Satisfiability Calculus: The Semantic Counterpart of a Proof Calculus in General Logics

  • Carlos Gustavo López Pombo
  • Pablo F. Castro
  • Nazareno M. Aguirre
  • Thomas S. E. Maibaum
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7841)


Since its introduction by Goguen and Burstall in 1984, the theory of institutions has been one of the most widely accepted formalizations of abstract model theory. This work was extended by a number of researchers, José Meseguer among them, who presented General Logics, an abstract framework that complements the model theoretical view of institutions by defining the categorical structures that provide a proof theory for any given logic. In this paper we intend to complete this picture by providing the notion of Satisfiability Calculus, which might be thought of as the semantical counterpart of the notion of proof calculus, that provides the formal foundations for those proof systems that use model construction techniques to prove or disprove a given formula, thus “implementing” the satisfiability relation of an institution.


Modal Logic Natural Transformation Logical System Proof Theory Kripke Structure 
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  1. 1.
    Goguen, J.A., Burstall, R.M.: Introducing institutions. In: Clarke, E., Kozen, D. (eds.) Logic of Programs 1983. LNCS, vol. 164, pp. 221–256. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  2. 2.
    Meseguer, J.: General logics. In: Ebbinghaus, H.D., Fernandez-Prida, J., Garrido, M., Lascar, D., Artalejo, M.R. (eds.) Proceedings of the Logic Colloquium 1987, Granada, Spain, vol. 129, pp. 275–329. North Holland (1989)Google Scholar
  3. 3.
    Fiadeiro, J.L., Maibaum, T.S.E.: Generalising interpretations between theories in the context of π-institutions. In: Burn, G., Gay, D., Ryan, M. (eds.) Proceedings of the First Imperial College Department of Computing Workshop on Theory and Formal Methods, London, UK, pp. 126–147. Springer (1993)Google Scholar
  4. 4.
    Goguen, J.A., Burstall, R.M.: Institutions: abstract model theory for specification and programming. Journal of the ACM 39(1), 95–146 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Tarlecki, A.: Moving between logical systems. In: Haveraaen, M., Owe, O., Dahl, O.J. (eds.) Abstract Data Types 1995 and COMPASS 1995. LNCS, vol. 1130, pp. 478–502. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  6. 6.
    Sannella, D., Tarlecki, A.: Specifications in an arbitrary institution. Information and Computation 76(2-3), 165–210 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Tarlecki, A.: Abstract specification theory: an overview. In: Broy, M., Pizka, M. (eds.) Proceedings of the NATO Advanced Study Institute on Models, Algebras and Logic of Engineering Software, Marktoberdorf, Germany. NATO Science Series, pp. 43–79. IOS Press (2003)Google Scholar
  8. 8.
    Mossakowski, T.: Comorphism-based Grothendieck logics. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 593–604. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Diaconescu, R., Futatsugi, K.: Logical foundations of CafeOBJ. Theoretical Computer Science 285(2), 289–318 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Diaconescu, R.: Grothendieck institutions. Applied Categorical Structures 10(4), 383–402 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Diaconescu, R. (ed.): Institution-independent Model Theory. Studies in Universal Logic, vol. 2. Birkhäuser (2008)Google Scholar
  13. 13.
    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
  14. 14.
    Tarlecki, A.: Towards heterogeneous specifications. In: Gabbay, D., de Rijke, M. (eds.) Frontiers of Combining Systems. Studies in Logic and Computation, vol. 2, pp. 337–360. Research Studies Press (2000)Google Scholar
  15. 15.
    Cengarle, M.V., Knapp, A., Tarlecki, A., Wirsing, M.: A heterogeneous approach to UML semantics. In: Degano, P., De Nicola, R., Meseguer, J. (eds.) Concurrency, Graphs and Models. LNCS, vol. 5065, pp. 383–402. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Beth, E.W.: The Foundations of Mathematics. North Holland (1959)Google Scholar
  17. 17.
    Beth, E.W.: Semantic entailment and formal derivability. In: Hintikka, J. (ed.) The Philosophy of Mathematics, pp. 9–41. Oxford University Press (1969) (reprinted from [34])Google Scholar
  18. 18.
    Herbrand, J.: Investigation in proof theory. In: Goldfarb, W.D. (ed.) Logical Writings, pp. 44–202. Harvard University Press (1969) (translated to English from [35])Google Scholar
  19. 19.
    Gentzen, G.: Investigation into logical deduction. In: Szabo, M.E. (ed.) The Collected Papers of Gerhard Gentzen, pp. 68–131. North Holland (1969) (translated to English from [36])Google Scholar
  20. 20.
    Smullyan, R.M.: First-order Logic. Dover Publishing (1995)Google Scholar
  21. 21.
    Robinson, J.A.: A machine-oriented logic based on the resolution principle. Journal of the ACM 12(1), 23–41 (1965)zbMATHCrossRefGoogle Scholar
  22. 22.
    McLane, S.: Categories for working mathematician. Graduate Texts in Mathematics. Springer, Berlin (1971)Google Scholar
  23. 23.
    Fiadeiro, J.L.: Categories for software engineering. Springer (2005)Google Scholar
  24. 24.
    Lopez Pombo, C.G., Castro, P., Aguirre, N.M., Maibaum, T.S.E.: Satisfiability calculus: the semantic counterpart of a proof calculus in general logics. Technical report, McMaster University, Centre for Software Certification (2011)Google Scholar
  25. 25.
    Fitting, M.: Tableau methods of proof for modal logics. Notre Dame Journal of Formal Logic 13(2), 237–247 (1972) (Lehman College) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Goguen, J.A., Rosu, G.: Institution morphisms. Formal Asp. Comput. 13(3-5), 274–307 (2002)zbMATHCrossRefGoogle Scholar
  27. 27.
    Castro, P.F., Aguirre, N.M., López Pombo, C.G., Maibaum, T.S.E.: Towards managing dynamic reconfiguration of software systems in a categorical setting. In: Cavalcanti, A., Deharbe, D., Gaudel, M.-C., Woodcock, J. (eds.) ICTAC 2010. LNCS, vol. 6255, pp. 306–321. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  28. 28.
    Castro, P.F., Aguirre, N., López Pombo, C.G., Maibaum, T.: A categorical approach to structuring and promoting Z specifications. In: Păsăreanu, C.S., Salaün, G. (eds.) FACS 2012. LNCS, vol. 7684, pp. 73–91. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  29. 29.
    Blackburn, P., de Rijke, M., Venema, Y.: Modal logic. Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press (2001)Google Scholar
  30. 30.
    Kozen, D.: Kleene algebra with tests. ACM Transactions on Programming Languages and Systems 19(3), 427–443 (1997)CrossRefGoogle Scholar
  31. 31.
    Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.: All About Maude. LNCS, vol. 4350. Springer, Heidelberg (2007)Google Scholar
  32. 32.
    Jackson, D.: Alloy: a lightweight object modelling notation. ACM Transactions on Software Engineering and Methodology 11(2), 256–290 (2002)CrossRefGoogle Scholar
  33. 33.
    Moura, L.D., Bjørner, N.: Satisfiability modulo theories: introduction and applications. Communications of the ACM 54(9), 69–77 (2011)CrossRefGoogle Scholar
  34. 34.
    Beth, E.W.: Semantic entailment and formal derivability. Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde 18(13), 309–342 (1955) (reprinted in [17])MathSciNetGoogle Scholar
  35. 35.
    Herbrand, J.: Recherches sur la theorie de la demonstration. PhD thesis, Université de Paris (1930) (English translation in [18])Google Scholar
  36. 36.
    Gentzen, G.: Untersuchungen tiber das logische schliessen. Mathematische Zeitschrijt 39, 176–210, 405–431 (1935) (English translation in [19])Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2013

Authors and Affiliations

  • Carlos Gustavo López Pombo
    • 1
    • 3
  • Pablo F. Castro
    • 2
    • 3
  • Nazareno M. Aguirre
    • 2
    • 3
  • Thomas S. E. Maibaum
    • 4
  1. 1.Departmento de Computación, FCEyNUniversidad de Buenos AiresArgentina
  2. 2.Departmento de Computación, FCEFQyNUniversidad Nacional de Río CuartoArgentina
  3. 3.Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)Argentina
  4. 4.Department of Computing & SoftwareMcMaster UniversityCanada

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