Equational Abstractions

  • José Meseguer
  • Miguel Palomino
  • Narciso Martí-Oliet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2741)


Abstraction reduces the problem of whether an infinite state system satisfies a temporal logic property to model checking that property on a finite state abstract version. The most common abstractions are quotients of the original system. We present a simple method of defining quotient abstractions by means of equations collapsing the set of states. Our method yields the minimal quotient system together with a set of proof obligations that guarantee its executability and can be discharged with tools such as those in the Maude formal environment.


Model Check Theorem Prove Mutual Exclusion Linear Temporal Logic 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abdulla, P., Annichini, A., Bouajjani, A.: Symbolic verification of lossy channel systems: Application to the bounded retransmission protocol. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, p. 208. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  2. 2.
    Bensalem, S., Lakhnech, Y., Owre, S.: Computing abstractions of infinite state systems compositionally and automatically. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 319–331. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  3. 3.
    Clarke, E.M., Grumberg, O., Long, D.E.: Model checking and abstraction. ACM Transactions on Programming Languages and Systems 16, 1512–1542 (1994)CrossRefGoogle Scholar
  4. 4.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (1999)Google Scholar
  5. 5.
    Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 154–169. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Colón, M.A., Uribe, T.E.: Generating finite-state abstractions of reactive systems using decision procedures. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 293–304. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Dams, D., Gerth, R., Grumberg, O.: Abstract interpretation of reactive systems. ACM Transactions on Programming Languages and Systems 19, 253–291 (1997)CrossRefGoogle Scholar
  8. 8.
    Havelund, K., Shankar, N.: Experiments in theorem proving and model checking for protocol verification. In: Gaudel, M.-C., Woodcock, J.C.P. (eds.) FME 1996. LNCS, vol. 1051, pp. 662–681. Springer, Heidelberg (1996)Google Scholar
  9. 9.
    Kesten, Y., Pnueli, A.: Control and data abstraction: The cornerstones of practical formal verification. International Journal on Software Tools for Technology Transfer 4, 328–342 (2000)CrossRefGoogle Scholar
  10. 10.
    Kesten, Y., Pnueli, A.: Verification by augmentary finitary abstraction. Information and Computation 163, 203–243 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Loiseaux, C., Graf, S., Sifakis, J., Bouajjani, A., Bensalem, S.: Property preserving abstractions for the verification of concurrent systems. Formal Methods in System Design 6, 1–36 (1995)CrossRefGoogle Scholar
  12. 12.
    Manolios, P.: Mechanical Verification of Reactive Systems. PhD thesis, Univ. of Texas at Austin (2001) Google Scholar
  13. 13.
    Müller, O., Nipkow, T.: Combining model checking and deduction for I/Oautomata. In: Brinksma, E., Steffen, B., Cleaveland, W.R., Larsen, K.G., Margaria, T. (eds.) TACAS 1995. LNCS, vol. 1019, pp. 1–16. Springer, Heidelberg (1995)Google Scholar
  14. 14.
    Saïdi, H., Shankar, N.: Abstract and model check while you prove. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 443–454. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  15. 15.
    Uribe Restrepo, T.E.: Abstraction-Based Deductive-Algorithmic Verification of Reactive Systems. PhD thesis, Dept. of Computer Science, Stanford Univ. (1998) Google Scholar
  16. 16.
    Clavel, M., Durán, F., Ecker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Quesada, J.F.: Maude: Specification and programming in rewriting logic. Theoretical Computer Science 285, 187–243 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Eker, S., Meseguer, J., Sridharanarayanan, A.: The Maude LTL model checker. In: Gadducci, F., Montanari, U. (eds.) Rewriting Logic and its Applications, WRLA 2004. ENTCS, vol. 71. Elsevier, Amsterdam (2002)Google Scholar
  18. 18.
    Clavel, M.: The ITP tool. In: Nepomuceno, A., et al. (eds.) Logic, Language, and Information, Kronos, pp. 55–62 (2001)Google Scholar
  19. 19.
    Durán, F., Meseguer, J.: A Church-Rosser checker tool for Maude equational specifications (2000),
  20. 20.
    Durán, F.: Coherence checker and completion tools for Maude specifications (2000),
  21. 21.
    Meseguer, J., Palomino, M., Martí-Oliet, N.: Notes on model checking and abstraction in rewriting logic (2002),
  22. 22.
    Meseguer, J.: Conditional rewriting logic as a unified model of concurrency. Theoretical Computer Science 96, 73–155 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Meseguer, J.: Membership algebra as a logical framework for equational specification. In: Parisi-Presicce, F. (ed.) WADT 1997. LNCS, vol. 1376, pp. 18–61. Springer, Heidelberg (1998)Google Scholar
  24. 24.
    Borovanský, P., Kirchner, C., Kirchner, H., Moreau, P.E.: ELAN from a rewriting logic point of view. Theoretical Computer Science 285, 155–185 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Futatsugi, K., Diaconescu, R.: CafeOBJ Report. World Scientific, Singapore (1998)zbMATHGoogle Scholar
  26. 26.
    Lamport, L.: A new solution of Dijkstra’s concurrent programming problem. Communications of the ACM 17, 453–455 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Dershowitz, N., Jouannaud, J.P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 243–320. North-Holland, Amsterdam (1990)Google Scholar
  28. 28.
    Viry, P.: Equational rules for rewriting logic. Theoretical Computer Science 285 (2002)Google Scholar
  29. 29.
    Contejean, E., Marché, C.: The CiME system: tutorial and user’s manual. Manuscript, Univ. Paris-Sud, Centre d’OrsayGoogle Scholar
  30. 30.
    Durán, F.: Termination checker and Knuth-Bendix completion tools for Maude equational specifications. Manuscript, Computer Science Laboratory, SRI International (2000),

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • José Meseguer
    • 1
  • Miguel Palomino
    • 1
    • 2
  • Narciso Martí-Oliet
    • 2
  1. 1.Computer Science DepartmentUniversity of Illinois at Urbana-Champaign 
  2. 2.Departamento de Sistemas InformáticosUniversidad Complutense de Madrid 

Personalised recommendations