Advertisement

On the role of category theory in the area of algebraic specifications

  • H. Ehrig
  • M. Große-Rhode
  • U. Wolter
Invited Presentations
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1130)

Abstract

The paper summarizes the main concepts and paradigms of category theory and explores some of their applications to the area of algebraic specifications.

In detail we discuss different approaches to an abstract theory of specification logics. Further we present a uniform framework for developing particular specification logics. We make use of ‘classifying categories’, to present categories of algebras as functor categories and to obtain necessary basic results for particular specification logics in a uniform manner. The specification logics considered are: equational logic for total algebras, conditional equational logic for partial algebras, and rewrite logic for concurrent systems.

Keywords

Category Theory Universal Property Forgetful Functor Partial Algebra Specification Logic 
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. [Abr93]
    S. Abramsky. Interaction categories (extended abstract). In J.L. Burn, S.J. Gay, and M.D. Ryan, editors, Theory and Formal Methods 1993: Proc. First Imperial College Department of Computing Workshop on Theory and Formal Methods, pages 57–70. Springer Verlag Workshops in Computer Science, 1993.Google Scholar
  2. [Abr94]
    S. Abramsky. Interaction categories and communicating sequential processes. In A.W. Roscoe, editor, A Classical Mind: Essays in Honour of C.A.R.Hoare. Prentice Hall International, 1994.Google Scholar
  3. [BW85]
    M. Barr and C. Wells. Toposes, triples, and theories. Springer Verlag, 1985.Google Scholar
  4. [BW90]
    M. Barr and C. Wells. Category Theory for Computing Science. Prentice Hall, 1990.Google Scholar
  5. [CGW95]
    I. Claßen, M. Große-Rhode, and U. Wolter. Categorical concepts for parameterized partial specifications. Math. Struct. in Comp. Science, 5(2):153–188, 1995.Google Scholar
  6. [CM93]
    M. Cerioli and J. Meseguer. May I Borrow Your Logic? Technical report, SRI International, Menlo Park, 1993.Google Scholar
  7. [EG94]
    H. Ehrig and M. Große-Rhode. Functorial theory of parameterized specifications in a general specification framework. TCS, (135):221–266, 1994.Google Scholar
  8. [Ehr89]
    H. Ehrig. Algebraic specification of modules and modular software systems within the framework of specification logics. Technical Report 89-17, TU Berlin, 1989.Google Scholar
  9. [EM45]
    S. Eilenberg and S. MacLane. General theory of natural equivalences. Trans. Am. Math. Soc., 58:231–294, 1945.Google Scholar
  10. [EM85]
    H. Ehrig and B. Mahr. Fundamentals of Algebraic Specification 1: Equations and Initial Semantics, volume 6 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1985.Google Scholar
  11. [EM90]
    H. Ehrig and B. Mahr. Fundamentals of Algebraic Specification 2: Module Specifications and Constraints, volume 21 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1990.Google Scholar
  12. [EM95]
    H. Ehrig and B. Mahr. A decade of TAPSOFT: Aspects of progress and prospects in theory and practice of software development. In Proc. TAP-SOFT'95, pages 3–24. LNCS 915, 1995.Google Scholar
  13. [Fre72]
    P. Freyd. Aspects of topoi. Bull. Austr. Math. Soc., (7):1–72, 1972.Google Scholar
  14. [Fre73]
    P. Freyd. Aspects of topoi, corrections. Bull. Austr. Math. Soc., (8):467–480, 1973.Google Scholar
  15. [FS87]
    J. Fiadeiro and A. Sernadas. Structuring theories on consequence. In D. Sannella and A. Tarlecki, editors, Recent Trend in Data Type Specification, LNCS 332, pages 221–235. Springer Verlag, 1987.Google Scholar
  16. [GB84]
    J. A. Goguen and R. M. Burstall. Introducing institutions. In Proc. Logics of Programming Workshop, LNCS 164, pages 221–256. Carnegie-Mellon, Springer, 1984.Google Scholar
  17. [GB85]
    J. Goguen and R.M. Burstall. Institutions: Abstract model theory for computer science. Technical Report CSLI-85-30, Stanford University, 1985.Google Scholar
  18. [GB92]
    J. Goguen and R.M. Burstall. Institutions: Abstract model theory for specification and programming. Journal of the ACM, 39(1):95–146, 1992.Google Scholar
  19. [Gog91]
    J. Goguen. A categorical manifesto. Mathematical Structures in Computer Science, 1(1):49–67, 1991.Google Scholar
  20. [GTW78]
    J. A. Goguen, J. W. Thatcher, and E. G. Wagner. An initial algebra approach to the specification, correctness and implementation of abstract data types. In R. Yeh, editor, Current Trends in Programming Methodology IV: Data Structuring, pages 80–144. Prentice Hall, 1978.Google Scholar
  21. [HP89]
    J. M. E. Hyland and A. M. Pitts. The theory of constructions: Categorical semantics and topos-theoretic models. In Categories in Computer Science and Logic, pages 137–200. AMS-IMS-SIAM Joint Summer Research Conference, University of Colorado, Boulder, 1989.Google Scholar
  22. [Joh77]
    P. T. Johnstone. Topos Theory. Academic Press, 1977.Google Scholar
  23. [Kel74]
    G. M. Kelly. Review of the elements of 2-categories. Lecture Notes in Mathematics, (420):74–103, 1974.Google Scholar
  24. [KR72]
    H. Kaphengst and H. Reichel. Operative Theorien und Kategorien von operativen Systemen. In Studien zur Algebra und ihren Anwendungen, volume 16, pages 41–56. Akademie-Verlag, 1972.Google Scholar
  25. [KR77]
    A. Kock and G. E. Reyes. Doctrines in categorical logic. In J. Barwise, editor, Handbook of Mathematical Logic, pages 283–313. Elsevier Science Publishers B. V., North Holland, 1977.Google Scholar
  26. [Law63]
    F. W. Lawvere. Functorial semantics of algebraic theories. In Proc. National Academy of Science, U.S.A., 50, pages 869–872. Columbia University, 1963.Google Scholar
  27. [LS86]
    J. Lambek and P.J. Scott. Introduction to higher order categorical logic. Cambridge University Press, 1986.Google Scholar
  28. [Mac71]
    S. Mac Lane. Categories for the Working Mathematician, volume 5 of Graduate Texts in Mathematics. Springer, New York, 1971.Google Scholar
  29. [Man76]
    E. Manes. Algebraic Theories. Springer Verlag, 1976.Google Scholar
  30. [Mes89]
    J. Meseguer. General logics. In H.-D. Ebbinghaus et. al., editor, Logic colloquium '87, pages 275–329. Elsevier Science Publishers B. V.,North Holland, 1989.Google Scholar
  31. [Mes90]
    J. Meseguer. Rewriting as a unified model of concurrency. Technical Report SRI-CSL-90-02, SRI International, Computer Science Laboratory, 1990.Google Scholar
  32. [Mes92]
    J. Meseguer. Conditional rewriting logic as a unified model of concurrency. TCS, 96:73–155, 1992.Google Scholar
  33. [MG85]
    J. Meseguer and J. A. Goguen. Initiality, induction, and computability. In M. Nivat and J. Reynolds, editors, Algebraic Methods in Semantics, chapter 14, pages 459–541. Cambridge University Press, 1985.Google Scholar
  34. [Poi85]
    A. Poigné. Algebra categorically. In D. Pitt, S. Abramsky, A. Poigné, and D. Rydeheard, editors, Category Theory and Computer Programming, pages 77–102. LNCS 240, Springer, 1985.Google Scholar
  35. [Rei87]
    H. Reichel. Initial Computability, Algebraic Specifications, and Partial Algebras. Oxford University Press, Oxford, 1987.Google Scholar
  36. [See84]
    R. A. G. Seely. Locally cartesian closed categories and type theory. Mathematical Proceedings Cambridge Philosophical Society, (95):33–48, 1984.Google Scholar
  37. [See87]
    R. A. G. Seely. Categorical semantics for higher order polymorphic lambda calculus. Journal of Symbolic Logic, 52(4):969–989, December 1987.Google Scholar
  38. [Tar96]
    A. Tarlecki. Moving between logical systems. In M. Haveraaen, O. Owe, and O.-J. Dahl, editors, Recent Trends in Data Type Specifications. WADT11. Oslo Norway, September 1995. LNCS (this volume), Springer, 1996.Google Scholar
  39. [Wec92]
    W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer, Berlin, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • H. Ehrig
    • 1
  • M. Große-Rhode
    • 1
  • U. Wolter
    • 1
  1. 1.FB InformatikTechnische Universität BerlinBerlinGermany

Personalised recommendations