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Theory of Computing Systems

, Volume 43, Issue 3–4, pp 410–424 | Cite as

Division Safe Calculation in Totalised Fields

  • J. A. BergstraEmail author
  • J. V. Tucker
Open Access
Article
First Online:

Abstract

A 0-totalised field is a field in which division is a total operation with 0−1=0. Equational reasoning in such fields is greatly simplified but in deriving a term one still wishes to know whether or not the calculation has invoked 0−1. If it has not then we call the derivation division safe. We propose three methods of guaranteeing division safe calculations in 0-totalised fields.

Keywords

Rational number Meadow Zero totalised field Elementary algebraic specification 
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References

  1. 1.
    Bergstra, J.A.: Elementary algebraic specifications of the rational function field. In: Beckmann, A., et al. (eds.) Logical Approaches to Computational Barriers. Proceedings of Computability in Europe 2006. Lecture Notes in Computer Science, vol. 3988, pp. 40–54. Springer, New York (2006) CrossRefGoogle Scholar
  2. 2.
    Bergstra, J.A., Tucker, J.V.: The completeness of the algebraic specification methods for data types. Inf. Control 54, 186–200 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bergstra, J.A., Tucker, J.V.: Initial and final algebra semantics for data type specifications: two characterisation theorems. SIAM J. Comput. 12, 366–387 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bergstra, J.A., Tucker, J.V.: Algebraic specifications of computable and semicomputable data types. Theor. Comput. Sci. 50, 137–181 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bergstra, J.A., Tucker, J.V.: Equational specifications, complete term rewriting systems, and computable and semicomputable algebras. J. ACM 42, 1194–1230 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bergstra, J.A., Tucker, J.V.: The data type variety of stack algebras. Ann. Pure Appl. Log. 73, 11–36 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bergstra, J.A., Tucker, J.V.: Elementary algebraic specifications of the rational complex numbers. In: Futatsugi, K., et al. (eds.) Algebra, Meaning and Computation. Goguen Festschrift. Lecture Notes in Computer Science, vol. 4060, pp. 459–475. Springer, New York (2006) CrossRefGoogle Scholar
  8. 8.
    Bergstra, J.A., Tucker, J.V.: The rational numbers as an abstract data type. J. ACM 54(2), Article 7 (April 2007), 25 pages Google Scholar
  9. 9.
    Calkin, N., Wilf, H.S.: Recounting the rationals. Am. Math. Mon. 107, 360–363 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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
  11. 11.
    Hirschfeld, Y.: Personal communication (August 2006) Google Scholar
  12. 12.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993) zbMATHGoogle Scholar
  13. 13.
    Kamin, S.: Some definitions for algebraic data type specifications. SIGLAN Not. 14(3), 28 (1979) CrossRefMathSciNetGoogle Scholar
  14. 14.
    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
  15. 15.
    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
  16. 16.
    Moss, L.: Simple equational specifications of rational arithmetic. Discret. Math. Theor. Comput. Sci. 4, 291–300 (2001) zbMATHMathSciNetGoogle Scholar
  17. 17.
    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
  18. 18.
    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
  19. 19.
    Terese, K.: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003) Google Scholar
  20. 20.
    Wechler, W.: Universal Algebra for Computer Scientists. EATCS Monographs in Computer Science. Springer, New York (1992) zbMATHGoogle Scholar
  21. 21.
    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 Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Informatics InstituteUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Department of Computer ScienceSwansea UniversitySwanseaUK

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