Advertisement

The formal description of data types using sketches

  • Charles Wells
  • Michael Barr
Part VI New Directions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 298)

Abstract

This paper is an exposition of the basic ideas of the mathematical theory of sketches and a detailed description of some of the ways in which this theory can be used in theoretical computer science to specify datatypes. In particular, this theory provides a convenient way of introducing datatypes which have variants, for example in case of errors or nil pointers. The semantics is a generalization of initial algebra semantics which in some cases allows initial algebras depending on a parameter such as a bound for overflow.

Keywords

Commutative Diagram Context Free Grammar Derivation Tree Finite Limit Abstract Data Type 
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.

14. Bibliography

  1. M. Barr, Models of sketches, Cahiers de Topologie et Géometrie Différentielle, 27 (1986), 93–107.Google Scholar
  2. M. Barr and C. Wells. Toposes, Triples and Theories. Springer-Verlag, 1985.Google Scholar
  3. M. Barr and C. Wells. Category Theory for Computer Scientists, in preparation.Google Scholar
  4. A. Bastiani C. Ehresmann, Categories of sketched structures. Cahiers de Topologie et Géometrie Différentielle 10 (1968), 104–213.Google Scholar
  5. R. M. Burstall, Electronic category theory, in Mathematical Foundations of Computer Science. Springer Lecture Notes in Computer Science 88 (1980).Google Scholar
  6. N. Chomsky, M. P. Schützenberger, The algebraic theory of context-free languages, in P. Braffort, D. Hirschberg, eds. Computer Programming and Formal Systems, North-Holland, 1963.Google Scholar
  7. Y. Diers, Catégories Localizables. Thèse de doctorat, Université de Paris, 1977.Google Scholar
  8. J. Donahue and A. Demers, Data types are values. To appear in ACM Transactions on Programming Languages and Systems.Google Scholar
  9. H. Ehrig, J. W. Thatcher, P. Lucas and S. N. Zilles, Denotational and initial algebra semantics of the algebraic specification language Look. Preprint: Technische Universität Berlin, 1982.Google Scholar
  10. H. Ehrig, H.-J. Kreowski, J. Thatcher, E. Wagner and J. Wright, Parameter passing in algebraic specification languages. Theoretical Computer Science 28 (1984), 45–81.CrossRefGoogle Scholar
  11. H. Ehrig and B. Mahr, Fundamentals of algebraic specifications I. Springer-Verlag, 1985.Google Scholar
  12. M. Fourman and S. Vickers, Theories as categories, in Category Theory and Computer Programming, Springer Lecture Notes in Computer Science 240, Springer-Verlag, 1986.Google Scholar
  13. J. A. Goguen, Abstract errors for abstract data types, In E. J. Neuhold, ed. Formal Description of Programming Concepts, North-Holland, 1978.Google Scholar
  14. J. A. Goguen and J. Meseguer, Eqlog: equality, types and generic modules for logic programming. To appear in DeGroot and Lindstrom, eds., Functional and Logic Programming, Prentice-Hall, 1985a.Google Scholar
  15. J. Goguen and J. Meseguer, Order sorted algebra I: partial and overloaded operators, errors and inheritance. Preprint, SRI International, Menlo Park, CA 94025, 1985b.Google Scholar
  16. J. Goguen, J. W. Thatcher, E. G. Wagner and J. B. Wright, Initial algebra semantics and continuous algebras. J. ACM, 24 (1977), 68–95.CrossRefGoogle Scholar
  17. J. Gray, Categorical aspects of parametric data types. Preprint: University of Illinois, 1985.Google Scholar
  18. R. Guitart, On the geometry of computations. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 (1986), 107–136.Google Scholar
  19. R. Guitart and C. Lair, Calcul syntaxique des modèles et calcul des formules internes. Diagrammes, 4 (1980), 1–106.Google Scholar
  20. J. Lambek and P. Scott, Cartesian Closed Categories and λ-Calculus. Cambridge Studies in Advanced Mathematics 7. Cambridge University Press, 1986.Google Scholar
  21. S. Mac Lane, Categories for the Working Mathematician. Graduate Texts in Mathematics 5, Springer-Verlag, 1971.Google Scholar
  22. J. Makowsky, Why Horn formulas matter in computer science: initial structures and generic examples. Technical Report #329, Department of Computer Science, Technion, Haifa, Israel, 1984.Google Scholar
  23. P. Mateti and F. Hunt, Precision descriptions of software designs: an example. IEEE Compsac, 1985, 130–136.Google Scholar
  24. C. McLarty, Left exact logic. Journal of Pure and Applied Algebra, 41 (1986), 63–66.CrossRefGoogle Scholar
  25. J. Meseguer and J. Goguen, Initiality, induction and computability, in M. Nivat and J. C. Reynolds, eds., Algebraic Methods in Semantics, Cambridge University Press, 1985.Google Scholar
  26. D. E. Rydehead and R. M. Burstall, The unification of terms: a category-theoretic algorithm, in Category Theory and Computer Programming, Springer Lecture Notes in Computer Science 240. Springer-Verlag, 1986.Google Scholar
  27. J. W. Thatcher, E. G. Wagner and J. B. Wright, Specification of abstract data types using conditional axioms. (Extended abstract). IBM T. J. Watson Research Center Research Report RC 6214 (#26679), 1976.Google Scholar
  28. J. W. Thatcher, E. G. Wagner and J. B. Wright, Data type specification: parametrization and the power of specification techniques. ACM Transactions on Programming Languages and Systems, Vol. 4 no. 4, October, 1982.Google Scholar
  29. H. Volger, On theories which admit initial structures. Preprint: Universität Tübingen, 1985.Google Scholar
  30. E. Wagner, S. Bloom and J. W. Thatcher, Why algebraic theories? in M. Nivat and J. C. Reynolds, eds., Algebraic Methods in Semantics, Cambridge University Press, 1985.Google Scholar
  31. S. N. Zilles, P. Lucas and J. W. Thatcher, A look at algebraic specifications. IBM T. J. Watson Research Center Research Report RJ 3568 (#41985), 1982.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Charles Wells
    • 1
  • Michael Barr
    • 2
  1. 1.Department of Mathematics and StatisticsCase Western Reserve UniversityClevelandUSA
  2. 2.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada

Personalised recommendations