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Unification and Matching in Hierarchical Combinations of Syntactic Theories

  • Serdar Erbatur
  • Deepak Kapur
  • Andrew M. Marshall
  • Paliath Narendran
  • Christophe Ringeissen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9322)

Abstract

We investigate a hierarchical combination approach to the unification problem in non-disjoint unions of equational theories. In this approach, the idea is to extend a base theory with some additional axioms given by rewrite rules in such way that the unification algorithm known for the base theory can be reused without loss of completeness. Additional techniques are required to solve a combined problem by reducing it to a problem in the base theory. In this paper we show that the hierarchical combination approach applies successfully to some classes of syntactic theories, such as shallow theories since the required unification algorithms needed for the combination algorithm can always be obtained. We also consider the matching problem in syntactic extensions of a base theory. Due to the more restricted nature of the matching problem, we obtain several improvements over the unification problem.

Keywords

Inference Rule Combination Method Equational Theory Match Problem Combination Algorithm 
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.
    Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, New York (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Boudet, A.: Combining unification algorithms. Journal of Symbolic Computation 16(6), 597–626 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boudet, A., Contejean, E.: On n-syntactic equational theories. In: Kirchner, H., Levi, G. (eds.) ALP 1992. LNCS, vol. 632, pp. 446–457. Springer, Heidelberg (1992)CrossRefGoogle Scholar
  6. 6.
    Comon, H., Haberstrau, M., Jouannaud, J.: Syntacticness, cycle-syntacticness, and shallow theories. Inf. Comput. 111(1), 154–191 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erbatur, S., Kapur, D., Marshall, A.M., Narendran, P., Ringeissen, C.: Hierarchical combination. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 249–266. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  8. 8.
    Erbatur, S., Kapur, D., Marshall, A.M., Narendran, P., Ringeissen, C.: Hierarchical combination of matching algorithms. In: Twentyeighth International Workshop on Unification (UNIF 2014), Vienna, Austria (2014)Google Scholar
  9. 9.
    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)zbMATHGoogle Scholar
  10. 10.
    Gallier, J.H., Snyder, W.: Complete sets of transformations for general E-unification. Theoretical Computer Science 67(2–3), 203–260 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jouannaud, J.-P.: Syntactic theories. In: Rovan, B. (ed.) MFCS 1990. LNCS, vol. 452, pp. 15–25. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  12. 12.
    Kirchner, C., Klay, F.: Syntactic theories and unification. In: Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science Logic in Computer Science, LICS 1990, pp. 270–277, June1990Google Scholar
  13. 13.
    Lynch, C., Morawska, B.: Basic syntactic mutation. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 471–485. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  14. 14.
    Nieuwenhuis, R.: Decidability and complexity analysis by basic paramodulation. Inf. Comput. 147(1), 1–21 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nipkow, T.: Proof transformations for equational theories. In: Proceedings of the Fifth Annual IEEE Symposium on Logic in Computer Science Logic in Computer Science, LICS 1990, pp. 278–288, June 1990Google Scholar
  16. 16.
    Nipkow, T.: Combining matching algorithms: The regular case. J. Symb. Comput. 12(6), 633–654 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ringeissen, C.: Combining decision algorithms for matching in the union of disjoint equational theories. Inf. Comput. 126(2), 144–160 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schmidt-Schauß, M.: Unification in a combination of arbitrary disjoint equational theories. Journal of Symbolic Computation 8, 51–99 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Snyder, W.: A Proof Theory for General Unification. Progress in Computer Science and Applied Logic, vol. 11. Birkhäuser (1991)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Serdar Erbatur
    • 1
  • Deepak Kapur
    • 2
  • Andrew M. Marshall
    • 3
  • Paliath Narendran
    • 4
  • Christophe Ringeissen
    • 5
  1. 1.Ludwig-Maximilians-UniversitätMünchenGermany
  2. 2.University of New MexicoAlbuquerqueUSA
  3. 3.University of Mary WashingtonFredericksburgUSA
  4. 4.University at AlbanySUNYUSA
  5. 5.LORIA – INRIANancy-Grand EstFrance

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