Advertisement

Equivalences among various logical frameworks of partial algebras

  • Till Mossakowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1092)

Abstract

We examine a variety of liberal logical frameworks of partial algebras. Therefore we use simple, conjunctive and weak embeddings of institutions which preserve model categories and may map sentences to sentences, finite sets of sentences, or theory extensions using unique-existential quantifiers, respectively. They faithfully represent theories, model categories, theory morphisms, colimit of theories, reducts etc. Moreover, along simple and conjunctive embeddings, theorem provers can be re-used in a way that soundness and completeness is preserved. Our main result states the equivalence of all the logical frameworks with respect to weak embeddability. This gives us compilers between all frameworks. Thus it is a chance to unify the different branches of specification using liberal partial logics.

This is important for reaching the goal of formal interoperability of different specification languages for software development. With formal interoperability, a specification can contain parts written in different logical frameworks using a multiparadigm specification language, and one can re-use tools which are available for one framework also for other frameworks.

Keywords

Atomic Formula Logical Framework Total Operation Partial Algebra Partial Operation 
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. [1]
    J. Adámek, J. Rosický. Locally Presentable and Accessible Categories. Cambridge University Press, 1994.Google Scholar
  2. [2]
    E. Astesiano, M. Cerioli. Multiparadigm specification languages: a first attempt at foundations. In J.F. Groote C.M.D.J. Andrews, ed., Semantics of Specification Languages (SoSl 93), Workshops in Computing, 168–185. Springer Verlag, 1994.Google Scholar
  3. [3]
    M. Barr, C. Wells. Toposes, Triples and Theories, Grundlehren der mathematischen Wissenschaften 278. Springer Verlag, 1985.Google Scholar
  4. [4]
    P. Burmeister. Partial algebras — survey of a unifying approach towards a two-valued model theory for partial algebras. Algebra Universalis 15, 306–358, 1982.Google Scholar
  5. [5]
    P. Burmeister. A model theoretic approach to partial algebras. Akademie Verlag, Berlin, 1986.Google Scholar
  6. [6]
    R. M. Burstall, J. A. Goguen. Putting theories together to make specifications. In Proceedings of the 5th International Joint Conference on Artificial Intelligence, 1045–1058. Cambridge, 1977.Google Scholar
  7. [7]
    M. Cerioli. Relationships between Logical Formalisms. PhD thesis, TD-4/93, Università di Pisa-Genova-Udine, 1993.Google Scholar
  8. [8]
    M. Cerioli, J. Meseguer. May I borrow your logic? In A.M. Borzyszkowski, S. Sokolowski, eds., Proc. MFCS'93 (Mathematical Foundations of Computer Science), LNCS 711, 342–351. Springer Verlag, Berlin, 1993. To appear in Theoretical Computer Science.Google Scholar
  9. [9]
    M. Coste. Localisation, spectra and sheaf representation. In M.P. Fourman, C.J. Mulvey, D.S. Scott, eds., Application of Sheaves, Lecture Notes in Mathematics 753, 212–238. Springer Verlag, 1979.Google Scholar
  10. [10]
    H. Ehrig, B. Mahr. Fundamentals of Algebraic Specification 1. Springer Verlag, Heidelberg, 1985.Google Scholar
  11. [11]
    H. Ehrig, B. Mahr. Fundamentals of Algebraic Specification 2. Springer Verlag, Heidelberg, 1990.Google Scholar
  12. [12]
    H. Ehrig, P. Pepper, F. Orejas. On recent trends in algebraic specification. In Proc. ICALP'89, LNCS 372, 263–288. Springer Verlag, 1989.Google Scholar
  13. [13]
    W. A. Farmer. A partial functions version of Church's simple type theory. Journal of Symbolic Logic 55, 1269–1291, 1991.Google Scholar
  14. [14]
    P. Freyd. Aspects of topoi. Bull. Austral. Math. Soc. 7, 1–76, 1972.Google Scholar
  15. [15]
    P. Gabriel, F. Ulmer. Lokal präsentierbare Kategorien, Lecture Notes in Mathematics 221. Springer Verlag, Heidelberg, 1971.Google Scholar
  16. [16]
    J. A. Goguen, R. M. Burstall. Institutions: Abstract model theory for specification and programming. Journal of the Association for Computing Machinery 39, 95–146, 1992. Predecessor in: LNCS 164(1984):221–256.Google Scholar
  17. [17]
    J. A. Goguen, J. Meseguer. Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations. Theoretical Computer Science 105, 217–273, 1992.Google Scholar
  18. [18]
    Joseph Goguen, Jean-Pierre Jouannaud, José Meseguer. Operational semantics of order-sorted algebra. In Wilfried Brauer, ed., Proceedings, 1985 International Conference on Automata, Languages and Programming. Springer, 1985. Lecture Notes in Computer Science, Volume 194.Google Scholar
  19. [19]
    J. W. Gray. Categorical aspects of data type constructors. Theoretical Computer Science 50, 103–135, 1987.Google Scholar
  20. [20]
    H.-J. Hoehnke. On partial algebras. In Universal Algebra (Proc. Coll. Esztergom 1977), Colloq. Math. Soc. J. Bolyai 29, 373–412. North Holland, Amsterdam, 1981.Google Scholar
  21. [21]
    G. Jarzembski. Weak varieties of partial algebras. Algebra Universalis 25, 247–262, 1988.Google Scholar
  22. [22]
    G. Jarzembski. Programs in partial algebras. Theoretical Computer Science 115, 131–149, 1993.Google Scholar
  23. [23]
    S.C. Kleene. Introduction to Metamathematics. North Holland, 1952.Google Scholar
  24. [24]
    H.-J. Kreowski, T. Mossakowski. Equivalence and difference of institutions: Simulating horn clause logic with based algebras. Mathematical Structures in Computer Science 5, 189–215, 1995.Google Scholar
  25. [25]
    J. Meseguer. General logics. In Logic Colloquium 87, 275–329. North Holland, 1989.Google Scholar
  26. [26]
    J. Meseguer, J. Goguen. Order-sorted algebra solves the constructor, selector, multiple representation and coercion problems. Information and Computation 103(1), 114–158, March 1993.Google Scholar
  27. [27]
    T. Mossakowski. Simulations between various institutions of partial and total algebras. Talk at the Workshop of the ESPRIT Basic Research Working Group COMPASS, Dresden, September 1993.Google Scholar
  28. [28]
    T. Mossakowski. Parameterized recursion theory — a tool for the systematic classification of specification methods. In M. Nivat, C. Rattray, T. Rus, G. Scollo, eds., Proceedings of the Third International Conference on Algebraic Methodology and Software Technology, 1993, Workshops in Computing, 139–146. Springer-Verlag, London, 1993. Also to appear in Theoretical Computer Science.Google Scholar
  29. [29]
    T. Mossakowski. A hierarchy of institutions separated by properties of parameterized abstract data types. In Recent Trends in Data Type Specification. Proceedings, Lecture Notes in Computer Science 906, 389–405. Springer Verlag, London, 1995.Google Scholar
  30. [30]
    T. Mossakowski. Different types of arrow between logical frameworks. In ICALP 96, LNCS. To appear. Springer Verlag, 1996.Google Scholar
  31. [31]
    J. Slominski. Peano-algebras and quasi-algebras, Dissertationes Mathematicae (Rozprawy Mat.) 62. 1968.Google Scholar
  32. [32]
    A. Poigné. Algebra categorically. In D. Pitt et al., ed., Category Theory and Computer Programming, Lecture Notes in Computer Science 240, 76–102. Springer Verlag, 1986.Google Scholar
  33. [33]
    H. Reichel. Initial Computability, Algebraic Specifications and Partial Algebras, Oxford Science Publications, 1987.Google Scholar
  34. [34]
    D. Sannella, A. Tarlecki. Specifications in an arbitrary institution. Information and Computation 76, 165–210, 1988.Google Scholar
  35. [35]
    J.R. Shoenfield. Mathematical Logic. Addison-Wesley, Reading, Massachusetts, 1967.Google Scholar
  36. [36]
    A. Tarlecki. Working with multiple logical systems. Unpublished manuscript.Google Scholar
  37. [37]
    A. Tarlecki. On the existence of free models in abstract algebraic institutions. Theoretical Computer Science 37, 269–304, 1985.Google Scholar
  38. [38]
    M. Wirsing. Structured algebraic specifications: A kernel language. Theoretical Computer Science 42, 123–249, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Till Mossakowski
    • 1
  1. 1.Dept. of Computer ScienceUniversity of BremenBremen

Personalised recommendations