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Rule-Based Unification in Combined Theories and the Finite Variant Property

  • Ajay K. Eeralla
  • Serdar Erbatur
  • Andrew M. Marshall
  • Christophe RingeissenEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11417)

Abstract

We investigate the unification problem in theories defined by rewrite systems which are both convergent and forward-closed. These theories are also known in the context of protocol analysis as theories with the finite variant property and admit a variant-based unification algorithm. In this paper, we present a new rule-based unification algorithm which can be seen as an alternative to the variant-based approach. In addition, we define forward-closed combination to capture the union of a forward-closed convergent rewrite system with another theory, such as the Associativity-Commutativity, whose function symbols may occur in right-hand sides of the rewrite system. Finally, we present a combination algorithm for this particular class of non-disjoint unions of theories.

Keywords

Term rewriting Unification Combination Forward-closure 

References

  1. 1.
    Abadi, M., Cortier, V.: Deciding knowledge in security protocols under equational theories. Theor. Comput. Sci. 367(1–2), 2–32 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, New York (1998)CrossRefGoogle Scholar
  3. 3.
    Baader, F., Schulz, K.U.: Unification in the union of disjoint equational theories: combining decision procedures. J. Symb. Comput. 21(2), 211–243 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Basin, D., Mödersheim, S., Viganò, L.: An on-the-fly model-checker for security protocol analysis. In: Snekkenes, E., Gollmann, D. (eds.) ESORICS 2003. LNCS, vol. 2808, pp. 253–270. Springer, Heidelberg (2003).  https://doi.org/10.1007/978-3-540-39650-5_15CrossRefGoogle Scholar
  5. 5.
    Blanchet, B.: Modeling and verifying security protocols with the applied Pi calculus and ProVerif. Found. Trends Priv. Secur. 1(1–2), 1–135 (2016)Google Scholar
  6. 6.
    Bouchard, C., Gero, K.A., Lynch, C., Narendran, P.: On forward closure and the finite variant property. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS (LNAI), vol. 8152, pp. 327–342. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40885-4_23CrossRefGoogle Scholar
  7. 7.
    Ciobâcă, S., Delaune, S., Kremer, S.: Computing knowledge in security protocols under convergent equational theories. J. Autom. Reasoning 48(2), 219–262 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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).  https://doi.org/10.1007/978-3-540-32033-3_22CrossRefGoogle Scholar
  9. 9.
    Durán, F., Eker, S., Escobar, S., Martí-Oliet, N., Meseguer, J., Talcott, C.: Built-in variant generation and unification, and their applications in Maude 2.7. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 183–192. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40229-1_13CrossRefGoogle Scholar
  10. 10.
    Eeralla, A.K., Erbatur, S., Marshall, A.M., Ringeissen, C.: Unification in non-disjoint combinations with forward-closed theories. http://hal.inria.fr
  11. 11.
    Eeralla, A.K., Lynch, C.: Bounded ACh Unification. CoRR abs/1811.05602 (2018). http://arxiv.org/abs/1811.05602
  12. 12.
    Erbatur, S., Kapur, D., Marshall, A.M., Narendran, P., Ringeissen, C.: Hierarchical combination. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 249–266. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-38574-2_17CrossRefGoogle Scholar
  13. 13.
    Escobar, S., Meadows, C., Meseguer, J.: Maude-NPA: cryptographic protocol analysis modulo equational properties. In: Aldini, A., Barthe, G., Gorrieri, R. (eds.) FOSAD 2007-2009. LNCS, vol. 5705, pp. 1–50. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-03829-7_1CrossRefzbMATHGoogle Scholar
  14. 14.
    Escobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. J. Log. Algebr. Program. 81(7–8), 898–928 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Jouannaud, J., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM J. Comput. 15(4), 1155–1194 (1986).  https://doi.org/10.1137/0215084MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kirchner, C., Klay, F.: Syntactic theories and unification. In: Logic in Computer Science 1990 Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, LICS 1990, pp. 270–277, June 1990.  https://doi.org/10.1109/LICS.1990.113753
  17. 17.
    Lynch, C., Morawska, B.: Basic syntactic mutation. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 471–485. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45620-1_37CrossRefGoogle Scholar
  18. 18.
    Meier, S., Schmidt, B., Cremers, C., Basin, D.: The TAMARIN prover for the symbolic analysis of security protocols. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 696–701. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39799-8_48CrossRefGoogle Scholar
  19. 19.
    Meseguer, J.: Variant-based satisfiability in initial algebras. Sci. Comput. Program. 154, 3–41 (2018)CrossRefGoogle Scholar
  20. 20.
    Nipkow, T.: Proof transformations for equational theories. In: Logic in Computer Science 1990 Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science, LICS 1990, pp. 278–288 June 1990Google Scholar
  21. 21.
    Schmidt-Schauß, M.: Unification in a combination of arbitrary disjoint equational theories. J. Symb. Comput. 8, 51–99 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Ajay K. Eeralla
    • 1
  • Serdar Erbatur
    • 2
  • Andrew M. Marshall
    • 3
  • Christophe Ringeissen
    • 4
    Email author
  1. 1.University of MissouriColumbiaUSA
  2. 2.Ludwig-Maximilians-UniversitätMünchenGermany
  3. 3.University of Mary WashingtonFredericksburgUSA
  4. 4.Université de Lorraine, CNRS, Inria, LORIANancyFrance

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