Satisfiability of ECTL* with Tree Constraints

  • Claudia Carapelle
  • Shiguang Feng
  • Alexander Kartzow
  • Markus Lohrey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9139)

Abstract

Recently, we proved that satisfiability for \(\mathsf {ECTL}^*\) with constraints over \(\mathbb {Z}\) is decidable using a new technique based on weak monadic second-order logic with the bounding quantifier (\(\mathsf {WMSO\!+\!B}\)). Here we apply this approach to concrete domains that are tree-like. We show that satisfiability of \(\mathsf {ECTL}^*\) with constraints is decidable over (i) semi-linear orders, (ii) ordinal trees (semi-linear orders where the branches form ordinals), and (iii) infinitely branching order trees of height h for each fixed \(h\in \mathbb {N}\). In contrast, we introduce Ehrenfeucht-Fraïssé-games for \(\mathsf {WMSO\!+\!B}\) (weak \(\mathsf {MSO}\) with the bounding quantifier) and use them to show that our approach cannot deal with the class of order trees. Missing proofs and details can be found in the long version [6].

References

  1. 1.
    Bojańczyk, M., Toruńczyk, S.: Weak MSO+U over infinite trees. In: Proceedings STACS 2012, vol. 14, pp. 648–660. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)Google Scholar
  2. 2.
    Bozzelli, L., Gascon, R.: Branching-time temporal logic extended with qualitative presburger constraints. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 197–211. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  3. 3.
    Bozzelli, L., Pinchinat, S.: Verification of gap-order constraint abstractions of counter systems. Theor. Comput. Sci. 523, 1–36 (2014)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Carapelle, C., Kartzow, A., Lohrey, M.: Satisfiability of CTL\(^{*}\) with constraints. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013 – Concurrency Theory. LNCS, vol. 8052, pp. 455–469. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  5. 5.
    Carapelle, C., Kartzow, A., Lohrey, M.: Satisfiability of ECTL\({}^*\) with constraints. submitted for publication. http://www.eti.uni-siegen.de/ti/veroeffentlichungen/ectl-with-constraints.pdf
  6. 6.
    Carapelle, C., Kartzow, A., Lohrey, M., Feng, S.: Satisfiability of \({\sf ECTL} ^{*}\) with tree constraints. http://arXiv.org/abs/1412.2905
  7. 7.
    Demri, S., Deters,M.: Temporal logics on strings with prefix relation. Research Report LSV-14-13, ENS Cachan. http://www.lsv.ens-cachan.fr/Publis/RAPPORTS_LSV/PDF/rr-lsv-2014-13.pdf
  8. 8.
    Demri, S., Gascon, R.: Verification of qualitative Z constraints. Theor. Comput. Sci. 409(1), 24–40 (2008)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Ebbinghaus, H.D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic, 1st edn. Springer, Heidelberg (1995) MATHCrossRefGoogle Scholar
  10. 10.
    Gascon, R.: An automata-based approach for CTL\(^{*}\) with constraints. Electron. Notes Theor. Comput. Sci. 239, 193–211 (2009)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Thomas, W.: Computation tree logic and regular omega-languages. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency. LNCS, vol. 354, pp. 690–713. Springer, Heidelberg (1989) CrossRefGoogle Scholar
  12. 12.
    Vardi, M.Y., Wolper, P.: Yet another process logic. In: Clarke, E., Kozen, D. (eds.) Logics of Programs. LNCS, vol. 164, pp. 501–512. Springer, Heidelberg (1984) CrossRefGoogle Scholar
  13. 13.
    Wolk, E.S.: The comparability graph of a tree. Proc. Am. Math. Soc. 13(5), 789–795 (1962)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Wolk, E.S.: A note on “the comparability graph of a tree”. Proc. Am. Math. Soc. 16(1), 17–20 (1965)MATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Claudia Carapelle
    • 1
  • Shiguang Feng
    • 1
  • Alexander Kartzow
    • 2
  • Markus Lohrey
    • 2
  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany
  2. 2.Department für Elektrotechnik Und InformatikUniversität SiegenSiegenGermany

Personalised recommendations