Hierarchical Combination

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7898)


A novel approach is described for the combination of unification algorithms for two equational theories E 1 and E 2 which share function symbols. We are able to identify a set of restrictions and a combination method such that if the restrictions are satisfied the method produces a unification algorithm for the union of non-disjoint equational theories. Furthermore, we identify a class of theories satisfying the restrictions. The critical characteristics of the class is the hierarchical organization and the shared symbols being restricted to “inner constructors”.


Combination Method Equational Theory Symbolic Computation Commutative Theory Cipher Block Chain 
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  1. 1.
    Anantharaman, S., Bouchard, C., Narendran, P., Rusinowitch, M.: Unification modulo chaining. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 70–82. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Anantharaman, S., Erbatur, S., Lynch, C., Narendran, P., Rusinowitch, M.: Unification modulo synchronous distributivity. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 14–29. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Baader, F., Ghilardi, S., Tinelli, C.: A new combination procedure for the word problem that generalizes fusion decidability results in modal logics. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 183–197. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, New York (1998)Google Scholar
  5. 5.
    Baader, F., Schulz, K.U.: Unification in the union of disjoint equational theories: Combining decision procedures. Journal of Symbolic Computation 21(2), 211–243 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Baader, F., Snyder, W.: Unification theory. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 445–532. Elsevier and MIT Press (2001)Google Scholar
  7. 7.
    Baader, F., Tinelli, C.: Combining equational theories sharing non-collapse-free constructors. In: Kirchner, H. (ed.) FroCos 2000. LNCS (LNAI), vol. 1794, pp. 260–274. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Baader, F., Tinelli, C.: Combining decision procedures for positive theories sharing constructors. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, pp. 352–366. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Boudet, A.: Combining unification algorithms. Journal of Symbolic Computation 16(6), 597–626 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bürckert, H.-J., Herold, A., Schmidt-Schauß, M.: On equational theories, unification, and (un)decidability. Journal of Symbolic Computation 8(1-2), 3–49 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Domenjoud, E., Klay, F., Ringeissen, C.: Combination techniques for non-disjoint equational theories. In: Bundy, A. (ed.) CADE 1994. LNCS, vol. 814, pp. 267–281. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  12. 12.
    Dougherty, D.J., Johann, P.: An improved general E-unification method. Journal of Symbolic Computation 14(4), 303–320 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Erbatur, S., Marshall, A.M., Kapur, D., Narendran, P.: Unification over distributive exponentiation (sub)theories. Journal of Automata, Languages and Combinatorics (JALC) 16(2-4), 109–140 (2011)Google Scholar
  14. 14.
    Gallier, J.H., Snyder, W.: Complete sets of transformations for general E-unification. Theoretical Computer Science 67(2-3), 203–260 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Huet, G.P.: Confluent reductions: Abstract properties and applications to term rewriting systems. Journal of the ACM (JACM) 27(4), 797–821 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Jouannaud, J.-P., Kirchner, C.: Solving equations in abstract algebras: A rule-based survey of unification. In: Computational Logic - Essays in Honor of Alan Robinson, pp. 257–321 (1991)Google Scholar
  17. 17.
    Morawska, B.: General E-unification with eager variable elimination and a nice cycle rule. Journal of Automated Reasoning 39, 77–106 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Ringeissen, C.: Unification in a combination of equational theories with shared constants and its application to primal algebras. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 261–272. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  19. 19.
    Schmidt-Schauß, M.: Unification in a combination of arbitrary disjoint equational theories. Journal of Symbolic Computation 8, 51–99 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Snyder, W.: A Proof Theory for General Unification, Birkhauser. Progress in Computer Science and Applied Logic, vol. 11 (1991)Google Scholar
  21. 21.
    Tidén, E.: Unification in combinations of collapse-free theories with disjoint sets of function symbols. In: Siekmann, J.H. (ed.) CADE 1986. LNCS, vol. 230, pp. 431–449. Springer, Heidelberg (1986)CrossRefGoogle Scholar
  22. 22.
    Yelick, K.A.: Unification in combinations of collapse-free regular theories. Journal of Symbolic Computation 3(1-2), 153–181 (1987)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.University of New MexicoUSA
  2. 2.Naval Research LaboratoryUSA
  3. 3.University at Albany, SUNYUSA
  4. 4.LORIA – INRIA Nancy-Grand EstFrance
  5. 5.Dipartimento di InformaticaUniversità degli Studi di VeronaItaly

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