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Abstract

For an infinite-state concurrent system \(\mathcal{S}\) with a set AP of state predicates, its predicate abstraction defines a finite-state system whose states are subsets of AP, and its transitions s → s′ are witnessed by concrete transitions between states in \(\mathcal{S}\) satisfying the respective sets of predicates s and s′. Since it is not always possible to find such witnesses, an over-approximation adding extra transitions is often used. For systems \(\mathcal{S}\) described by formal specifications, predicate abstractions are typically built using various automated deduction techniques. This paper presents a new method—based on rewriting, semantic unification, and variant narrowing—to automatically generate a predicate abstraction when the formal specification of \(\mathcal{S}\) is given by a conditional rewrite theory. The method is illustrated with concrete examples showing that it naturally supports abstraction refinement and is quite accurate, i.e., it can produce abstractions not needing over-approximations.

Keywords

Model Check Mutual Exclusion Linear Temporal Logic Kripke Structure Tree Automaton 
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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References

  1. 1.
    Abdulla, P.A., Chen, Y.-F., Delzanno, G., Haziza, F., Hong, C.-D., Rezine, A.: Constrained monotonic abstraction: A CEGAR for parameterized verification. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 86–101. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Avenhaus, J., Loría-Sáenz, C.: On conditional rewrite systems with extra variables and deterministic logic programs. In: Pfenning, F. (ed.) LPAR 1994. LNCS, vol. 822, pp. 215–229. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  3. 3.
    Baader, F., Snyder, W.: Unification theory. In: Handbook of Automated Reasoning, pp. 445–532. Elsevier and MIT Press (2001)Google Scholar
  4. 4.
    Bae, K., Escobar, S., Meseguer, J.: Abstract logical model checking of infinite-state systems using narrowing. In: RTA. LIPIcs, vol. 21, pp. 81–96 (2013)Google Scholar
  5. 5.
    Ball, T., Majumdar, R., Millstein, T., Rajamani, S.K.: Automatic predicate abstraction of C programs. ACM SIGPLAN Notices 36(5), 203–213 (2001)CrossRefGoogle Scholar
  6. 6.
    Clarke, E., 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
  7. 7.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. The MIT Press (2001)Google Scholar
  8. 8.
    Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Talcott, C.: All About Maude - A High-Performance Logical Framework. LNCS, vol. 4350. Springer, Heidelberg (2007)zbMATHGoogle Scholar
  9. 9.
    Comon-Lundh, H., Delaune, S.: The finite variant property: How to get rid of some algebraic properties. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 294–307. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Das, S., Dill, D.L., Park, S.: Experience with predicate abstraction. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 160–171. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Dershowitz, N., Jouannaud, J.P.: Rewrite systems. In: Handbook of Theoretical Computer Science, vol. B, pp. 243–320. North-Holland (1990)Google Scholar
  12. 12.
    Durán, F., Meseguer, J.: A Church-Rosser checker tool for conditional order-sorted equational Maude specifications. In: Ölveczky, P.C. (ed.) WRLA 2010. LNCS, vol. 6381, pp. 69–85. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Escobar, S., Meseguer, J.: Symbolic model checking of infinite-state systems using narrowing. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 153–168. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Escobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. J. Algebraic and Logic Programming 81, 898–928 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Genet, T., Rusu, V.: Equational approximations for tree automata completion. Journal of Symbolic Computation 45(5), 574–597 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Graf, S., Saïdi, H.: Construction of abstract state graphs with PVS. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 72–83. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  17. 17.
    Lahiri, S.K., Bryant, R.E., Cook, B.: A symbolic approach to predicate abstraction. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 141–153. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    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)CrossRefGoogle Scholar
  19. 19.
    Meseguer, J.: Conditional rewriting logic as a unified model of concurrency. Theor. Comp. Sci. 96(1), 73–155 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Meseguer, J.: Twenty years of rewriting logic. J. Algebraic and Logic Programming 81, 721–781 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Meseguer, J., Palomino, M., Martí-Oliet, N.: Equational abstractions. Theor. Comp. Sci. 403(2-3), 239–264 (2008)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ohsaki, H., Seki, H., Takai, T.: Recognizing boolean closed A-tree languages with membership conditional rewriting mechanism. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 483–498. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. 23.
    Palomino, M.: A predicate abstraction tool for maude (2005), http://maude.sip.ucm.es/~miguelpt/bibliography.html
  24. 24.
    Viry, P.: Equational rules for rewriting logic. Theor. Comp. Sci. 285 (2002)Google Scholar
  25. 25.
    Visser, W., Havelund, K., Brat, G., Park, S., Lerda, F.: Model checking programs. Automated Software Engineering 10(2), 203–232 (2003)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Kyungmin Bae
    • 1
  • José Meseguer
    • 1
  1. 1.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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