Formal Aspects of Computing

, Volume 27, Issue 2, pp 309–334 | Cite as

Abstraction and approximation in fuzzy temporal logics and models

  • Gholamreza SotudehEmail author
  • Ali Movaghar
Original Article


Recently, by defining suitable fuzzy temporal logics, temporal properties of dynamic systems are specified during model checking process, yet a few numbers of fuzzy temporal logics along with capable corresponding models are developed and used in system design phase, moreover in case of having a suitable model, it suffers from the lack of a capable model checking approach. Having to deal with uncertainty in model checking paradigm, this paper introduces a fuzzy Kripke model (FzKripke) and then provides a verification approach using a novel logic called Fuzzy Computation Tree Logic* (FzCTL*). Not only state space explosion is handled using well-known concepts like abstraction and bisimulation, but an approximation method is also devised as a novel technique to deal with this problem. Fuzzy program graph, a generalization of program graph and FzKripke, is also introduced in this paper in consideration of higher level abstraction in model construction. Eventually modeling, and verification of a multi-valued flip-flop is studied in order to demonstrate capabilities of the proposed models.


Abstraction Approximation Fuzzy Kripke model Fuzzy temporal logic Model checking Fuzzy program graph Multi-valued flip-flop 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. AD94.
    Alur R, Dill DL (1994) A theory of timed automata. Theor Comput Sci 126: 183–235CrossRefzbMATHMathSciNetGoogle Scholar
  2. BCL+94.
    Burch JR, Clarke EM, Long DE, McMillan KL, Dill DL (1994) Symbolic model checking for sequential circuit verification. IEEE Trans Comput -Aided Des Integr Circuits Syst 13(4): 401–424CrossRefGoogle Scholar
  3. BG04.
    Bruns G, Godefroid P (2004) Model checking with multi-valued logics. In: Lecture notes in computer science, vol 3142. pp 281–293Google Scholar
  4. BK.
    Baier C, Katoen J-P (2008) Principles of model checking. MIT Press, Cambridge, p 975. doi: 10.1093/comjnl/bxp025
  5. BPS10.
    Baresi L, Pasquale L, Spoletini P (2010) Fuzzy goals for requirements-driven adaptation. In: 2010 18th IEEE international requirements engineering conference. IEEE, New York, pp 125–134Google Scholar
  6. CGD+06.
    Chechik M, Gurfinkel A, Devereux B, Lai A, Easterbrook S (2006) Data structures for symbolic multi-valued model-checking. Form Methods Syst Des 29(3): 295–344CrossRefzbMATHGoogle Scholar
  7. CGP99.
    Clarke EM Jr, Grumberg O, Peled DA (1999) Model checking. MIT Press, CambridgeGoogle Scholar
  8. Cun04.
    Cunningham PA (2004) Verification of asynchronous circuits. Ph.D. Thesis, University of Cambridge, CambridgeGoogle Scholar
  9. ECD+03.
    Easterbrook S, Chechik M, Devereux B, Gurfinkel A, Lai A, Petrovykh V, Tafliovich A, Thompson-Walsh C (2003) \({\chi}\) Chek: a model checker for multi-valued reasoning. In: ICSE ’03: Proceedings of the 25th International Conference on Software Engineering. pp 804–805Google Scholar
  10. EFR74.
    Epstein G, Frieder G, Rine DC (1974) The development of multiple-valued logic as related to computer science. Computer 7(9): 20–32CrossRefzbMATHGoogle Scholar
  11. Fai05.
    Fainekos GE (2005) An introduction to multi-valued model checking. Technical Reports (CIS)Google Scholar
  12. FPS12.
    Frigeri A, Pasquale L, Spoletini P (2012) Fuzzy time in ltl. arXiv:1203.6278 (arXiv preprint)
  13. FSGC05.
    Fern A, Solla AG, Garc J, Cabrer MR (2005) Multi-valued model checking in dense-time. In: Lecture notes in computer science, vol 3751. pp 638–649Google Scholar
  14. Gur03.
    Gurfinkel A (2003) Multi-valued symbolic model-checking: fairness, counter-examples, running time. University of TorontoGoogle Scholar
  15. Gur10.
    Gurumurthy KS (2010) Quaternary sequential circuits. IJCSNS 10(7): 110–117Google Scholar
  16. HO89.
    Hirota K, Ozawa K (1989) The concept of fuzzy flip-flop. IEEE Trans Syst Man Cybern 19(5): 980–997CrossRefMathSciNetGoogle Scholar
  17. HP95.
    Hirota K, Pedrycz W (1995) Design of fuzzy systems with fuzzy flip-flops. IEEE Trans Syst Man Cybern 25(1). doi: 10.1109/21.362956
  18. HR04.
    Huth M, Ryan M (2004) Logic in computer science: modeling and reasoning about systems Cambridge University Press, New York.
  19. Hur84.
    Hurst SL (1984) Multiple-valued logic? Its status and its future. IEEE Trans Comput C-33(12):1160–1179Google Scholar
  20. IMMT05.
    Intrigila B, Magazzeni D, Melatti I, Tronci E (2005) A model checking technique for the verification of fuzzy control systems. In: Proceedings of the international conference on computational intelligence for modelling, control and automation and international conference on intelligent agents, web technologies and internet commerce vol-1 (CIMCA-IAWTIC’06), vol 01. CIMCA ’05, Washington, DC, USA. IEEE Computer Society, Silver Spring, pp 536–542Google Scholar
  21. KHH87.
    Kameyama M, Hanyu T, Higuchi T (1987) Design and implementation of quaternary NMOS integrated circuits for pipelined image processing. IEEE J Solid-State Circuits 22(1): 20–27CrossRefGoogle Scholar
  22. KNSW07.
    Kwiatkowska M, Norman G, Sproston J, Wang F (2007) Symbolic model checking for probabilistic timed automata. Inf Comput 205(7): 1027–1077CrossRefzbMATHMathSciNetGoogle Scholar
  23. KP03.
    Konikowka B, Penczek W (2003) On designated values in multi-valued ctl* model checking. Fundam Inf 60(1–4): 211–224Google Scholar
  24. LN96.
    Lind-Nielsen J (1996) BuDDy—a binary decision diagram package. IT-TR. Department of Information Technology, Technical University of DenmarkGoogle Scholar
  25. MLL04.
    Moon S, Lee KH, Lee D (2004) Fuzzy branching temporal logic. IEEE Trans Syst Man Cybern Part B Cybern 34(2): 1045–1055CrossRefGoogle Scholar
  26. MP95.
    Maler O, Pnueli A (1995) Timing analysis of asynchronous circuits using timed automata. In: Camurati PE, Eveking H (eds) Proceedings of CHARME’95, LNCS 987. Springer, Berlin, pp 189–205Google Scholar
  27. MSSY95.
    Mateescu A, Salomaa A, Salomaa K, Yu S (1995) Lexical analysis with a simple finite-fuzzy-automaton model. J Univers Comput Sci 1: 292–311zbMATHMathSciNetGoogle Scholar
  28. OHK+95.
    Ozawa K, Hirota K, Koczy LT, Pedrycz W, Ikoma N (1995) Summary of fuzzy flip-flop. In: Proceedings of 1995 IEEE International Conference on Fuzzy Systems, vol 3. doi: 10.1109/FUZZY.1995.409897
  29. OHK96.
    Ozawa K, Hirota K, Koczy LT (1996) Fuzzy flip-flop. In: Patyra MJ, Mlynek DM (eds) Fuzzy logic. Implementation and applications. Chichester, Wiley, pp 197–236Google Scholar
  30. Pal00.
    Palshikar GK (2000) Representing fuzzy temporal knowledge. In: International conference on knowledge-based systems (KBCS-2000), Mumbai, India. pp 252–263Google Scholar
  31. RA11.
    Rajaretnam T, Ayyaswamy SK (2011) Fuzzy monoids in a fuzzy finite state automaton with unique membership transition on an input symbol. Int J Math Sci Comput I(I):48–51Google Scholar
  32. Sla05.
    Sladoje N (2005) On analysis of discrete spatial fuzzy sets in 2 and 3 dimensions. Ph.D. thesis, Swedish University of Agricultural Sciences UppsalaGoogle Scholar
  33. Smi81.
    Smith KC (1981) The prospects for multivalued logic: a technology and applications view. IEEE Trans Comput C-30(9):619–634Google Scholar
  34. Smi88.
    Smith KC (1988) A multiple valued logic: a tutorial and appreciation. Computer 21(4): 17–27CrossRefGoogle Scholar
  35. Tar55.
    Tarski A (1955) A lattice-theoretical fixpoint theorem and its applications. Pac J Math 5: 285–309CrossRefzbMATHMathSciNetGoogle Scholar
  36. Vaz01.
    Vazirani VV (2001) Approximation algorithms. Springer, Berlin, p 68Google Scholar
  37. Wan06.
    Wang F (2006) Symbolic model checking for probabilistic real-time systems. Ph.D. thesis, University of Birmingham, BirminghamGoogle Scholar
  38. Wie10.
    Wierman MJ (2010) An introduction to the mathematics of uncertainty. Creighton University, p 133.
  39. WSB+09.
    Whittle J, Sawyer P, Bencomo N, Cheng BHC, Bruel J-M (2009) Relax: incorporating uncertainty into the specification of self-adaptive systems. In: 2009 17th IEEE international requirements engineering conference, Atalanta, Georgia, August. IEEE, New York, pp 79–88Google Scholar
  40. WwWG07.
    Wang L, wen Wang B, Guo Y (2007) Cell mapping description for digital control system with quantization effect. Technical reportGoogle Scholar

Copyright information

© British Computer Society 2014

Authors and Affiliations

  1. 1.Department of Computer EngineeringIslamic Azad University (Science and Research Branch)TehranIran
  2. 2.Department of Computer EngineeringSharif University of TechnologyTehranIran

Personalised recommendations