Acta Informatica

, Volume 45, Issue 1, pp 43–78 | Cite as

A decision procedure for propositional projection temporal logic with infinite models

  • Zhenhua DuanEmail author
  • Cong Tian
  • Li Zhang
Original Article


This paper investigates the satisfiability of Propositional Projection Temporal Logic (PPTL) with infinite models. A decision procedure for PPTL formulas is given. To this end, Normal Form (NF) and Labeled Normal Form Graph (LNFG) for PPTL formulas are defined, and algorithms for transforming a formula to its normal form and constructing the LNFG for the given formula are presented. Further, the finiteness of LNFGs is proved in details. Moreover, the decision procedure is extended to check the satisfiability of the formulas of Propositional Interval Temporal Logic. In addition, examples are also given to illustrate how the decision procedure works.


Normal Form Model Check Temporal Logic Decision Procedure Atomic Proposition 
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.


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  1. 1.
    Moszkowski, B.C.: Reasoning about digital circuits. PhD Thesis, Stanford University. TRSTAN-CS-83-970 (1983)Google Scholar
  2. 2.
    Rosner, R., Pnueli, A.: A choppy logic. In: First annual IEEE symposium on logic in computer science, LICS, pp. 306–314 (1986)Google Scholar
  3. 3.
    Moszkowski, B.C.: A complete axiomatization of interval temporal logic with infinite time. In: 15th Annual IEEE symposium on logic in computer science (LICS’00), LICS, p. 241, (2000)Google Scholar
  4. 4.
    Chaochen Z., Hoare C.A.R. and Ravn A.P. (1991). A calculus of duration. Inf. Process. Lett. 40(5): 269–275 zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bowman H. and Thompson S. (2003). A decision procedure and complete axiomatization of interval temporal logic with projection. J. Logic Comput. 13(2): 195–239 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dutertre, B.: Complete proof systems for first order interval temporal logic. In: Proceedings of LICS’95, pp. 36–43 (1995)Google Scholar
  7. 7.
    Wang, H., Xu, Q.: Temporal logics over infinite intervals. Technical Report 158, UNU/IIST, Macau (1999)Google Scholar
  8. 8.
    Halpern, J., Manna, Z., Moszkowski, B.: A hardware semantics based on temporal intervals. In: Proceedings of the 10th international conlloquium on automata, Languages and Programming, vol. 154. Springer, LNCS, Barcelona (1983)Google Scholar
  9. 9.
    Kono, S.: A combination of clausal and non-clausal temporal logic programs. In: Lecture notes in artificial intelligence, vol. 897, pp. 40–57. Springer, Heidelberg (1995)Google Scholar
  10. 10.
    Bowman, H., Thompson, S.: A Tableau method for interval temporal logic with projection. In: de Swart, H. (ed.) TABLEAUX98, LNAI 1397, Springer, Berlin (1998)Google Scholar
  11. 11.
    Duan, Z.: An extended interval temporal logic and a framing technique for temporal logic programming. PhD thesis, University of Newcastle Upon Tyne (1996)Google Scholar
  12. 12.
    Duan Z. (2006). Temporal Logic and Temporal Logic Programming Language. Science press, Beijing Google Scholar
  13. 13.
    Duan, Z., Koutny, M., Holt, C.: Projection in temporal logic programming. In: Pfenning, F. (ed.) Proceedings of logic programming and automatic reasoning, Lecture Notes in Artificial Intelligence, vol. 822, pp. 333–344. Springer, Heidelberg (1994)Google Scholar
  14. 14.
    Duan Z. and Koutny M. (2004). A framed temporal logic programming language. J. Comput. Sci. Technol. 19: 333–344 MathSciNetGoogle Scholar
  15. 15.
    Duan, Z., Yang, X., Kounty, M.: Semantics of framed temporal logic programs. In: Proceedings of ICLP 2005, vol. 3668, pp. 256–270. LNCS, Barcelona (2005)Google Scholar
  16. 16.
    Moszkowski, B.C.: Compositional reasoning about projected and infinite time. In: Proceeding of the first IEEE international conference on engineering of complex computer systems (ICECCS’95), pp. 238–245. IEEE Computer Society Press (1995)Google Scholar
  17. 17.
    Manna Z. and Pnueli A. (1992). The Temporal Logic of Reactive and Concurrent Systems. Springer, Heidelberg Google Scholar
  18. 18.
    Duan, Z., Zhang, L.: A decision procedure for propositional projection temporal logic. Technical Report No.1, Institute of computing Theory and Technology, Xidian University, Xi’an, People’s Republic of China, (2005)
  19. 19.
    Kripke S.A. (1963). Semantical analysis of modal logic I: normal propositional calculi. Z. Math. Logik Grund. Math. 9: 67–96 zbMATHMathSciNetGoogle Scholar
  20. 20.
    Winskel, G.: The Formal Semantics of Programming Languages. Foundations of Computing. MIT, CambridgeGoogle Scholar
  21. 21.
    Holzmann G.J. (1997). The Model Checker Spin. IEEE Trans. Softw. Eng. 23(5): 279–295 CrossRefMathSciNetGoogle Scholar
  22. 22.
    McMillan, K.L.: Symbolic Model Checking. Kluwer (1993)Google Scholar
  23. 23.
    Harel D., Kozen D. and Parikh R. (1982). Process logic: expressiveness, decidability, completeness. J. Comput. Syst. Sci. 25(2): 144–170 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Chandra A., Halpern J., Meyer A. and Parikh R. (1985). Equations between regular terms and an application to process logic. SIAM J. Comput. 14(4): 935–942 zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Moszkowski B. (1986). Executing Temporal Logic Programs. Cambridge University Press, Cambridge Google Scholar
  26. 26.
    Duan Z. (2005). Modelling and Analysis of Hybrid Systems. Science Press, Beijing Google Scholar
  27. 27.
    Duan Z., Holcombe M. and Bell A. (2000). A logic for biosystems. Biosystems 55(1-3): 93–105 CrossRefGoogle Scholar
  28. 28.
    Paech, B.: Gentzen-systems for propositional temporal logics. In: Borger, E., Kleine Buning, H., Richter, M.M. (eds.) Proceedings of the 2nd workshop on computer science logic, Duisburg (FRG), vol. 385, pp. 240–253. Springer, Heidelberg (1988)Google Scholar
  29. 29.
    Pnueli, A.: The temporal logic of programs. In: Proceedings of 18th IEEE symposium on foundations of computer science, pp. 46–57 (1977)Google Scholar
  30. 30.
    Tian, C., Duan, Z.: Model Checking Propositional Projection Temporal Logic Based on SPIN, ICFEM 2007, LNCS4789, pp. 246-265, Springer, Heidelberg (2007)Google Scholar
  31. 31.
    Kröger, F.: Temporal Logic of Programs. EATCS Monographs on Theoretical Computer Science, vol. 8. Springer, Heidelberg (1987)Google Scholar
  32. 32.
    Wolper P.L. (1983). Temporal logic can be more expressive. Inf. Control 56: 72–99 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of Computing Theory and TechnologyXidian UniversityXi’anPeople’s Republic of China

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