Advertisement

Elementary Algebraic Specifications of the Rational Complex Numbers

  • Jan A. Bergstra
  • John V. Tucker
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4060)

Abstract

From the range of techniques available for algebraic specifications we select a core set of features which we define to be the elementary algebraic specifications. These include equational specifications with hidden functions and sorts and initial algebra semantics. We give an elementary equational specification of the field operations and conjugation operator on the rational complex numbers ℚ(i) and discuss some open problems.

Keywords

Data Type Rational Number Universal Algebra Galois Theory Rational Complex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adamek, J., Hebert, M., Rosicky, J.: On abstract data types presented by multiequations. Theoretical Computer Science 275, 427–462 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bergstra, J.A., Tucker, J.V.: The completeness of the algebraic specification methods for data types. Information and Control 54, 186–200 (1982)MATHCrossRefMathSciNetGoogle Scholar
  3. 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)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bergstra, J.A., Tucker, J.V.: Algebraic specifications of computable and semicomputable data types. Theoretical Computer Science 50, 137–181 (1987)MATHCrossRefMathSciNetGoogle Scholar
  5. 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)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bergstra, J.A., Tucker, J.V.: The data type variety of stack algebras. Annals of Pure and Applied Logic 73, 11–36 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bergstra, J.A., Tucker, J.V.: The rational numbers as an abstract data type (submitted)Google Scholar
  8. 8.
    Bergstra, J.A., Tucker, J.V.: On fields and meadows of finite characteristic (submitted)Google Scholar
  9. 9.
    Bergstra, J.A.: Elementary algebraic specifications of the rational function field. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 40–54. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Calkin, N., Wilf, H.S.: Recounting the rationals. American Mathematical Monthly 107, 360–363 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    Edwards, H.: Galois theory. Springer, Heidelberg (1984)MATHGoogle Scholar
  13. 13.
    Goguen, J.A.: Memories of ADJ. Bulletin of the European Association for Theoretical Computer Science 36, 96–102 (1989)Google Scholar
  14. 14.
    Goguen, J.A.: A categorical manifesto. Mathematical Structures in Computer Science, vol. 1, pp. 49–67 (1991)Google Scholar
  15. 15.
    Goguen, J.A.: Tossing algebraic flowers down the great divide. In: Calude, C.S. (ed.) People and ideas in theoretical computer science, Singapore, pp. 93–129. Springer, Heidelberg (1999)Google Scholar
  16. 16.
    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
  17. 17.
    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)MATHGoogle Scholar
  18. 18.
    Goguen, J.A., Diaconescu, R.: An Oxford Survey of Order Sorted Algebra. Mathematical Structures in Computer Science 4, 363–392 (1994)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    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)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    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, Englewood Cliffs (1978)Google Scholar
  21. 21.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)MATHCrossRefGoogle Scholar
  22. 22.
    Kamin, S.: Some definitions for algebraic data type specifications. SIGLAN Notices 14(3), 28 (1979)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Khoussainov, B.: Randomness, computability, and algebraic specifications. Annals of Pure and Applied Logic, 1–15 (1998)Google Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    Klop, J.W.: Term rewriting systems. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science Mathematical Structures, vol. 2, pp. 1–116 (1992)Google Scholar
  26. 26.
    Marongiu, G., Tulipani, S.: On a conjecture of Bergstra and Tucker. Theoretical Computer Science 67, 87–97 (1989)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Meinke, K.: Universal algebra in higher types. Theoretical Computer Science 100, 385–417 (1992)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    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
  29. 29.
    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
  30. 30.
    Moss, L.: imple equational specifications of rational arithmetic. Discrete Mathematics and Theoretical Computer Science 4, 291–300 (2001)MATHMathSciNetGoogle Scholar
  31. 31.
    Moss, L., Meseguer, J., Goguen, J.A.: Final algebras, cosemicomputable algebras, and degrees of unsolvability. Theoretical Computer Science 100, 267–302 (1992)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Mosses, P.: Unified algebras and institutions. In: Proceedings 4th Logic in Computer Science, pp. 304–312. IEEE Press, Los Alamitos (1989)Google Scholar
  33. 33.
    Rees, D., Stephenson, K., Tucker, J.V.: The algebraic structure of interfaces. Science of Computer Programming 49, 47–88 (2003)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Stewart, I.: Galois theory. Chapman and Hall, Boca Raton (1973)MATHGoogle Scholar
  35. 35.
    Stoltenberg-Hansen, V., Tucker, J.V.: Effective algebras. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science Semantic Modelling, pp. 357–526. Oxford University Press, Oxford (1995)Google Scholar
  36. 36.
    Stoltenberg-Hansen, V., Tucker, J.V.: Computable rings and fields. In: Griffor, E. (ed.) Handbook of Computability Theory, pp. 363–447. Elsevier, Amsterdam (1999)CrossRefGoogle Scholar
  37. 37.
    Terese, Term Rewriting Systems, Cambridge Tracts in Theoretical Computer Science 55, Cambridge University Press, Cambridge (2003)Google Scholar
  38. 38.
    Wagner, E.: Algebraic specifications: some old history and new thoughts. Nordic Journal of Computing 9, 373–404 (2002)MATHMathSciNetGoogle Scholar
  39. 39.
    Wechler, W.: Universal algebra for computer scientists. In: EATCS Monographs in Computer Science, Springer, Heidelberg (1992)Google Scholar
  40. 40.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jan A. Bergstra
    • 1
  • John V. Tucker
    • 2
  1. 1.Informatics InstituteUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Computer ScienceUniversity of Wales SwanseaSingleton Park, SwanseaUnited Kingdom

Personalised recommendations