Skip to main content
SpringerLink
Log in

A Combined Superposition and Model Evolution Calculus

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

We present a new calculus for first-order theorem proving with equality, \( \mathcal{ME}+\) Sup, which generalizes both the Superposition calculus and the Model Evolution calculus (with equality) by integrating their inference rules and redundancy criteria in a non-trivial way. The main motivation is to combine the advantageous features of these two rather complementary calculi in a single framework. In particular, Model Evolution, as a lifted version of the propositional DPLL procedure, contributes a non-ground splitting rule that effectively permits to split a clause into non variable disjoint subclauses. In the paper we present the calculus in detail. Our main result is its completeness under semantically justified redundancy criteria and simplification rules. We also show how under certain assumptions the model representation computed by a (finite and fair) derivation can be queried in an effective way.

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. Baumgartner, P.: Logical engineering with instance-based methods. In: Pfenning, F. (ed.) CADE-21—The 21st International Conference on Automated Deduction. Lecture Notes in Artificial Intelligence, vol. 4603, pp. 404–409. Springer (2007)

  2. Bachmair, L., Ganzinger, H.: Chapter 11: Equational reasoning in saturation-based theorem proving. In: Bibel, W., Schmitt, P.H. (eds.) Automated Deduction. A Basis for Applications. Vol. I: Foundations. Calculi and Refinements, pp. 353–398. Kluwer Academic Publishers (1998)

  3. Bachmair, L., Ganzinger, H., Waldmann, U.: Refutational theorem proving for hierarchic first-order theories. Appl. Algebra Eng. Commun. Comput. 5(3/4), 193–212 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  5. Baumgartner, P., Tinelli, C.: The model evolution calculus. In: Baader, F. (ed.) CADE-19—The 19th International Conference on Automated Deduction. Lecture Notes in Artificial Intelligence, vol. 2741, pp. 350–364. Springer (2003)

  6. Baumgartner, P., Tinelli, C.: The model evolution calculus with equality. In: Nieuwenhuis, R. (ed.) CADE-20—The 20th International Conference on Automated Deduction. Lecture Notes in Artificial Intelligence, vol. 3632, pp. 392–408. Springer (2005)

  7. Comon, H.: Disunification: a survey. In: Lassez, J.-L., Plotkin, G. (eds.) Computational Logic: Essays in Honor of Alan Robinson. MIT Press (1991)

  8. Ganzinger, H., Korovin, K.: New directions in instantiation-based theorem proving. In: Proc. 18th IEEE Symposium on Logic in Computer Science, (LICS’03), pp. 55–64. IEEE Computer Society Press (2003)

  9. Ganzinger, H., Korovin, K.: Integrating equational reasoning into instantiation-based theorem proving. In: Computer Science Logic (CSL’04). Lecture Notes in Computer Science, vol. 3210, pp. 71–84. Springer (2004)

  10. Jacobs, S., Waldmann, U.: Comparing instance generation methods for automated reasoning. J. Autom. Reason. 38(1–3), 57–78 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Korovin, K.: Iprover—an instantiation-based theorem prover for first-order logic (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) Automated Reasoning, 4th International Joint Conference, IJCAR 2008, Sydney, Australia, August 2008. Lecture Notes in Computer Science, vol. 5195, pp. 292–298. Springer (2008)

  12. Korovin, K.: Instantiation-based automated reasoning: from theory to practice. In: Schmidt, R.A. (ed.) Automated Deduction—CADE-22, 22nd International Conference on Automated Deduction, Montreal, Canada, August 2–7, 2009. Proceedings. Lecture Notes in Computer Science, vol. 5663, pp. 163–166. Springer (2009)

  13. Lynch, C., McGregor, R.E.: Combining instance generation and resolution. In: Ghilardi, S., Sebastiani, R. (eds.) Frontiers of Combining Systems (FroCoS 2009). LNAI, vol. 5749, pp. 304–318. Springer (2009)

  14. Nieuwenhuis, R., Rubio, A.: Theorem proving with ordering and equality constrained clauses. J. Symb. Comput. 19, 321–351 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Plaisted, D.A., Zhu, Y.: Ordered semantic hyper linking. J. Autom. Reason. 25(3), 167–217 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Weidenbach, C., Schmidt, R., Hillenbrand, T., Rusev, R., Topic, D.: System description: Spass version 3.0. In: Pfenning, F. (ed.) CADE-21—21st International Conference on Automated Deduction. Lecture Notes in Artificial Intelligence, vol. 4603, pp. 514–520. Springer (2007)

  17. Zhang, T., Sipma, H.B., Manna, Z.: The decidability of the first-order theory of term algebras with Knuth-Bendix order. In: Nieuwenhuis, R. (ed.) 20th International Conference on Automated Deduction (CADE’05). Lecture Notes in Artificial Intelligence, vol. 3632, pp. 131–148. Springer (2005)

Download references

Author information

Authors and Affiliations

  1. NICTA and Australian National University, Canberra, Australia

    Peter Baumgartner

  2. MPI für Informatik, Saarbrücken, Germany

    Uwe Waldmann

Authors
  1. Peter Baumgartner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Uwe Waldmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Baumgartner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Baumgartner, P., Waldmann, U. A Combined Superposition and Model Evolution Calculus. J Autom Reasoning 47, 191–227 (2011). https://doi.org/10.1007/s10817-010-9214-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-010-9214-x

Keywords

Search

Navigation