Graded Computation Tree Logic with Binary Coding

  • Alessandro Bianco
  • Fabio Mogavero
  • Aniello Murano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6247)


Graded path quantifiers have been recently introduced and investigated as a useful framework for generalizing standard existential and universal path quantifiers in the branching-time temporal logic CTL (GCTL), in such a way that they can express statements about a minimal and conservative number of accessible paths. These quantifiers naturally extend to paths the concept of graded world modalities, which has been deeply investigated for the μ- Calculus (Gμ- Calculus) where it allows to express statements about a given number of immediately accessible worlds. As for the ”non-graded” case, it has been shown that the satisfiability problem for GCTL and the Gμ- Calculus coincides and, in particular, it remains solvable in ExpTime. However, GCTL has been only investigated w.r.t. graded numbers coded in unary, while Gμ- Calculus uses for this a binary coding, and it was left open the problem to decide whether the same result may or may not hold for binary GCTL. In this paper, by exploiting an automata theoretic-approach, which involves a model of alternating automata with satellites, we answer positively to this question. We further investigate the succinctness of binary GCTL and show that it is at least exponentially more succinct than Gμ- Calculus.


Model Check Binary Code Atomic Proposition Acceptance Condition Kripke Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, Heidelberg (1976)Google Scholar
  2. 2.
    Bianco, A., Mogavero, F., Murano, A.: Graded Computation Tree Logic. In: LICS 2009, pp. 342–351 (2009)Google Scholar
  3. 3.
    Bonatti, P.A., Lutz, C., Murano, A., Vardi, M.Y.: The Complexity of Enriched μ-Calculi. LMCS 4(3:11), 1–27 (2008)MathSciNetGoogle Scholar
  4. 4.
    Clarke, E.M., Emerson, E.A.: Design and Synthesis of Synchronization Skeletons Using Branching-Time Temporal Logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  5. 5.
    Emerson, E.A., Halpern, J.Y.: “Sometimes” and “Not Never” Revisited: On Branching Versus Linear Time. JACM 33(1), 151–178 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ferrante, A., Napoli, M., Parente, M.: CTL Model-Checking with Graded Quantifiers. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 18–32. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Ferrante, A., Napoli, M., Parente, M.: Graded-CTL: Satisfiability and Symbolic Model Checking. In: Breitman, K., Cavalcanti, A. (eds.) ICFEM 2009. LNCS, vol. 5885, pp. 306–325. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Fine, K.: In So Many Possible Worlds. NDJFL 13, 516–520 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    De Giacomo, G., Lenzerini, M.: Concept Language with Number Restrictions and Fixpoints, and its Relationship with Mu-calculus. In: ECAI 1994, pp. 411–415 (1994)Google Scholar
  10. 10.
    Grädel, E.: On The Restraining Power of Guards. JSL 64(4), 1719–1742 (1999)zbMATHCrossRefGoogle Scholar
  11. 11.
    Kupferman, O., Sattler, U., Vardi, M.Y.: The Complexity of the Graded μ-Calculus. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 423–437. Springer, Heidelberg (2002)Google Scholar
  12. 12.
    Kupferman, O., Vardi, M.Y.: Memoryful Branching-Time Logic. In: LICS 2006, pp. 265–274. IEEE Computer S., Los Alamitos (2006)Google Scholar
  13. 13.
    Kupferman, O., Vardi, M.Y., Wolper, P.: An Automata Theoretic Approach to Branching-Time Model Checking. JACM 47(2), 312–360 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lamport, L.: “Sometime” is Sometimes “Not Never”: On the Temporal Logic of Programs. In: POPL 1980, pp. 174–185 (1980)Google Scholar
  15. 15.
    Lange, M.: A Purely Model-Theoretic Proof of the Exponential Succinctness Gap between CTL+ and CTL. IPL 108(5), 308–312 (2008)CrossRefGoogle Scholar
  16. 16.
    Lutz, C.: Complexity and Succinctness of Public Announcement Logic. In: AAMAS 2006, pp. 137–143 (2006)Google Scholar
  17. 17.
    Miyano, S., Hayashi, T.: Alternating Finite Automata on ω-Words. TCS 32, 321–330 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Muller, D.E., Schupp, P.E.: Alternating Automata on Infinite Trees. TCS 54(2-3), 267–276 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pnueli, A.: The Temporal Logic of Programs. In: FOCS 1977, pp. 46–57 (1977)Google Scholar
  20. 20.
    Pnueli, A.: The Temporal Semantics of Concurrent Programs. TCS 13, 45–60 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Rabin, M.O.: Decidability of Second-Order Theories and Automata on Infinite Trees. BAMS 74, 1025–1029 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Tobies, S.: PSpace Reasoning for Graded Modal Logics. JLC 11(1), 85–106 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Vardi, M.Y., Wolper, P.: An Automata-Theoretic Approach to Automatic Program Verification. In: LICS 1986, pp. 332–344 (1986)Google Scholar
  24. 24.
    Vardi, M.Y., Wolper, P.: Automata-Theoretic Techniques for Modal Logics of Programs. JCSS 32(2), 183–221 (1986)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Wilke, T.: CTL+ is Exponentially More Succinct than CTL. In: Pandu Rangan, C., Raman, V., Sarukkai, S. (eds.) FST TCS 1999. LNCS, vol. 1738, pp. 110–121. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Alessandro Bianco
    • 1
  • Fabio Mogavero
    • 1
  • Aniello Murano
    • 1
  1. 1.Universitá degli Studi di Napoli ”Federico II”NapoliItaly

Personalised recommendations