A Logic on Subobjects and Recognizability

  • H. J. Sander Bruggink
  • Barbara König
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 323)


We introduce a simple logic that allows to quantify over the subobjects of a categorical object. We subsequently show that, for the category of graphs, this logic is equally expressive as second-order monadic graph logic (msogl). Furthermore we show that for the more general setting of hereditary pushout categories, a class of categories closely related to adhesive categories, we can recover Courcelle’s result that every msogl-expressible property is recognizable. This is done by giving an inductive translation of formulas of our logic into so-called automaton functors which accept recognizable languages of cospans.


Atomic Formula Tree Automaton Categorical Logic Typing Morphism Category Graph 
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  1. 1.
    Blume, C.: Graphsprachen für die Spezifikation von Invarianten bei verteilten und dynamischen Systemen. Master’s thesis, Universität Duisburg-Essen (November 2008)Google Scholar
  2. 2.
    Blume, C., Sander Bruggink, H.J., König, B.: Recognizable graph languages for checking invariants. In: Proc. of GT-VMT ’10 (Workshop on Graph Transformation and Visual Modeling Techniques), Electronic Communications of the EASST (2010)Google Scholar
  3. 3.
    Bouajjani, A., Jonsson, B., Nilsson, M., Touili, T.: Regular model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 403–418. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    Bozapalidis, S., Kalampakas, A.: Recognizability of graph and pattern languages. Acta Informatica 42(8/9), 553–581 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bozapalidis, S., Kalampakas, A.: Graph automata. Theoretical Computer Science 393, 147–165 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bruggink, H.J.S., König, B.: On the recognizability of arrow and graph languages. In: Ehrig, H., Heckel, R., Rozenberg, G., Taentzer, G. (eds.) ICGT 2008. LNCS, vol. 5214, pp. 336–350. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Courcelle, B.: The monadic second-order logic of graphs I. Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Courcelle, B.: The expression of graph properties and graph transformations in monadic second-order logic. In: Rozenberg, G. (ed.) Handbook of Graph Grammars and Computing by Graph Transformation, ch. 5, Foundations, vol. 1, World Scientific, Singapore (1997)Google Scholar
  9. 9.
    Courcelle, B., Durand, I.: Verifying monadic second order graph properties with tree automata. In: European Lisp Symposium (May 2010)Google Scholar
  10. 10.
    Geser, A., Hofbauer, D., Waldmann, J.: Match-bounded string rewriting systems. Applicable Algebra in Engineering, Communication and Computing 15(3-4), 149–171 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Griffing, G.: Composition-representative subsets. Theory and Applications of Categories 11(19), 420–437 (2003)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Grohe, M.: Logic, graphs and algorithms. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata – History and Perspectives. Amsterdam University Press (2007)Google Scholar
  13. 13.
    Heindel, T.: A category theoretical approach to the concurrent semantics of rewriting. PhD thesis, Universität Duisburg–Essen (September 2009),
  14. 14.
    Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundation of Mathematics, vol. 141. Elsevier, Amsterdam (1999)zbMATHGoogle Scholar
  15. 15.
    Lack, S., Sobociński, P.: Adhesive and quasiadhesive categories. RAIRO – Theoretical Informatics and Applications 39(3) (2005)Google Scholar
  16. 16.
    Pitts, A.M.: Categorical logic. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science V. Oxford University Press, Oxford (2001)Google Scholar
  17. 17.
    Rensink, A.: Representing first-order logic using graphs. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 319–335. Springer, Heidelberg (2004)Google Scholar

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© IFIP 2010

Authors and Affiliations

  • H. J. Sander Bruggink
    • 1
  • Barbara König
    • 1
  1. 1.Universität Duisburg-EssenGermany

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