Counterpart Semantics for a Second-Order μ-Calculus

  • Fabio Gadducci
  • Alberto Lluch Lafuente
  • Andrea Vandin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6372)


We propose a novel approach to the semantics of quantified μ-calculi, considering models where states are algebras; the evolution relation is given by a counterpart relation (a family of partial homomorphisms), allowing for the creation, deletion, and merging of components; and formulas are interpreted over sets of state assignments (families of substitutions, associating formula variables to state components). Our proposal avoids the limitations of existing approaches, usually enforcing restrictions of the evolution relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of.


Quantified μ-calculi counterpart semantics graph transformation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Baldan, P., Corradini, A., König, B., Lluch Lafuente, A.: A temporal graph logic for verification of graph transformation systems. In: Fiadeiro, J.L., Schobbens, P.-Y. (eds.) WADT 2006. LNCS, vol. 4409, pp. 1–20. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Belardinelli, F.: Quantified Modal Logic and the Ontology of Physical Objects. Ph.D. thesis, Scuola Normale Superiore of Pisa (2006)Google Scholar
  3. 3.
    Caires, L.: Behavioral and spatial observations in a logic for the π-calculus. In: Walukiewicz, I. (ed.) FOSSACS 2004. LNCS, vol. 2987, pp. 72–89. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Caires, L., Cardelli, L.: A spatial logic for concurrency (part I). Information and Computation 186(2), 194–235 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cardelli, L., Gardner, P., Ghelli, G.: A spatial logic for querying graphs. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 597–610. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Cardelli, L., Gardner, P., Ghelli, G.: Manipulating trees with hidden labels. In: Gordon, A.D. (ed.) FOSSACS 2003. LNCS, vol. 2620, pp. 216–232. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Cardelli, L., Ghelli, G.: TQL: a query language for semistructured data based on the ambient logic. Mathematical Structures in Computer Science 14(3), 285–327 (2004)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, pp. 313–400. World Scientific, Singapore (1997)CrossRefGoogle Scholar
  9. 9.
    Dawar, A., Gardner, P., Ghelli, G.: Expressiveness and complexity of graph logic. Information and Compututation 205(3), 263–310 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Distefano, D., Katoen, J.P., Rensink, A.: Who is pointing when to whom? In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 250–262. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Distefano, D., Rensink, A., Katoen, J.P.: Model checking birth and death. In: Baeza-Yates, R.A., Montanari, U., Santoro, N. (eds.) IFIP International Conference on Theoretical Computer Science (TCS 2002). IFIP Conference Proceedings, vol. 223, pp. 435–447. Kluwer, Dordrecht (2002)Google Scholar
  12. 12.
    Franconi, E., Toman, D.: Fixpoint extensions of temporal description logics. In: Calvanese, D., Giacomo, G.D., Franconi, E. (eds.) 16th International Workshop on Description Logics (DL 2003). CEUR Workshop Proceedings, vol. 81 (2003),
  13. 13.
    Gadducci, F., Heckel, R., Koch, M.: A fully abstract model for graph-interpreted temporal logic. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) TAGT 1998. LNCS, vol. 1764, pp. 310–322. Springer, Heidelberg (2000)Google Scholar
  14. 14.
    Gadducci, F., Lluch Lafuente, A.: Graphical encoding of a spatial logic for the π-calculus. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) CALCO 2007. LNCS, vol. 4624, pp. 209–225. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Hazen, A.: Counterpart-theoretic semantics for modal logic. The Journal of Philosophy 76(6), 319–338 (2004)CrossRefGoogle Scholar
  16. 16.
    Hodkinson, I., Wolter, F., Zakharyaschev, M.: Monodic fragments of first-order temporal logics: 2000-2001 a.d. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 1–23. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    Huth, M., Ryan, M.: Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  18. 18.
    Reif, J., Sistla, A.P.: A multiprocess network logic with temporal and spatial modalities. International Journal of Computer and System Sciences 30(1), 41–53 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rensink, A.: Towards model checking graph grammars. In: Leuschel, M., Gruner, S., Lo Presti, S. (eds.) 3rd Workshop on Automated Verification of Critical Systems (AvoCS 2003). University of Southampton Technical Reports, vol. DSSE–TR–2003–2, pp. 150–160. University of Southampton (2003)Google Scholar
  20. 20.
    Rensink, A.: Model checking quantified computation tree logic. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 110–125. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Reynolds, J.: Separation logic: A logic for shared mutable data structures. In: 17th IEEE Symposium on Logic in Computer Science (LICS 2002), pp. 55–74. IEEE Computer Society, Los Alamitos (2002)CrossRefGoogle Scholar
  22. 22.
    Vandin, A.: Algebraic models for a second-order modal logic. Master’s thesis, University of Pisa (2009),
  23. 23.
    Yahav, E., Reps, T.W., Sagiv, S., Wilhelm, R.: Verifying temporal heap properties specified via evolution logic. Logic Journal of the IGPL 14(5), 755–783 (2006)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Fabio Gadducci
    • 1
  • Alberto Lluch Lafuente
    • 2
  • Andrea Vandin
    • 2
  1. 1.Department of Computer ScienceUniversity of PisaItaly
  2. 2.IMT Institute for Advanced Studies LuccaItaly

Personalised recommendations