Skip to main content
SpringerLink
Log in

On deciding subsumption problems

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Subsumption is an important redundancy elimination method in automated deduction. A clause D is subsumed by a set \(\mathcal{C}\) of clauses if there is a clause C ∈ \(\mathcal{C}\) and a substitution σ such that the literals of Cσ are included in D. In the field of automated model building, subsumption has been modified to an even stronger redundancy elimination method, namely the so-called clausal H-subsumption. Atomic H-subsumption emerges from clausal H-subsumption by restricting D to an atom and \(\mathcal{C}\) to a set of atoms. Both clausal and atomic H-subsumption play an indispensable key role in automated model building. Moreover, problems equivalent to atomic H-subsumption have been studied with different terminologies in many areas of computer science. Both clausal and atomic H-subsumption are known to be intractable, i.e., Π 2 p -complete and NP-complete, respectively. In this paper, we present a new approach to deciding (clausal and atomic) H-subsumption that is based on a reduction to QSAT2 and SAT, respectively.

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

Similar content being viewed by others

References

  1. F. Baader and J.H. Siekmann, Unification theory, in: Handbook of Logic in Artificial Intelligence and Logic Programming, eds. D.M. Gabbay, C.J. Hogger and J.A. Robinson (1994) pp. 41–125.

  2. M. Cadoli, A. Giovanardi and M. Schaerf, An algorithm to evaluate quantified Boolean formulae, in: Proceedings of the 15th National Conference on Artificial Intelligence (AAAI’98) (1998) pp. 262-267.

  3. R. Caferra and N. Peltier, Decision procedures using model building techniques, in: Proceedings of the 9th International Workshop for Computer Science Logic (CSL’95), ed. H. Kleine Büning, Lecture Notes in Computer Science, Vol. 1092 (1996) pp. 130–144.

  4. R. Caferra and N. Zabel, Extending resolution for model construction, in: Logics in Artificial Intelligence (JELIA’90), Proceedings, ed. J. van Eijck, Lecture Notes in Computer Science, Vol. 478 (1991) pp. 153–169.

  5. A. Colmerauer, Equations and inequations on finite and infinite trees, in: Proceedings of the International Conference on Fifth Generation Computer Systems (FGCS’84) (1984) pp. 85–99.

  6. H. Comon and C. Delor, Equational formulae with membership constraints, Information and Computation 112(2) (1994) 167–216.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Comon and P. Lescanne, Equational problems and disunification, Journal of Symbolic Computation 7(3–4) (1989) 371–425.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Delgrande, T. Schaub, H. Tompits and S. Woltran, On computing solutions to belief change scenarios, in: Proceedings of the 6th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’01), eds. S. Benferhat and P. Besnard, Lecture Notes in Computer Science, Vol. 2143 (2001) pp. 510–521.

  9. U. Egly, T. Eiter, H. Tompits and S. Woltran, Solving advanced reasoning tasks using quantified Boolean formulas, in: Proceedings of the Seventeenth National Conference on Artificial Intelligence (AAAI’00) (2000) pp. 417–422.

  10. U. Egly, H. Tompits and S. Woltran, On quantifier shifting for quantified Boolean formulas, in: Proceedings of the SAT-02 Workshop on Theory and Applications of Quantified Boolean Formulas (QBF-02) (2002) pp. 48–61.

  11. R. Feldmann, B. Monien and S. Schamberger, A distributed algorithm to evaluate quantified Boolean formulas, in: Proceedings of the 17th National Conference on Artificial Intelligence (AAAI’00) (2000) pp. 285–290.

  12. C. Fermüller and A. Leitsch, Hyperresolution and automated model building, Journal of Logic and Computation 6(2) (1996) 173–203.

    Article  MATH  MathSciNet  Google Scholar 

  13. M.R. Garey and D.S. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness (Freeman, 1979).

  14. E. Giunchiglia, M. Narizzano and A. Tacchella, QuBE: A system for deciding quantified Boolean formulas satisfiability, in: Proceedings of the International Joint Conference on Automated Reasoning (IJCAR’01) eds. R. Goré, A. Leitsch and T. Nipkow (2001) pp. 364–369.

  15. E. Goldberg and Y. Novikov, BerkMin: A fast and robust sat-solver, in: Proceedings of the Design, Automation and Test in Europe Conference (DATE’02) (2002) pp. 142–149.

  16. G. Gottlob, S. Marcus, A. Nerode, G. Salzer and V.S. Subrahmanian, A non-ground realization of the stable and well-founded semantics, Theoretical Computer Science 166(1–2) (1996) 221–262.

    Article  MATH  MathSciNet  Google Scholar 

  17. G. Gottlob and R. Pichler, Working with ARMs: Complexity results on atomic representations of Herbrand models, Information and Computation 165 (2001) 183–207.

    Article  MATH  MathSciNet  Google Scholar 

  18. D. Kapur, P. Narendran, D. Rosenkrantz and H. Zhang, Sufficient-completeness, ground-reducibility and their complexity, Acta Informatica 28(4) (1991) 311–350.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Kautz and B. Selman, Planning as satisfiability, in: Proceedings of the 10th European Conference on Artificial Intelligence (ECAI’92), ed. B. Neumann (1992) pp. 359–363.

  20. H. Kleine Büning, M. Karpinski and A. Flögel, Resolution for quantified Boolean formulas, Information and Computation 117(1) (1995) 12–18.

    Article  MATH  MathSciNet  Google Scholar 

  21. K. Kunen, Answer sets and negation as failure, in: Proceedings of the 4th International Conference on Logic Programming (ICLP’87), ed. J.-L. Lassez (1987) pp. 219–228.

  22. G. Kuper, K. McAloon, K. Palem and K. Perry, Efficient parallel algorithms for anti-unification and relative complement, in: Proceedings of the 3rd Annual Symposium on Logic in Computer Science (LICS’88) (1988) pp. 112–120.

  23. J.-L. Lassez, M. Maher and K. Marriott, Unification revisited, in: Foundations of Logic and Functional Programming, Proceedings (1986) pp. 67–113.

  24. J.-L. Lassez, M. Maher and K. Marriott, Elimination of negation in term algebras, in: Proceedings of the 16th International Symposium on Mathematical Foundations of Computer Science (MFCS’91), ed. A. Tarlecki (1991) pp. 1–16.

  25. J.-L. Lassez and K. Marriott, Explicit representation of terms defined by counter examples, Journal of Automated Reasoning 3(3) (1987) 301–317.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Letz, Lemma and model caching in decision procedures for quantified Boolean formulas, in: Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX 2002, eds. U. Egly and C.G. Fermüller, Lecture Notes in Computer Science, Vol. 2381 (2002) pp. 160–175.

  27. A. Martelli and U. Montanari, An efficient unification algorithm, ACM Transactions on Programming Languages and Systems 4(2) (1982) 258–282.

    Article  MATH  Google Scholar 

  28. M.W. Moskewicz, C.F. Madigan, Y. Zhao, L. Zhang and S. Malik, Chaff: Engineering an efficient SAT solve, in: Proceedings of the 38th Design Automation Conference (DAC’01) (2001).

  29. C.H. Papadimitriou, Computational Complexity (Addison-Wesley, 1994).

  30. D. Pearce, H. Tompits and S. Woltran, Encodings for equilibrium logic and logic programs with nested expressions, in: Progress in Artificial Intelligence, Knowledge Extraction, Multi-agent Systems, Logic Programming and Constraint Solving, 10th Portuguese Conference on Artificial Intelligence (EPIA’01), eds. P. Brazdil and A. Jorge, Lecture Notes in Computer Science, Vol. 2258 (2001) pp. 306–320.

  31. R. Pichler, On the complexity of H-subsumption, in: Proceedings of the 12th International Workshop for Computer Science Logic (CSL’98), eds. G. Gottlob, E. Grandjean and K. Seyr, Lecture Notes in Computer Science, Vol. 1584 (1998) pp. 355–371.

  32. R. Pichler, Completeness and redundancy in constrained clause logic, in: Automated Deduction in Classical and Non-Classical Logics, Selected Papers, eds. R. Caferra and G. Salzer, Lecture Notes in Computer Science, Vol. 1761 (2000) pp. 221–235.

  33. R. Pichler, The explicit representability of implicit generalizations, in: Proceedings of the 11th International Conference on Rewriting Techniques and Applications (RTA’00), ed. L. Bachmair (2000) pp. 187–202.

  34. D.A. Plaisted and S. Greenbaum, A structure preserving clause form translation, Journal of Symbolic Computation 2(3) (1986) 293–304.

    Article  MATH  MathSciNet  Google Scholar 

  35. J. Rintanen, Constructing conditional plans by a theorem prover, Journal of Artificial Intelligence Research 10 (1999) 323–352.

    MATH  Google Scholar 

  36. J. Rintanen, Improvements to the evaluation of quantified Boolean formulae, in: Proceedings of the 16th International Joint Conference on Artificial Intelligence (IJCAI’99), ed. T. Dean (1999) pp. 1192–1197.

  37. J. Siekmann and G. Wrightson, eds., Automation of Reasoning: Classical Papers in Computational Logic 1967–1970, Vol. 2 (Springer, Berlin, 1983).

    Book  Google Scholar 

  38. L. Stockmeyer, The polynomial-time hierarchy, Theoretical Computer Science 3 (1977) 1–22.

    MATH  MathSciNet  Google Scholar 

  39. G. Tseitin, On the complexity of proofs in propositional logics, Seminars in Mathematics 8 (1970), reprinted in [37].

  40. H. Zhang, SATO: An efficient propositional prover, in: Proceedings of the 17th Internationonal Conference on Automated Deduction (CADE-17), ed. W. McCune (1997) pp. 272-275.

Download references

Author information

Authors and Affiliations

  1. Abteilung für Wissensbasierte Systeme E184/3, TU Wien, Favoritenstr. 9-11, A-1040, Vienna, Austria

    Uwe Egly & Stefan Woltran

  2. Siemens AG Österreich, Siemensstr. 92, 1210, Vienna, Austria

    Reinhard Pichler

Authors
  1. Uwe Egly
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Reinhard Pichler
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefan Woltran
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This was partially supported by the Austrian Science Foundation under grant P15068. We thank the anonymous referees for valuable comments.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Egly, U., Pichler, R. & Woltran, S. On deciding subsumption problems. Ann Math Artif Intell 43, 255–294 (2005). https://doi.org/10.1007/s10472-005-0434-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10472-005-0434-4

Keywords

Search

Navigation