Skip to main content
SpringerLink
Log in

Sentence-Normalized Conditional Narrowing Modulo in Rewriting Logic and Maude

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

This work studies the relationship between verifiable and computable answers for reachability problems in rewrite theories with an underlying membership equational logic. A new definition for R, A-rewriting that allows us to solve a bigger class of reachability problems, and a calculus that solves this class of problems always working with canonical terms and normalized 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 a more general one that can be instantiated to that answer.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Antoy, S., Echahed, R., Hanus, M.: A needed narrowing strategy. In: Boehm, H.-J., Lang, B., Yellin, D. M. (eds.) Conference Record of POPL’94: 21st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Portland, Oregon, USA, January 17–21, 1994, pp. 268–279. ACM Press (1994)

  2. Aguirre, L., Martí-Oliet, N., Palomino, M., Pita, I.: Conditional narrowing modulo in rewriting logic and Maude. In: Escobar [19], pp. 80–96

  3. Boolos, G., Jeffrey, R.C.: Computability and Logic, 2nd edn. Cambridge University Press, New York (1987)

    MATH  Google Scholar 

  4. Bruni, R., Meseguer, J.: Semantic foundations for generalized rewrite theories. Theoret. Comput. Sci. 360(1–3), 386–414 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bae, K., Meseguer, J.: Model checking LTLR formulas under localized fairness. In: Durán, F. (ed.) Rewriting Logic and Its Applications—9th International Workshop, WRLA 2012, Held as a Satellite Event of ETAPS, Tallinn, Estonia, March 24–25, 2012, Revised Selected Papers, volume 7571 of Lecture Notes in Computer Science, pp. 99–117. Springer (2012)

  6. Bae, K., Meseguer, J.: Infinite-state model checking of LTLR formulas using narrowing. In: Escobar [19], pp. 113–129

  7. Bockmayr, A.: Conditional narrowing modulo a set of equations. Appl. Algebra Eng. Commun. Comput. 4, 147–168 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Quesada, J.F.: Maude: specification and programming in rewriting logic. Theoret. Comput. Sci. 285(2), 187–243 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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, vol. 4350 of Lecture Notes in Computer Science. Springer (2007)

  10. Clavel, M., Durán, F., Hendrix, J., Lucas, S., Meseguer, J., Ölveczky, P.C.: The maude formal tool environment. In: Mossakowski, T., Montanari, U., Haveraaen, M. (eds.) Algebra and Coalgebra in Computer Science, Second International Conference, CALCO 2007, Bergen, Norway, August 20–24, 2007, Proceedings, vol. 4624 of Lecture Notes in Computer Science, pp. 173–178. Springer (2007)

  11. 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. http://hdl.handle.net/2142/50289 (August 2014)

  12. Comon, H.: Completion of rewrite systems with membership constraints. Part I: deduction rules. J. Symb. Comput. 25(4), 397–419 (1998)

    Article  MATH  Google Scholar 

  13. Comon, H.: Completion of rewrite systems with membership constraints. Part II: constraint solving. J. Symb. Comput. 25(4), 421–453 (1998)

    Article  MATH  Google Scholar 

  14. Durán, F., Lucas, S., Marché, C., Meseguer, J., Urbain, X.: Proving operational termination of membership equational programs. High. Order Symb. Comput. 21(1–2), 59–88 (2008)

    Article  MATH  Google Scholar 

  15. Durán, F., Lucas, S., Meseguer, J.: MTT: The Maude termination tool (system description). In: Armando, A., Baumgartner, P., Dowek, G.(eds.) Automated Reasoning, 4th International Joint Conference, IJCAR 2008, Sydney, Australia, August 12–15, 2008, Proceedings, vol. 5195 of Lecture Notes in Computer Science, pp. 313–319. Springer (2008)

  16. Durán, F., Meseguer, J.: On the Church–Rosser and coherence properties of conditional order-sorted rewrite theories. J. Logic Algebr. Program. 81(7–8), 816–850 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Escobar, S., Meadows, C., Meseguer, J.: Maude-NPA: cryptographic protocol analysis modulo equational properties. In: Aldini, A., Barthe, G., Gorrieri, R. (eds.) Foundations of Security Analysis and Design V, vol. 5705 of Lecture Notes in Computer Science, pp. 1–50. Springer (2009)

  19. Escobar, S. (ed.): Rewriting Logic and Its Applications—10th International Workshop, WRLA 2014, Held as a Satellite Event of ETAPS, Grenoble, France, April 5–6, 2014, Revised Selected Papers, vol. 8663 of Lecture Notes in Computer Science. Springer (2014)

  20. Escobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. J. Log. Algebr. Program. 81(7–8), 898–928 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fay, M.: First-order unification in Equational Theory. In: Proceedings of 4th Workshop on Automated Deduction. Academic Press, Austin, pp. 161–167 (1979)

  22. Feuillade, G., Genet, T.: Reachability in conditional term rewriting systems. Electron. Notes Theoret. Comput. Sci. 86(1), 133–146 (2003)

    Article  MATH  Google Scholar 

  23. Fribourg, L.: SLOG: a logic programming language interpreter based on clausal superposition and rewriting. In: Proceedings of the 1985 Symposium on Logic Programming, Boston, Massachusetts, USA, July 15–18, 1985, pp. 172–184. IEEE-CS (1985)

  24. Gnaedig, I., Kirchner, H.: Narrowing, abstraction and constraints for proving properties of reduction relations. In: Comon-Lundh, H., Kirchner, C., Kirchner, H. (eds.) Rewriting, Computation and Proof, Essays Dedicated to Jean-Pierre Jouannaud on the Occasion of His 60th Birthday, vol. 4600 of Lecture Notes in Computer Science, pp. 44–67. Springer (2007)

  25. 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, Workshop, Trento, Italy, December 15–19, 1986, Proceedings, vol. 306 of Lecture Notes in Computer Science, pp. 157–167. Springer (1986)

  26. Hamada, M.: Strong completeness of a narrowing calculus for conditional rewrite systems with extra variables. Electron. Notes Theoret. Comput. Sci. 31, 89–103 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Hullot, J.-M.: Canonical forms and unification. In: Bibel, W., Kowalski, R.A. (eds.) 5th Conference on Automated Deduction, Les Arcs, France, July 8–11, 1980, Proceedings, vol. 87 of Lecture Notes in Computer Science, pp. 318–334. Springer (1980)

  28. Kirchner, C., Kirchner, H., Nahon, F.: Narrowing based inductive proof search. In: Voronkov, A., Weidenbach, C. (eds.) Programming Logics—Essays in Memory of Harald Ganzinger, vol. 7797 of Lecture Notes in Computer Science, pp. 216–238. Springer (2013)

  29. Kirchner, H., Ringeissen, C.: Constraint solving by narrowing in combined algebraic domains. In: Van Hentenryck, P. (ed) Logic Programming, Proceedings of the Eleventh International Conference on Logic Programming, Santa Marherita Ligure, Italy, June 13–18, 1994, pp. 617–631. MIT Press (1994)

  30. Lucas, S., Meseguer, J.: Operational termination of membership equational programs: the order-sorted way. Electron. Notes Theoret. Comput. Sci. 238(3), 207–225 (2009)

    Article  MATH  Google Scholar 

  31. Lucas, S., Marché, C., Meseguer, J.: Operational termination of conditional term rewriting systems. Inf. Process. Lett. 95(4), 446–453 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Meseguer, J.: Rewriting as a unified model of concurrency. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR ’90 Theories of Concurrency: Unification and Extension, vol. 458 of Lecture Notes in Computer Science, pp. 384–400. Springer (1990)

  33. Meseguer, J.: Membership algebra as a logical framework for equational specification. In: Parisi-Presicce, F. (ed) Recent Trends in Algebraic Development Techniques, 12th International Workshop, WADT’97, Tarquinia, Italy, June 1997, Selected Papers, vol. 1376 of Lecture Notes in Computer Science, pp. 18–61. Springer (1997)

  34. Meseguer, J.: Twenty years of rewriting logic. J. Log. Algebr. Program. 81(7–8), 721–781 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Meseguer, J.: Strict Coherence of Conditional Rewriting Modulo Axioms. Technical report, C.S. Department, University of Illinois at Urbana-Champaign. http://hdl.handle.net/2142/50288 (August 2014)

  36. Middeldorp, A., Hamoen, E.: Completeness results for basic narrowing. Appl. Algebra Eng. Commun. Comput. 5, 213–253 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Martí-Oliet, N., Meseguer, J.: Rewriting logic: roadmap and bibliography. Theoret. Comput. Sci. 285(2), 121–154 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  MATH  Google Scholar 

  39. Plotkin, G.: Building in equational theories. Mach. Intell. 7, 73–90 (1972)

    MATH  Google Scholar 

  40. Rocha, C., Meseguer, J., Muñoz, C.: Rewriting modulo SMT and open system analysis. In: Escobar [19], pp. 247–262

  41. Viry, P.: Rewriting: an effective model of concurrency. In: Halatsis, C., Maritsas, D. G., Philokyprou, G., Theodoridis, S. (eds.) PARLE ’94: Parallel Architectures and Languages Europe, 6th International PARLE Conference, Athens, Greece, July 4–8, 1994, Proceedings, vol. 817 of Lecture Notes in Computer Science, pp. 648–660. Springer (1994)

Download references

Acknowledgements

We are very grateful to the referees for their comments to improve the paper, to Santiago Escobar for all his advice, and to José Meseguer for inspiration.

Author information

Authors and Affiliations

  1. Facultad de Informática, Universidad Complutense de Madrid, Madrid, Spain

    Luis Aguirre, Narciso Martí-Oliet, Miguel Palomino & Isabel Pita

Authors
  1. Luis Aguirre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Narciso Martí-Oliet
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miguel Palomino
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Isabel Pita
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Miguel Palomino.

Additional information

Partially supported by MINECO Spanish project StrongSoft (TIN2012-39391-C04-04) and Comunidad de Madrid program N-GREENS Software (S2013/ICE-2731).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aguirre, L., Martí-Oliet, N., Palomino, M. et al. Sentence-Normalized Conditional Narrowing Modulo in Rewriting Logic and Maude. J Autom Reasoning 60, 421–463 (2018). https://doi.org/10.1007/s10817-017-9417-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-017-9417-5

Keywords

Search

Navigation