On the Complexity of H-Subsumption

  • Reinhard Pichler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)


The importance of subsumption as a redundancy elimination method in automated theorem proving is generally acknowledged. For a given Herbrand universe H, it can be further strengthened to the so-called H-subsumption, i.e.: A clause D is H-subsumed by a clause set \({\cal C}\), iff for every H-ground instance of D there is a clause \(C \in {\cal C}\), s.t. C subsumes . In recent time, H-subsumption has gained increasing importance especially in the field of automated model building (cf. e.g. [5], [4], [6]). Furthermore, it can be easily shown that H-subsumption may be incorporated as a redundancy deletion rule into many familiar (resolution- and paramodulation-based) inference systems without destroying the refutational completeness.

However, no satisfactory algorithm for checking H-subsumption has been presented so far. We therefore have to investigate the inherent complexity of H-subsumption in order to explain this lack of efficient algorithms: The main result of this work is a Π\(_{\rm 2}^{p}\)-completeness proof for H-subsumption even if it is subjected to some strong restrictions. Hence, unless the polynomial hierarchy collapses to the first level, H-subsumption is non-polynomially more complex than ordinary subsumption.

Finally we present a new algorithm for H-subsumption whose complexity is compared with previously known algorithms (i.e.: from [5] and [4] on the one hand and from [6] on the other hand). The main advantage of our approach is that the total size of an H-subsumption problem (i.e.: in particular, the term depth of the expressions involved) only has polynomial influence on the overall (time and space) complexity. This is in great contrast to the other two approaches.


Function Symbol Propositional Variable Automate Theorem Prove Ground Instance Semantic Tree 
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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  1. 1.
    Bachmair, L., Ganzinger, H.: Rewrite-based Equational Theorem Proving with Selection and Simplification. Journal of Logic and Computation 4(3), 217–247 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baader, F., Siekmann, J.H.: Unification Theory. In: Handbook of Logic in Artificial Intelligence and Logic Programming. Oxford University Press, Oxford (1994)Google Scholar
  3. 3.
    Comon, H., Lescanne, P.: Equational Problems and Disunification. Journal of Symbolic Computation 7, 371–425 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Caferra, R., Peltier, N.: Decision Procedures using Model Building Techniques. In: Kleine Büning, H. (ed.) CSL 1995. LNCS, vol. 1092, pp. 130–144. Springer, Heidelberg (1996)Google Scholar
  5. 5.
    Caferra, R., Zabel, N.: Extending Resolution for Model Construction. In: van Eijck, J. (ed.) JELIA 1990. LNCS, vol. 478, pp. 153–169. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  6. 6.
    Fermüller, C., Leitsch, A.: Hyperresolution and Automated Model Building. Journal of Logic and Computation 6(2), 173–230 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability, A guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  8. 8.
    Hsiang, J., Rusinowitch, M.: Proving Refutational Completeness of Theorem- Proving Strategies: The Transfinite Semantic Tree Method. Journal of the ACM 38(3), 559–587 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kowalski, R., Hayes, P.J.: Semantic Trees in Automated Theorem Proving. Machine Intelligence 4, 87–101 (1969)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Leitsch, A.: The Resolution Calculus. Texts in Theoretical Computer Science. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  11. 11.
    Nieuwenhuis, R., Rubio, A.: Theorem Proving with ordering and equality constrained clauses. Journal of Symbolic Computation 11, 1–32 (1995)Google Scholar
  12. 12.
    Pichler, R.: Algorithms on atomic representations of herbrand models. In: Dix, J., Fariñas del Cerro, L., Furbach, U. (eds.) JELIA 1998. LNCS (LNAI), vol. 1489, pp. 199–215. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Pichler, R.: Completeness and redundancy in constrained clause logic. In: Caferra, R., Salzer, G. (eds.) FTP 1998. LNCS (LNAI), vol. 1761, pp. 193–203. Springer, Heidelberg (1998), Google Scholar
  14. 14.
    Stockmeyer, L.J.: The Polynomial Time Hierarchy. Journal of Theoretical Computer Science 3, 1–12 (1976)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Reinhard Pichler
    • 1
  1. 1.Technische Universität Wien 

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