initiality, induction, and computability. In: Nivat, M. (ed.) Algebraic methods in semantics, pp. 459–541. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  14. Meseguer, J., Goguen, J.A.: Remarks on remarks on many-sorted algebras with possibly emtpy carrier sets. Bulletin of the EATCS 30, 66–73 (1986)

    MATH  Google Scholar 

  15. Goguen, J.A., Diaconescu, R.: An Oxford Survey of Order Sorted Algebra. Mathematical Structures in Computer Science 4, 363–392 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goguen, J.A., Thatcher, J.W., Wagner, E.G., Wright, J.B.: Initial algebra semantics and continuous algebras. Journal of ACM 24, 68–95 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goguen, J.A., Thatcher, J.W., Wagner, E.G.: An initial algebra approach to the specification, correctness and implementation of abstract data types. In: Yeh, R.T. (ed.) Current trends in programming methodology. IV. Data structuring, pp. 80–149. Prentice-Hall, Engelwood Cliffs (1978)

    Google Scholar 

  18. Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  19. Kamin, S.: Some definitions for algebraic data type specifications. SIGLAN Notices 14(3), 28 (1979)

    Article  Google Scholar 

  20. Khoussainov, B.: Randomness, computability, and algebraic specifications. Annals of Pure and Applied Logic, 1–15 (1998)

    Google Scholar 

  21. Khoussainov, B.: On algebraic specifications of abstract data types. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 299–313. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  22. Marongiu, G., Tulipani, S.: On a conjecture of Bergstra and Tucker. Theoretical Computer Science 67 (1989)

    Google Scholar 

  23. Meinke, K., Tucker, J.V.: Universal algebra. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, Mathematical Structures, vol. I, pp. 189–411. Oxford University Press, Oxford (1992)

    Google Scholar 

  24. Meseguer, J., Goguen, J.A.: Initiality, induction and computability. In: Nivat, M., Reynolds, J. (eds.) Algebraic methods in semantics, pp. 459–541. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  25. Moss, L.: Simple equational specifications of rational arithmetic. Discrete Mathematics and Theoretical Computer Science 4, 291–300 (2001)

    MathSciNet  MATH  Google Scholar 

  26. Moss, L., Meseguer, J., Goguen, J.A.: Final algebras, cosemicomputable algebras, and degrees of unsolvability. Theoretical Computer Science 100, 267–302 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stoltenberg-Hansen, V., Tucker, J.V.: Effective algebras. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, Semantic Modelling, vol. IV, pp. 357–526. Oxford University Press, Oxford (1995)

    Google Scholar 

  28. Stoltenberg-Hansen, V., Tucker, J.V.: Computable rings and fields. In: Griffor, E. (ed.) Handbook of Computability Theory, pp. 363–447. Elsevier, Amsterdam (1999)

    Chapter  Google Scholar 

  29. Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  30. Wagner, E.: Algebraic specifications: some old history and new thoughts. Nordic Journal of Computing 9, 373–404 (2002)

    MathSciNet  MATH  Google Scholar 

  31. Wechler, W.: Universal algebra for computer scientists. EATCS Monographs in Computer Science. Springer, Heidelberg (1992)

    Book  MATH  Google Scholar 

  32. Wirsing, M.: Algebraic specifications. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, Formal models and semantics, vol. B, pp. 675–788. North-Holland, Amsterdam (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Programming Research Group, University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, Netherlands

    J. A. Bergstra

  2. Applied Logic Group, Utrecht University, Heidelberglaan 8, 3584 CS, Utrecht, Netherlands

    J. A. Bergstra

Authors
  1. J. A. Bergstra
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bergstra, J.A. (2006). Elementary Algebraic Specifications of the Rational Function Field. 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_5

Download citation

  • DOI: https://doi.org/10.1007/11780342_5

  • 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)

Publish with us

Policies and ethics

Search

Navigation