Journal of Logic, Language and Information

, Volume 18, Issue 4, pp 593–624

Branching-Time Logics Repeatedly Referring to States



While classical temporal logics lose track of a state as soon as a temporal operator is applied, several branching-time logics able to repeatedly refer to a state have been introduced in the literature. We study such logics by introducing a new formalism, hybrid branching-time logics, subsuming the other approaches and making the ability to refer to a state more explicit by assigning a name to it. We analyze the expressive power of hybrid branching-time logics and the complexity of their satisfiability problem. As main result, the satisfiability problem for the hybrid versions of several branching-time logics is proved to be 2EXPTIME-complete. To prove the upper bound, the automata-theoretic approach to branching-time logics is extended to hybrid logics. As a result of independent interest, the nonemptiness problem for alternating one-pebble Büchi tree automata is shown to be 2EXPTIME-complete. A common property of the logics studied is that they refer to only one state. This restriction is crucial: The ability to refer to more than one state causes a nonelementary blow-up in complexity. In particular, we prove that satisfiability for NCTL* has nonelementary complexity.


Branching time logic Hybrid logic Temporal logic Complexity CTL Pebble automata Alternating tree automata Symmetric automata 


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  1. Adler M., Immerman N. (2003) An n! lower bound on formula size. ACM Transactions on Computational Logic 4(3): 296–314CrossRefGoogle Scholar
  2. Areces, C., Blackburn, P., & Marx, M. (1999). A road-map on complexity for hybrid logics. In Proceedings of the 13th international workshop on computer science logic (CSL ’99), LNCS (Vol. 1683, pp. 307–321). Springer.Google Scholar
  3. Areces C., Blackburn P., Marx M. (2001) Hybrid logics: Characterization, interpolation and complexity. Journal of Symbolic Logic 66(3): 977–1010CrossRefGoogle Scholar
  4. Areces, C., & ten Cate, B. (2007). Hybrid logics. In Handbook of modal logic, studies in logic (Vol. 3, pp. 821–868). New York: Elsevier.Google Scholar
  5. Ben-Ari M., Pnueli A., Manna Z. (1983) The temporal logic of branching time. Acta Informatica 20: 207–226CrossRefGoogle Scholar
  6. Bozzelli, L. (2008). The complexity of CTL* + linear past. In Proceedings of the 11th international conference on foundations of software science and computational structures (FOSSACS 2008), LNCS (Vol. 4962, pp. 186–200). Springer.Google Scholar
  7. Chlebus B.S. (1986) Domino-tiling games. Journal of Computer and System Sciences 32(3): 374–392CrossRefGoogle Scholar
  8. Clarke, E. M., & Emerson, E. A. (1981). Design and synthesis of synchronization skeletons using branching- time temporal logic. In Proceedings logic of programs, LNCS (Vol. 131, pp. 52–71). Springer.Google Scholar
  9. Clarke E.M., Grumberg O., Peled D.A. (1999) Model checking. MIT Press, CambridgeGoogle Scholar
  10. Demri, S., & Lazić, R. (2006). LTL with the freeze quantifier and register automata. In Proceedings of the 21th IEEE symposium on logic in computer science (LICS 2006) (pp. 17–26). IEEE.Google Scholar
  11. Emerson E.A., Halpern J.Y. (1986) “Sometimes” and “not never” revisited: On branching versus linear time temporal logic. Journal of the ACM 33(1): 151–178CrossRefGoogle Scholar
  12. Emerson, E. A., & Jutla, C. S. (1991). Tree automata, mu-calculus and determinacy. In Proceedings of the 32nd IEEE annual symposium on foundations of computer science (FOCS ’91) (pp. 368–377). IEEE.Google Scholar
  13. Franceschet M., de Rijke M. (2006) Model checking hybrid logics (with an application to semistructured data). Journal of Applied Logic 4(3): 279–304CrossRefGoogle Scholar
  14. Franceschet, M., de Rijke, M., & Schlingloff, B. H. (2003). Hybrid logics on linear structures: Expressivity and complexity. In Proceedings of the 10th international symposium on temporal representation and reasoning/4th international conference on temporal logic (TIME-ICTL 2003) (pp. 192–202). IEEE.Google Scholar
  15. Goranko, V. (1994). Temporal logic with reference pointers. In Proceedings of the first international conference on temporal logic (ICTL ’94), LNCS (Vol. 827, pp. 133–148). Springer.Google Scholar
  16. Grumberg, O., & Veith, H. (Eds.). (2008). 25 Years of model checking—history, achievements, perspectives, LNCS (Vol. 5000). Springer.Google Scholar
  17. Hafer, T., & Thomas, W. (1987). Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree. In Proceedings of the 14th international colloquium on automata, languages and programming (ICALP ’87), LNCS (Vol. 267, pp. 269–279). Springer.Google Scholar
  18. Jurdziński, M., & Lazić, R. (2007). Alternation-free mu-calculus for data trees. In Proceedings of the 22th IEEE symposium on logic in computer science (LICS 2007), IEEE.Google Scholar
  19. Kupferman, O., & Vardi, M. Y. (2006). Memoryful branching-time logic. In Proceedings of the 21st IEEE symposium on logic in computer science (LICS 2006) (pp. 265–274). IEEE.Google Scholar
  20. Laroussinie, F., Markey, N., & Schnoebelen, P. (2002). Temporal logic with forgettable past. In Proceedings of the 17th IEEE symposium on logic in computer science (LICS 2002) (pp. 383–392). IEEE.Google Scholar
  21. Laroussinie F., Schnoebelen P. (1995) A hierarchy of temporal logics with past. Theoretical Computer Science 148(2): 303–324CrossRefGoogle Scholar
  22. Laroussinie F., Schnoebelen P. (2000) Specification in CTL + Past for verification in CTL. Logic in Computer Science 156(1-2): 236–263Google Scholar
  23. Moller, F., & Rabinovich, A. M. (1999). On the expressive power of CTL*. In Proceedings of the 14th annual IEEE symposium on logic in computer science (LICS ’99) (pp. 360–369). IEEE.Google Scholar
  24. Mundhenk, M., Schneider, T., Schwentick, T., & Weber, V. (2005). Complexity of hybrid logics over transitive frames. In Proceedings of M4M-4, Humbold-Universität Berlin, Informatik-Berichte (Vol. 194, pp. 62–78).Google Scholar
  25. Rabin, M. (1970). Weakly definable relations and special automata. In Proceedings of symposium mathematical logic and foundations of set theory, North Holland (pp. 1–23).Google Scholar
  26. Schwentick, T., & Weber, V. (2007). Bounded-variable fragments of hybrid logics. In Proceedings of the 24th annual symposium on theoretical aspects of computer science (STACS 2007), LNCS (Vol. 4393, pp. 561–572). Springer.Google Scholar
  27. Stockmeyer, L. J. (1974). The complexity of decision problems in automata theory and logic. PhD thesis, MIT.Google Scholar
  28. ten Cate, B., & Franceschet, M. (2005). On the complexity of hybrid logics with binders. In Proceedings of the 19th international workshop on computer science logic (CSL 2005), LNCS (Vol. 3634, pp. 339–354). Springer.Google Scholar
  29. Thomas W. (1990) Automata on infinite objects. In: van Leeuwen J. (eds) Handbook of theoretical computer science, Vol. B: Formal models and sematics. Elsevier, MIT Press, pp 133–192Google Scholar
  30. Vardi, M. Y. (1995). Alternating automata and program verification. In Computer science today, LNCS (Vol. 1000, pp. 471–485). Heidelberg: Springer.Google Scholar
  31. Vardi, M. Y. (1998). Reasoning about the past with two-way automata. In Proceedings of the 25th international colloquium on automata, languages and programming (ICALP ’98), LNCS (Vol. 1443, pp. 628–641). Springer.Google Scholar
  32. Vardi, M. Y. (2007). Automata-theoretic techniques for temporal reasoning. In Handbook of modal logic, studies in logic (Vol. 3, pp. 971–989). Elsevier.Google Scholar
  33. Vardi, M. Y., & Stockmeyer, L. J. (1985). Improved upper and lower bounds for modal logics of programs: Preliminary report. In Proceedings of the 17th annual ACM symposium on theory of computing (STOC ’85), ACM (pp. 240–251).Google Scholar
  34. Wilke, T. (1999). CTL+ is exponentially more succinct than CTL. In Proceedings of the 19th conference on foundations of software technology and theoretical computer science (FSTTCS), LNCS (Vol. 1738, pp. 110–121). Springer.Google Scholar
  35. Zielonka W. (1998) Infinite games on finitely coloured graphs with applications to automata and infinite trees. Theoretical Computer Science 200: 135–183CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Fakultät für InformatikTechnische Universität DortmundDortmundGermany

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