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Toward formal development of programs from algebraic specifications: Implementations revisited

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The program development process is viewed as a sequence of implementation steps leading from a specification to a program. Based on an elementary notion of refinement, two notions of implementation are studied: constructor implementations which involve a construction “on top of” the implementing specification, and abstractor implementations which additionally provide for abstraction from some details of the implemented specification. These subsume most formal notions of implementation in the literature. Both kinds of implementations satisfy a vertical composition and a (modified) horizontal composition property. All the definitions and results are shown to generalise to the framework of an arbitrary institution, and a way of changing institutions during the implementation process is introduced. All this is illustrated by means of simple concrete examples.

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References

  1. Arbib, M.A., Manes, E.G.: Arrow, Structures and Functors: the Categorical Imperative. New York, London: Academic Press 1975

    Google Scholar 

  2. Astesiano, E., Mascari, G.F., Reggio, G., Wirsing, M.: On the parameterized algebraic specification of concurrent systems. Proc. 10th Colloq. on Trees in Algebra and Programming, Joint Conf. on Theory and Practice of Software Development (TAPSOFT). Lect. Notes Comput. Sci. Vol. 185, pp. 342–358. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  3. Astesiano, E., Reggio, G.: A unifying viewpoint for the constructive specification of cooperation, concurrency and distribution. Quaderni CNET no. 115, ETS Pisa (1983)

    Google Scholar 

  4. Barwise, K.J.: Axioms for abstract model theory. Ann. Math. Log. 7, 221–265 (1974)

    Google Scholar 

  5. Bauer, F.L., Broy, M., Dosch, W., Gnatz, R., Geiselbrechtinger, F., Hesse, W., Krieg-Brückner, B., Laut, A., Matzner, T., Möller, B., Partsch, H., Pepper, P., Samelson, K., Wirsing, M., Wössner, H.: Report on a wide spectrum language for program specification and development. Report TUM-I8104, Technische Univ. München (1981). See also: The Wide Spectrum Language CIP-L. Lect. Notes Comput. Sci. Vol. 183. Berlin, Heidelberg, New York: Springer 1985

  6. Bauer, F.L. Broy, M., Dosch, W., Gnatz, R., Krieg-Brückner, B., Laut, A., Luckmann, M., Matzner, T., Möller, B., Partsch, H., Papper, P., Samelson, K., Steinbrüggen, R., Wirsing, M., Wössner, H.: Programming in a wide spectrum language: a collection of examples. Sci. Comput. Prog. 1, 73–114 (1981)

    Google Scholar 

  7. Bergstra, J.A., Meyer, J.J.: I/O computable data structures. SIGPLAN Notices 16, 27–32 (1981)

    Google Scholar 

  8. Bernot, G., Bidoit, M., Choppy, C.: Abstract implementations and correctness proofs. Proc. Symp. on Theoretical Aspects of Computer Science, Saarbrücken. Lect. Notes Comput. Sci. Vol. 210, pp. 236–251. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  9. Bloom, S.L., Wagner, E.G.: Many-sorted theories and their algebras with some applications to data types. In: Algebraic Methods in Semantics. Nivat, M., Reynolds, J.C. (eds.), pp. 133–168. Cambridge: University Press 1985

    Google Scholar 

  10. Broy, M., Möller, B., Pepper, P., Wirsing, M.: Algebraic implementations preserve program correctness. Sci. Comput. Prog. 7, 35–53 (1986)

    Google Scholar 

  11. Broy, M., Wirsing, M.: Partial abstract types. Acta Inf. 18, 47–64 (1982)

    Google Scholar 

  12. Burmeister, P.: A Model Theoretic Approach to Partial Algebras. Berlin: Akademie-Verlag 1986

    Google Scholar 

  13. Burstall, R.M., Goguen, J.A.: Putting together theories to make specifications. Proc. 5th Intl. Joint Conf. on Artificial Intelligence, pp. 1045–1058. Cambridge 1977

  14. Burstall, R.M., Goguen, J.A.: The semantics of Clear, a specification language. Proc. of Advanced Course on Abstract Software Specifications, Copenhagen. Lect. Notes Comput. Sci. Vol. 86, pp. 292–332. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  15. Burstall, R.M., Goguen, J.A.: Algebras, theories and freeness: an introduction for computer scientists. Proc. 1981 Marktoberdorf NATO Summer School. Reidel (1982)

  16. Burstall, R.M., MacQueen, D.B., Sannella, D.T.: HOPE: an experimental applicative language. Proc. 1980 LISP Conference, pp. 136–143. Stanford 1980

  17. Dubois, E., Levy, N., Souquieres, J.: Formalising restructuring operators in a specification process. Proc. ESEC'87, Strasbourg 1987

  18. de Nicola, R., Hennessy, M.C.B.: Testing equivalences for processes. Theor. Comput. Sci. 34, 83–133 (1984)

    Google Scholar 

  19. Ehrich, H.-D.: On realization and implementation. Proc. 10th Intl. Symp. on Mathematical Foundations of Computer Science, Strbske Pleso, Czechoslovakia. Lect. Notes Comput. Sci. Vol. 118, pp. 271–280. Berlin, Heidelberg, New York: Springer 1981

    Google Scholar 

  20. Ehrich, H.-D.: On the theory of specification, implementation, and parametrization of abstract data types. J. Assoc. Comput. Mach. 29, 206–227 (1982)

    Google Scholar 

  21. Ehrig, H., Fey, W., Hansen, H.: ACT ONE: an algebraic specification language with two levels of semantics. Report Nr. 83-03, Institut für Software und Theoretische Informatik, Technische Univ. Berlin 1983

    Google Scholar 

  22. Ehrig, H., Kreowski, H.-J.: Parameter passing commutes with implementation of parameterized data types. Proc. 9th Intl. Colloq. on Automata, Languages and Programming, Aarhus, Denmark. Lect. Notes Comput. Sci. Vol. 140, pp. 197–211. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  23. Ehrig, H., Kreowski, H.-J., Mahr, B., Padawitz, P.: Algebraic implementation of abstract data types. Theor. Comput. Sci. 20, 209–263 (1982)

    Google Scholar 

  24. Ehrig, H., Kreowski, H.-J., Thatcher, J.W., Wagner, E.G., Wright, J.B.: Parameterized data types in algebraic specification languages (short version). Proc. 7th Intl. Colloq. on Automata, Languages and Programming, Noordwijkerhout, Netherlands. Lect. Notes Comput. Sci. Vol. 85, pp. 157–168. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  25. Ehrig, H., Mahr, B.: Fundamentals of Algebraic Specification I: Equations and Initial Semantics. EATCS Monographs on Theoretical Computer Science. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  26. Ehrig, H., Thatcher, J.W., Lucas, P., Zilles, S.N.: Denotational and initial algebra semantics of the algebraic specification language LOOK. Draft report, IBM research 1982

  27. Ehrig, H., Wagner, E.G., Thatcher, J.W.: Algebraic specifications with generating constraints. Proc. 10th Intl. Colloq. on Automata, Languages and Programming, Barcelona. Lect. Notes Comput. Sci. Vol. 154, pp. 188–202. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  28. Futatsugi, K., Goguen, J.A., Jouannaud, J.-P., Meseguer, J.: Principles of OBJ2. Proc. 12th ACM Symp. on Principles of Programming Languages, pp. 52–66. New Orleans 1985

  29. Ganzinger, H.: Parameterized specifications: parameter passing and implementation with respect to observability. Trans. Prog. Lang. Syst. 5, 318–354 (1983)

    Google Scholar 

  30. Giarratana, V., Gimona, F., Montanari, U.: Observability concepts in abstract data type specification. Proc. 5th Intl. Symp. on Mathematical Foundations of Computer Science, Gdansk. Lect. Notes Comput. Sci. Vol. 45, pp. 576–587. Berlin, Heidelberg, New York: Springer 1976

    Google Scholar 

  31. Gogolla, M.: Algebraic specifications with partially ordered sorts and declarations. Fb. 169, Abteilung Informatik, Universität Dortmund 1983

  32. Gogolla, M., Drosten, K., Lipeck, U., Ehrich, H.-D.: Algebraic and operational semantics of specifications allowing exceptions and errors. Theor. Comput. Sci. 34, 289–313 (1984)

    Google Scholar 

  33. Goguen, J.A., Burstall, R.M.: CAT, a system for the structured elaboration of correct programs from structured specifications. Technical report CSL-118, SRI International, 1980

  34. Goguen, J.A., Burstall, R.M.: Introducing institutions. Proc. Logics of Programming Workshop. Clarke, E., Kozen, D. (eds.), Carnegie-Mellon University. Lect. Notes Comput. Sci. Vol. 164, pp. 221–256. Berlin, Heidelberg, New York: Springer 1984

    Google Scholar 

  35. Goguen, J.A., Burstall, R. M.: Some fundamental algebraic tools for the semantics of computation. Part 1: Comma categories, colimits, signatures and theories. Theor. Comput. Sci. 31, 175–210 (1984)

    Google Scholar 

  36. Goguen, J.A., Burstall, R. M.: A study in the foundations of programming methodology: specifications, institutions, charters and parchments. Proc. Workshop on Category Theory and Computer Programming, Guildford. Lect. Notes Comput. Sci. Vol. 240, pp. 313–333. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  37. Goguen, J.A., Jouannaud, J.-P., Meseguer, J.: Operational semantics for order-sorted algebra. Proc. 12th Intl. Colloq. on Automata, Languages and Programming, Nafplion, Greece. Lect. Notes Comput. Sci. Vol. 194, pp. 221–231. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  38. Goguen, J.A., Meseguer, J.: Universal realization, persistent interconnection and implementation of abstract modules. Proc. 9th Intl. Colloq. on Automata, Languages and Programming, Aarhus. Lect. Notes Comput. Sci. Vol. 140, pp. 265–281. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  39. Goguen, J.A., Thatcher, J.W., Wagner, E.G.: An initial algebra approach to the specification, correctness, and implementation of abstract data types. IBM research report RC6487 (1976). Also in: Current Trends in Programming Methodology, Vol. 4: Data Structuring. Yeh, R.T. (ed.), pp. 80–149. Englewood Cliffs, NJ: Prentice-Hall 1978

  40. Goguen, J.A., Thatcher, J.W., Wagner, E.G., Wright, J.B.: Initial algebra semantics and continuous algebras. J. Assoc. Comput. Mach. 24, 68–95 (1977)

    Google Scholar 

  41. Gordon, M.J.: Denotational descriptions of Programming Languages. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  42. Guttag, J.V.: The specification and application to programming of abstract data types. Ph.D. thesis, University of Toronto 1975

  43. Guttag, J.V., Horning, J.J.: Preliminary report on the Larch Shared Language. Report CSL-83-6, Computer Science Laboratory, Xerox PARC, 1983

  44. Kamin, S.: Final data types and their specification. Trans. Prog. Lang. Syst. 5, 97–121 (1983)

    CAS  Google Scholar 

  45. Larsen, K.: Context-dependent bisimulation between processes. Ph.D. thesis, Dept. of Computer Science, University of Edinburgh, 1986

  46. Lipeck, U.: Ein algebraischer Kalkül für einer strukturierten Entwurf von Datenabstraktionen. Ph.D. thesis, Abteilung Informatik, Universität Dortmund 1983

  47. Liskov, B.H., Berzins, V.: An appraisal of program specifications. Computation Structures Group memo 141-1, Laboratory for Computer Science, MIT, 1977

  48. MacLane, S.: Categories for the Working Mathematician. Berlin, Heidelberg, New York: Springer 1971

    Google Scholar 

  49. MacQueen, D. B.: Modules for Standard ML. Polymorphism 2, 2 (1985). See also: Proc. 1984 ACM Symp. on LISP and Functional Programming, pp. 198–207. Austin, Texas, 1984

    Google Scholar 

  50. Manna, Z., Waldinger, R.: A deductive approach to program synthesis. ACM Trans. Prog. Lang. Syst. 2, 92–121 (1980)

    Google Scholar 

  51. Meseguer, J., Goguen, J.A.: Initiality, induction and computability. In: Algebraic Methods in Semantics. Nivat, M., Reynolds, J. (eds.), pp. 459–541. Cambridge: Univ. Press 1983

    Google Scholar 

  52. Milner, R.G.: A theory of type polymorphism in programming. J. Comput. Syst. Sci. 17, 348–375 (1978)

    Google Scholar 

  53. Milner, R. G.: The Standard ML core language. Polymorphism 2, 2 (1985). See also: A proposal for Standard ML. Proc. 1984 ACM Symp. on LISP and Functional Programming, pp. 184–197. Austin, Texas, 1984

    Google Scholar 

  54. Moore, E.F.: Gedanken-experiments on sequential machines. In: Automata Studies. Shannon, C.E., McCarthy, J. (eds.), pp. 129–153. Princeton: University Press 1956

    Google Scholar 

  55. Orejas, F.: Characterizing composability of abstract implementations. Proc. Intl. Conf. on Foundations of Computation Theory, Borgholm, Sweden. Lect. Notes Comput. Sci. Vol. 158, pp. 335–346. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  56. Parisi-Presicce, F., Blum, E.K.: The semantics of shared submodules specifications. Proc. 10th Colloq. on Trees in Algebra and Programming, Joint Conf. on Theory and Practice of Software Development (TAPSOFT). Lect. Notes Comput. Sci. Vol. 185, pp. 359–373. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  57. Pepper, P.: On the correctness of type transformations. Talk at 2nd Workshop on Theory and Applications of Abstract Data Types, Passau, 1983

  58. Plotkin, G.D.: LCF considered as a programming language. Theor. Comput. Sci. 5, 223–255 (1977)

    Google Scholar 

  59. Reichel, H.: Behavioural equivalence — a unifying concept for initial and final specification methods. Proc. 3rd Hungarian Computer Science Conference, pp. 27–39. Budapest, 1981

  60. Sannella, D.T., Tarlecki, A.: Some thoughts on algebraic specification. Proc. 3rd Workshop on Theory and Applications of Abstract Data Types, Bremen. Informatik-Fachberichte Vol. 116, pp. 31–38. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  61. Sannella, D.T., Tarlecki, A.: Program specification and development in Standard ML. Proc. 12th ACM Symp. on Principles of Programming Languages, pp. 67–77. New Orleans, 1985

  62. Sannella, D.T., Tarlecki, A.: Specifications in an arbitrary institution. Inf. Control. 76 (1988). See also: Building specifications in an arbitrary institution, Proc. Intl. Symposium on Semantics of Data Types, Sophia-Antipolis. Lect. Notes Comput. Sci. Vol. 173, pp. 337–356. Berlin, Heidelberg, New York: Springer 1984

  63. Sannella, D.T., Tarlecki, A.: Extended ML: an institution-independent framework for formal program development. Proc. Workshop on Category Theory and Computer Programming, Guildford. Lect. Notes Comput. Sci. Vol. 240, pp. 364–389. Berlin, Heidelberg, New York: Springer 1986

    Google Scholar 

  64. Sannella, D.T., Tarlecki, A.: On observational equivalence and algebraic specification. J. Comput. Syst. Sci. 34, 150–178 (1987). Extended abstract in: Proc. 10th Colloq. on Trees in Algebra and Programming, Joint Conf. on Theory and Practice of Software Development (TAPSOFT). Lect. Notes Comput. Sci. Vol. 185, pp. 308–322. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  65. Sannella, D.T., Tarlecki, A.: Toward formal development of programs from algebraic specifications: implementations revisited (extended abstract). Proc. 12th Colloq. on Trees in Algebra and Programming, Joint Conf. on Theory and Practice of Software Development (TAPSOFT), Pisa. Lect. Notes Comput. Sci. Vol. 249, pp. 96–110. Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  66. Sannella, D.T., Wirsing, M.: Implementation of parameterised specifications. Report CSR-103-82, Dept. of Computer Science, Univ. of Edinburgh. Extended abstract in: Proc. 9th Intl. Colloq. on Automata, Languages and Programming, Aarhus. Lect. Notes Comput. Sci. Vol. 140, pp. 473–488. Berlin, Heidelberg, New York: Springer 1982

    Google Scholar 

  67. Sannella, D.T., Wirsing, M.: A kernel language for algebraic specification and implementation. Report CSR-131-83, Dept. of Computer Science, Univ. of Edinburgh. Extended abstract in: Proc. Intl. Conf. on Foundations of Computation Theory, Borgholm, Sweden. Lect. Notes Comput. Sci. Vol. 158, pp. 413–427. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  68. Schoett, O.: Data abstraction and the correctness of modular programming. Ph.D. thesis, University of Edinburgh, 1986

  69. Tarlecki, A.: On the existence of free models in abstract algebraic institutions. Theor. Comput. Sci. 37, 269–304 (1985)

    Google Scholar 

  70. Tarlecki, A.: Quasi-varieties in abstract algebraic institutions. J. Comput. Syst. Sci. 33, 333–360 (1986)

    Google Scholar 

  71. Tarlecki, A.: Software-system development —an abstract view. Proc. 10th IFIP World Computer Congress, Dublin, pp. 685–688. Amsterdam: North-Holland 1986

    Google Scholar 

  72. Tarlecki, A., Wirsing, M.: Continuous abstract data types. Fundamenta Informaticae 9, pp. 95–126 (1986). Extended abstract: Continuous abstract data types —basic machinery and results. Proc. Intl. Conf. on Fundamentals of Computation Theory, Cottbus, GDR. Lect. Notes Comput. Sci. Vol. 199, pp. 431–441. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  73. Wand, M.: Specifications, models, and implementations of data abstractions. Theor. Comput. Sci. 20, 3–32 (1982)

    Google Scholar 

  74. Wirsing, M.: Structured algebraic specifications: a kernel language. Theor. Comput. Sci. 42, 123–249 (1986)

    Google Scholar 

  75. Zilles, S.N.: Algebraic specification of data types. Computation Structures Group memo 119, Laboratory for Computer Science, MIT (1974)

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Authors and Affiliations

  1. Department of Artificial Intelligence, University of Edinburgh, EH93JZ, Edingburgh, UK

    Donald Sannella

  2. Laboratory for Foundations of Computer Science, Department of Computer Science, University of Edinburgh, EH93JZ, Edingburgh, UK

    Donald Sannella

  3. Institute of Computer Science, Polish Academy of Sciences, Warsaw, Poland

    Andrzej Tarlecki

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  1. Donald Sannella
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  2. Andrzej Tarlecki
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An extended abstract of this paper appeared in [65]

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Sannella, D., Tarlecki, A. Toward formal development of programs from algebraic specifications: Implementations revisited. Acta Informatica 25, 233–281 (1988). https://doi.org/10.1007/BF00283329

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