Sentence-Normalized Conditional Narrowing Modulo in Rewriting Logic and Maude

  • Luis Aguirre
  • Narciso Martí-OlietEmail author
  • Miguel Palomino
  • Isabel Pita
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9200)


This work studies the relationship between verifiable and computable answers for reachability problems in rewrite theories with an underlying membership equational logic. These problems have the form
$$\begin{aligned} (\exists \bar{x})t(\bar{x})\rightarrow ^* t'(\bar{x}) \end{aligned}$$
with \(\bar{x}\) some variables, or a conjunction of several of these subgoals. A calculus that solves this kind of problems working always with normalized terms and substitutions has been developed. Given a reachability problem in a rewrite theory, this calculus can compute any normalized answer that can be checked by rewriting, or one that can be instantiated to that answer. Special care has been taken in the calculus to keep membership information attached to each term, to make use of it whenever possible.


Maude Narrowing Reachability Rewriting logic Unification Membership equational logic 



We are very grateful to the anonymous referees for all their helpful comments.


  1. 1.
    Aguirre, L., Martí-Oliet, N., Palomino, M., Pita, I.: Conditional narrowing modulo in rewriting logic and Maude. In: Escobar [11], pp. 80–96Google Scholar
  2. 2.
    Antoy, S., Echahed, R., Hanus, M.: A needed narrowing strategy. In: Boehm, H., Lang, B., Yellin, D.M. (eds.) Conference Record of POPL 1994: 21st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Portland, Oregon, USA, 17–21 January 1994, pp. 268–279. ACM Press (1994)Google Scholar
  3. 3.
    Bae, K., Meseguer, J.: Model checking LTLR formulas under localized fairness. In: Durán, F. (ed.) WRLA 2012. LNCS, vol. 7571, pp. 99–117. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  4. 4.
    Bae, K., Meseguer, J.: Infinite-state model checking of LTLR formulas using narrowing. In: Escobar [11], pp. 113–129Google Scholar
  5. 5.
    Bockmayr, A.: Conditional narrowing modulo a set of equations. Appl. Algebra Eng. Commun. Comput. 4, 147–168 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bruni, R., Meseguer, J.: Semantic foundations for generalized rewrite theories. Theor. Comput. Sci. 360(1–3), 386–414 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cholewa, A., Escobar, S., Meseguer, J.: Constrained narrowing for conditional equational theories modulo axioms. Technical report, C.S. Department, University of Illinois at Urbana-Champaign, August 2014.
  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: How to Specify, Program, and Verify Systems in Rewriting Logic. LNCS, vol. 4350. Springer, Heidelberg (2007)Google Scholar
  9. 9.
    Durán, F., Lucas, S., Marché, C., Meseguer, J., Urbain, X.: Proving operational termination of membership equational programs. High. Order Symb. Computat. 21(1–2), 59–88 (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Durán, F., Meseguer, J.: On the Church-Rosser and coherence properties of conditional order-sorted rewrite theories. J. Log. Algebr. Program. 81(7–8), 816–850 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Escobar, S. (ed.): WRLA 2014. LNCS, vol. 8663. Springer, Heidelberg (2014) zbMATHGoogle Scholar
  12. 12.
    Escobar, S., Meadows, C., Meseguer, J.: Maude-NPA: cryptographic protocol analysis modulo equational properties. In: Aldini, A., Barthe, G., Gorrieri, R. (eds.) FOSAD 2007/2008/2009. LNCS, vol. 5705, pp. 1–50. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  13. 13.
    Escobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. J. Log. Algebr. Program. 81(7–8), 898–928 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fay, M.: First-order Unification in an Equational Theory. University of California (1978)Google Scholar
  15. 15.
    Giovannetti, E., Moiso, C.: A completeness result for E-unification algorithms based on conditional narrowing. In: Boscarol, M., Aiello, L.C., Levi, G. (eds.) Foundations of Logic and Functional Programming. LNCS, vol. 306, pp. 157–167. Springer, Heidelberg (1986) CrossRefGoogle Scholar
  16. 16.
    Hamada, M.: Strong completeness of a narrowing calculus for conditional rewrite systems with extra variables. Electr. Notes Theor. Comput. Sci. 31, 89–103 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inf. Process. Lett. 95(4), 446–453 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lucas, S., Meseguer, J.: Operational termination of membership equational programs: the order-sorted way. Electr. Notes Theor. Comput. Sci. 238(3), 207–225 (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Martí-Oliet, N., Meseguer, J.: Rewriting logic: roadmap and bibliography. Theor. Comput. Sci. 285(2), 121–154 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Meseguer, J.: Rewriting as a unified model of concurrency. In: Baeten, J., Klop, J. (eds.) CONCUR 1990 Theories of Concurrency: Unification and Extension. LNCS, vol. 458, pp. 384–400. Springer, Heidelberg (1990) CrossRefGoogle Scholar
  21. 21.
    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
  22. 22.
    Meseguer, J.: Twenty years of rewriting logic. J. Log. Algebr. Program. 81(7–8), 721–781 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Meseguer, J.: Strict coherence of conditional rewriting Modulo axioms. Technical report, C.S. Department, University of Illinois at Urbana-Champaign, August 2014.
  24. 24.
    Meseguer, J., Thati, P.: Symbolic reachability analysis using narrowing and its application to verification of cryptographic protocols. High. Order Symb. Comput. 20(1–2), 123–160 (2007)CrossRefzbMATHGoogle Scholar
  25. 25.
    Middeldorp, A., Hamoen, E.: Completeness results for basic narrowing. Appl. Algebra Eng. Commun. Comput. 5, 213–253 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rocha, C.: Symbolic reachability analysis for rewrite theories. Ph.D. thesis, C.S. Department, University of Illinois at Urbana-Champaign, February 2013.
  27. 27.
    Viry, P.: Rewriting: an effective model of concurrency. In: Halatsis, C., Philokyprou, G., Maritsas, D., Theodoridis, S. (eds.) PARLE 1994. LNCS, vol. 817, pp. 648–660. Springer, Heidelberg (1994) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Luis Aguirre
    • 1
  • Narciso Martí-Oliet
    • 1
    Email author
  • Miguel Palomino
    • 1
  • Isabel Pita
    • 1
  1. 1.Facultad de InformáticaUniversidad Complutense de MadridMadridSpain

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