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A Proof Theoretic Interpretation of Model Theoretic Hiding

  • Mihai Codescu
  • Fulya Horozal
  • Michael Kohlhase
  • Till Mossakowski
  • Florian Rabe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7137)

Abstract

Logical frameworks like LF are used for formal representations of logics in order to make them amenable to formal machine-assisted meta-reasoning. While the focus has originally been on logics with a proof theoretic semantics, we have recently shown how to define model theoretic logics in LF as well. We have used this to define new institutions in the Heterogeneous Tool Set in a purely declarative way.

It is desirable to extend this model theoretic representation of logics to the level of structured specifications. Here a particular challenge among structured specification building operations is hiding, which restricts a specification to some export interface. Specification languages like ASL and CASL support hiding, using an institution-independent model theoretic semantics abstracting from the details of the underlying logical system.

Logical frameworks like LF have also been equipped with structuring languages. However, their proof theoretic nature leads them to a theory-level semantics without support for hiding. In the present work, we show how to resolve this difficulty.

Keywords

Type Family Logical Framework Signature Graph Signature Morphism Proof Theoretic Semantic 
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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References

  1. [AHMS99]
    Autexier, S., Hutter, D., Mantel, H., Schairer, A.: Towards an Evolutionary Formal Software-Development Using CASL. In: Bert, D., Choppy, C., Mosses, P.D. (eds.) WADT 1999. LNCS, vol. 1827, pp. 73–88. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  2. [AKKB99]
    Astesiano, E., Kreowski, H.-J., Krieg-Brückner, B.: Algebraic Foundations of Systems Specification. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  3. [Bar92]
    Barendregt, H.: Lambda calculi with types. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 2, Oxford University Press (1992)Google Scholar
  4. [BG80]
    Burstall, R., Goguen, J.: The semantics of Clear, a specification language. In: Bjorner, D. (ed.) Abstract Software Specifications. LNCS, vol. 86, pp. 292–332. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  5. [BHK90]
    Bergstra, J.A., Heering, J., Klint, P.: Module algebra. J. ACM 37(2), 335–372 (1990)CrossRefzbMATHGoogle Scholar
  6. [Bor02]
    Borzyszkowski, T.: Logical systems for structured specifications. Theor. Comput. Sci. 286(2), 197–245 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [CHK+10]
    Codescu, M., Horozal, F., Kohlhase, M., Mossakowski, T., Rabe, F., Sojakova, K.: Towards Logical Frameworks in the Heterogeneous Tool Set Hets. In: Workshop on Abstract Development Techniques (2010)Google Scholar
  8. [GB92]
    Goguen, J., Burstall, R.: Institutions: Abstract model theory for specification and programming. Journal of the Association for Computing Machinery 39(1), 95–146 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [GR04]
    Goguen, J., Rosu, G.: Composing Hidden Information Modules over Inclusive Institutions. In: Owe, O., Krogdahl, S., Lyche, T. (eds.) From Object-Orientation to Formal Methods. LNCS, vol. 2635, pp. 96–123. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. [HHP93]
    Harper, R., Honsell, F., Plotkin, G.: A framework for defining logics. Journal of the Association for Computing Machinery 40(1), 143–184 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [HR11]
    Horozal, F., Rabe, F.: Representing Model Theory in a Type-Theoretical Logical Framework. Theoretical Computer Science (to appear, 2011), http://kwarc.info/frabe/Research/HR_folsound_10.pdf
  12. [HST94]
    Harper, R., Sannella, D., Tarlecki, A.: Structured presentations and logic representations. Annals of Pure and Applied Logic 67, 113–160 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [IR11]
    Iancu, M., Rabe, F.: Formalizing Foundations of Mathematics. Mathematical Structures in Computer Science (to appear, 2011), http://kwarc.info/frabe/Research/IR_foundations_10.pdf
  14. [KMR09]
    Kohlhase, M., Mossakowski, T., Rabe, F.: The LATIN Project (2009), https://trac.omdoc.org/LATIN/
  15. [Koh06]
    Kohlhase, M.: OMDoc – An Open Markup Format for Mathematical Documents (version 1.2). LNCS (LNAI), vol. 4180. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  16. [KST97]
    Kahrs, S., Sannella, D., Tarlecki, A.: The definition of extended ML: A gentle introduction. Theoretical Computer Science 173(2), 445–484 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [MAH06]
    Mossakowski, T., Autexier, S., Hutter, D.: Development graphs - Proof management for structured specifications. J. Log. Algebr. Program. 67(1-2), 114–145 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [ML74]
    Martin-Löf, P.: An Intuitionistic Theory of Types: Predicative Part. In: Proceedings of the 1973 Logic Colloquium, pp. 73–118. North-Holland (1974)Google Scholar
  19. [MML07]
    Mossakowski, T., Maeder, C., Lüttich, K.: The Heterogeneous Tool Set, Hets. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 519–522. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  20. [Mos04]
    Mosses, P.D. (ed.): Casl Reference Manual. LNCS, vol. 2960. Springer, Heidelberg (2004)zbMATHGoogle Scholar
  21. [MT09]
    Mossakowski, T., Tarlecki, A.: Heterogeneous Logical Environments for Distributed Specifications. In: Corradini, A., Montanari, U. (eds.) WADT 2008. LNCS, vol. 5486, pp. 266–289. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  22. [Pau94]
    Paulson, L.C.: Isabelle: A Generic Theorem Prover. LNCS, vol. 828. Springer, Heidelberg (1994)CrossRefzbMATHGoogle Scholar
  23. [Rab10]
    Rabe, F.: A Logical Framework Combining Model and Proof Theory. Submitted to Mathematical Structures in Computer Science (2010), http://kwarc.info/frabe/Research/rabe_combining_09.pdf
  24. [RK10]
    Rabe, F., Kohlhase, M.: A Scalable Module System (2010), http://kwarc.info/frabe/Research/mmt.pdf
  25. [RS09]
    Rabe, F., Schürmann, C.: A Practical Module System for LF. In: Cheney, J., Felty, A. (eds.) Proceedings of the Workshop on Logical Frameworks: Meta-Theory and Practice (LFMTP), pp. 40–48. ACM Press (2009)Google Scholar
  26. [ST88]
    Sannella, D., Tarlecki, A.: Specifications in an arbitrary institution. Information and Computation 76, 165–210 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [ST11]
    Sannella, D., Tarlecki, A.: Foundations of Algebraic Specification and Formal Program Development. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  28. [SW83]
    Sannella, D., Wirsing, M.: A kernel language for algebraic specification and implementation. In: ADT (1983)Google Scholar
  29. [Wir86]
    Wirsing, M.: Structured algebraic specifications: A kernel language. Theor. Comput. Sci. 42, 123–249 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [ZK09]
    Zholudev, V., Kohlhase, M.: TNTBase: a Versioned Storage for XML. In: Proceedings of Balisage: The Markup Conference 2009. Balisage Series on Markup Technologies, vol. 3. Mulberry Technologies, Inc. (2009)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Mihai Codescu
    • 1
  • Fulya Horozal
    • 2
  • Michael Kohlhase
    • 2
  • Till Mossakowski
    • 1
  • Florian Rabe
    • 2
  1. 1.DFKIBremenGermany
  2. 2.Jacobs UniversityBremenGermany

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