Abstract
We provide several effective equivalent characterizations of EF (the modal logic of the descendant relation) on arbitrary trees. More specifically, we prove that, for EF-bisimulation invariant properties of trees, being definable by an EF formula, being a Borel set, and being definable in weak monadic second order logic, all coincide. The proof builds upon a known algebraic characterization of EF for the case of finitely branching trees due to Bojańczyk and Idziaszek. We furthermore obtain characterizations of modal logic on transitive Kripke structures as a fragment of weak monadic second order logic and of the μ-calculus.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Benedikt, M., Segoufin, L.: Regular Tree Languages Definable in FO and in FO mod . ACM Trans. on Computational Logic 11(1) (2009)
van Benthem, J.: Modal Correspondence Theory. PhD thesis, Mathematisch Instituut & Instituut voor Grondslagenonderzoek, University of Amsterdam (1976)
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Univ. Press, Cambridge (2001)
Bojańczyk, M.: Two-Way Unary Temporal Logic over Trees. In: LICS 2007, pp. 121–130 (2007)
Bojańczyk, M., Idziaszek, T.: Algebra for Infinite Forests with an Application to the Temporal Logic EF. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 131–145. Springer, Heidelberg (2009)
Bojańczyk, M., Segoufin, L., Straubing, H.: Piecewise Testable Tree Languages. In: LICS 2008, pp. 442–451 (2008)
Bojańczyk, M., Segoufin, L.: Tree languages defined in first-order logic with one quantifier alternation. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 233–245. Springer, Heidelberg (2008)
Bojańczyk, M., Walukiewicz, I.: Characterizing EF and EX Tree Logics. Theoretical Computer Science 358(2-3), 255–272 (2006)
Bojańczyk, M., Walukiewicz, I.: Forest Algebras. In: Automata and Logic: History and Perspectives, pp. 107–132. Amsterdam University Press, Amsterdam (2007)
Bradfield, J., Stirling, C.: Modal Logic and Mu-Calculi. In: Bergstra, J., et al. (eds.) Handbook of Process Algebra, pp. 293–332. Elsevier, North-Holland (2001)
Dawar, A., Otto, M.: Modal Characterisation Theorems over Special Classes of Frames. Ann. Pure Appl. Logic 161(1), 1–42 (2009)
Janin, D., Walukiewicz, I.: On the Expressive Completeness of the Propositional μ-Calculus with Respect to Monadic Second Order Logic. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 263–277. Springer, Heidelberg (1996)
Murlak, F.: Weak Index vs Borel Rank. In: STACS 2008, pp. 573–584 (2008)
Otto, M.: Eliminating recursion in the μ-calculus. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 531–540. Springer, Heidelberg (1999)
Place, T.: Characterization of logics over ranked tree languages. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 401–415. Springer, Heidelberg (2008)
Walukiewicz, I.: Monadic Second-Order Logic on Tree-Like Structures. Theoretical Computer Science 275(1-2), 311–346 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag GmbH Berlin Heidelberg
About this paper
Cite this paper
ten Cate, B., Facchini, A. (2011). Characterizing EF over Infinite Trees and Modal Logic on Transitive Graphs. In: Murlak, F., Sankowski, P. (eds) Mathematical Foundations of Computer Science 2011. MFCS 2011. Lecture Notes in Computer Science, vol 6907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22993-0_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-22993-0_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22992-3
Online ISBN: 978-3-642-22993-0
eBook Packages: Computer ScienceComputer Science (R0)