Domain algebras

  • Peter Dybjer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 172)


This paper proposes a way of relating domain-theoretic and algebraic interpretations of data types. It is different from Smyth, Plotkin, and Lehmann's T-algebra approach, and in particular the notion of homomorphism between higher-order algebras is not restricted in the same way, so that the usual initiality theorems of algebraic semantics, including one for inequational varieties, hold. Domain algebras are defined in terms of concepts from elementary category theory using Lambek's connection between cartesian closed categories and the typed λ-calculus. To this end axioms and inference rules for a theory of domain categories are given. Models of these are the standard categories of domains, such as Scott's information systems and Berry and Curien's sequential algorithms on concrete data structures. The set of axioms and inference rules are discussed and compared to the PPλ-logic of the LCF-system.


Inference Rule Denotational Semantic Algebraic Semantic Domain Equation Quotient Category 
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.


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7. References

  1. ADJ (= Goguen, J.A., Thatcher, J.W., Wagner, E.G., Wright, J.B.) (1976), "Rational Algebraic Theories and Fixed-Point Solutions", Proceedings 17th IEEE Symposium on Foundations of Computer Science, Houston, Texas, pp 147–158Google Scholar
  2. ADJ (= Goguen, J.A., Thatcher, J.W., Wagner, E.G., Wright, J.B.) (1977), "Initial Algebra Semantics and Continuous Algebras", JACM 24, 1, pp 68–95Google Scholar
  3. ADJ (= Goguen, J.A., Thatcher, J.W., Wagner, E.G.) (1978), "An Initial Algebra Approach to the Specification, Correctness and Implementation of Abstract Data Types", in "Current Trends in Programming Methodology", R.Yeh ed., Prentice-HallGoogle Scholar
  4. Benabou, J. (1968), "Structures algebraic dans les categories", Cahiers de topologie et geometrie differentiell 10, pp 1–24Google Scholar
  5. Berry, G. (1979), "Modèles complètement adéquats et stables des lambda-calculs typés", Thèse de doctorat d'etat ès sciences mathematiques, l'université Paris VIIGoogle Scholar
  6. Berry, G. (1981a), "On the Definition of Lambda Calculus Models", Proceedings International Colloquium on Formalization of Programming Concepts, Lecture Notes in Computer Science 107 (Springer Verlag, Berlin), pp 218–230Google Scholar
  7. Berry, G. (1981b), "Some Syntactic and Categorical Constructions of Lambda-Calculus Models", Rapport INRIA 80Google Scholar
  8. Berry, G. and Curien, P.L. (1982), "Sequential Algorithms on Concrete Data Structures", Theoretical Computer Science 20, pp 265–321Google Scholar
  9. Bloom, S.L. (1976), "Varieties of Ordered Algebras", Journal of Computer and System Sciences 13, pp 200–212Google Scholar
  10. Burstall, R.M. and Goguen, J.A. (1977), "Putting Theories Together to Make Specifications", Proceedings of the 5th IJCAI, pp 1045–1058Google Scholar
  11. Burstall, R.M. and Landin, P.J. (1969), "Programs and their Proofs: An Algebraic Approach", Machine Intelligence 4, Edinburgh University Press, pp 17–44Google Scholar
  12. Cohn, A.J. (1978), "High Level Proofs in LCF", Report CSR-35-78, Department of Computer Science, University of EdinburghGoogle Scholar
  13. Courcelle, B. and Nivat, M. (1976) "Algebraic Families of Interpretations", Proceedings of the 17th FOCS, HoustonGoogle Scholar
  14. Dybjer, P. (1983), "Category-Theoretic Logics and Algebras of Programs", Ph.D.thesis, CTHGoogle Scholar
  15. Ehrich, H.D. and Lipeck, U. (1983), "Algebraic Domain Equations", Theoretical Computer Science 27, pp 167–196Google Scholar
  16. Goguen, J.A. and Meseguer, J. (1981), "Completeness of Many-Sorted Equational Logic", SIGPLAN Notices 16, pp 24–32Google Scholar
  17. Guessarian, I. (1982) "Survey on some Classes of Interpretations and some of their applications", Laboratoire Informatique Theorieque et Programmation, 82–46, Univ. Paris VIIGoogle Scholar
  18. Karlsson, K. and Petersson, K., (eds) (1983), "Workshop on Semantics of Programming Languages", CTHGoogle Scholar
  19. Lambek, J. (1972), "Deductive Systems and Categories III", Proceedings Dalhousie Conference on Toposes, Algebraic Geometry and Logic, Lecture Notes in Mathematics 274, Springer-Verlag, pp 57–82Google Scholar
  20. Lambek, J. (1980), "From Lambda-Calculus to Cartesian Closed Categories", in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J.P. Seldin and J.R. Hindley (eds.), pp 376–402Google Scholar
  21. Lambek, J. and Scott, P.J. (1980), "Intuitionist Type Theory and the Free Topos", Journal of Pure and Applied Algebra 19, pp 215–257Google Scholar
  22. Lehmann, D.J. and Smyth, M.B. (1981), "Algebraic Specification of Data Types: A Synthetic Approach", Mathematical Systems Theory 14, pp 97–139Google Scholar
  23. MacLane, S. (1971), "Categories for the Working Mathematician", Springer-Verlag, BerlinGoogle Scholar
  24. Martin-Löf, P. (1979), "Constructive Mathematics and Computer Programming", 6th International Congress for Logic, Methodology and Philosophy of Science, HannoverGoogle Scholar
  25. Meseguer, J. (1977) "On Order-Complete Universal Algebra and Enriched Functorial Semantics", Proceedings of FCT, Lecture Notes in Computer Science 56 (Springer-Verlag, Berlin)Google Scholar
  26. Milner, R. (1979), "Flow Graphs and Flow Algebras", JACM 26, pp 794–818Google Scholar
  27. Milner, R., Morris, L., Newey, M. (1975), "A Logic for Computable Functions with Reflexive and Polymorphic Types", Proc. Conference on Proving and Improving Programs, Arc-et-SenansGoogle Scholar
  28. Morris, F.L. (1973), "Advice on Structuring Compilers and Proving them Correct", Proceedings, ACM Symposium on Principles of Programming Languages, Boston, pp 144–152Google Scholar
  29. Mosses, P.D. (1982), "Abstract Semantic Algebras!", DAIMI Report PB-145, Computer Science Department, Aarhus UniversityGoogle Scholar
  30. Obtułowicz, A. (1977), "Functorial Semantics of the λ-βη-calculus" in Proceedings of FCT, Lecture Notes in Computer Science 56 (Springer-Verlag, Berlin)Google Scholar
  31. Parsaye-Ghomi, K. (1982), "Higher Order Abstract Data Types", Ph.D. thesis, Department of Computer Science, UCLAGoogle Scholar
  32. Plotkin, G.D. (1976), "LCF Considered as a Programming Language", Theoretical Computer Science 5, pp 223–256Google Scholar
  33. Plotkin, G.D. (1980), "Domains", Edinburgh CS Dept, lecture notes.Google Scholar
  34. Poigne, A. (1983), "On Semantic Algebras Higher Order Structures", Forschungsbericht 156, Abt. Informatik, Universitat DortmundGoogle Scholar
  35. Scott, D.S. (1980), "Relating Theories of the Lambda-Calculus", in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J.P. Seldin and J.R. Hindley (eds), pp 404–450Google Scholar
  36. Scott, D.S. (1981), "Lectures on a Mathematical Theory of Computation", Technical Monograph PRG-19, Oxford University Computing LaboratoryGoogle Scholar
  37. Scott, D.S. (1982), "Domains for Denotational Semantics", Proceedings 9th International Colloquium on Automata, Languages and Programming, Aarhus, Springer-Verlag Lecture Notes in Computer Science, pp 577–613Google Scholar
  38. Smyth, M.B. (1978), "Effectively Given Domains", Theoretical Computer Science 5Google Scholar
  39. Smyth, M.B. (1982), "The Largest Cartesian Closed Category of Domains", Report CSR 108–82, Computer Science Department, University of EdinburghGoogle Scholar
  40. Smyth, M.B. and Plotkin, G.D. (1982), "The Category Theoretic Solution of Recursive Domain Equations", SIAM Journal on Computing 11Google Scholar
  41. Streicher, T. (1983), "Definability in Scott Domains", in Proc. Workshop on Semantics of Programming Languages, CTHGoogle Scholar
  42. Thatcher, J.W., Wagner, E.G., Wright, J.B. (1981), "More on Advice on Structuring Compilers and Proving them Correct", Theoretical Computer Science 15, pp 223–249Google Scholar
  43. Wand, M. (1977), "Fixed-Point Constructions in Order-Enriched Categories", Technical Report 23, Computer Science Department, Indiana University, BloomingtonGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Peter Dybjer
    • 1
  1. 1.Programming Methodology Group Department of Computer SciencesChalmers Technical UniversityGothenburgSweden

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