Skip to main content
SpringerLink
Log in
Book cover

International Conference on Automated Deduction

CADE 2013: Automated Deduction – CADE-24 pp 58–75Cite as

Completeness and Decidability Results for First-Order Clauses with Indices

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 7898))

Abstract

We define a proof procedure that allows for a limited form of inductive reasoning. The first argument of a function symbol is allowed to belong to an inductive type. We will call such an argument an index. We enhance the standard superposition calculus with a loop detection rule, in order to encode a particular form of mathematical induction. The satisfiability problem is not semi-decidable, but some classes of clause sets are identified for which the proposed procedure is complete and/or terminating.

This work has been partly funded by the project ASAP of the French Agence Nationale de la Recherche (ANR-09-BLAN-0407-01).

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aravantinos, V., Caferra, R., Peltier, N.: Decidability and undecidability results for propositional schemata. Journal of Artificial Intelligence Research 40, 599–656 (2011)

    MathSciNet  MATH  Google Scholar 

  2. Aravantinos, V., Caferra, R., Peltier, N.: Linear Temporal Logic and Propositional Schemata, Back and Forth. In: TIME 2011 (18th International Symposium on Temporal Representation and Reasoning) (2011)

    Google Scholar 

  3. Aravantinos, V., Echenim, M., Peltier, N.: A resolution calculus for first-order schemata. In: Fundamenta Informaticae (to appear 2013) (accepted for publication)

    Google Scholar 

  4. Armando, A., Ranise, S., Rusinowitch, M.: A rewriting approach to satisfiability procedures. Information and Computation 183(2), 140–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: CERES: An analysis of Fürstenberg’s proof of the infinity of primes. Theor. Comput. Sci. 403(2-3), 160–175 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation 3(4), 217–247 (1994)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Baelde, D., Miller, D., Snow, Z.: Focused inductive theorem proving. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 278–292. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  9. Bouhoula, A., Kounalis, E., Rusinowitch, M.: SPIKE, an automatic theorem prover. In: Voronkov, A. (ed.) LPAR 1992. LNCS, vol. 624, pp. 460–462. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  10. Bundy, A.: The automation of proof by mathematical induction. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 845–911. Elsevier and MIT Press (2001)

    Google Scholar 

  11. Bundy, A., Basin, D., Hutter, D., Ireland, A.: Rippling: meta-level guidance for mathematical reasoning. Cambridge University Press, New York (2005)

    Book  MATH  Google Scholar 

  12. Comon, H.: Inductionless induction. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 14, pp. 913–962. North-Holland (2001)

    Google Scholar 

  13. Comon, H., Lescanne, P.: Equational problems and disunification. Journal of Symbolic Computation 7, 371–475 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gupta, A., Fisher, A.L.: Parametric circuit representation using inductive boolean functions. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 15–28. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  15. Horbach, M., Weidenbach, C.: Deciding the inductive validity of ∀ ∃ * queries. In: Grädel, E., Kahle, R. (eds.) CSL 2009. LNCS, vol. 5771, pp. 332–347. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  16. Horbach, M., Weidenbach, C.: Superposition for fixed domains. ACM Trans. Comput. Logic 11(4), 1–35 (2010)

    Article  MathSciNet  Google Scholar 

  17. Kapur, D., Musser, D.: Proof by consistency. Artificial Intelligence 31 (1987)

    Google Scholar 

  18. Kersani, A., Peltier, N.: Completeness and Decidability Results for First-order Clauses with Indices (long version), Research Report (2013), http://membres-lig.imag.fr/peltier/kp13.pdf

  19. Leitsch, A.: Deciding clause classes by semantic clash resolution. Fundamenta Informaticae 18, 163–182 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, pp. 371–443. Elsevier and MIT Press (2001)

    Google Scholar 

  21. Peltier, N.: Some Techniques for Proving Termination of the Hyperresolution Calculus. Journal of Automated Reasoning 35, 391–427 (2006)

    Article  MathSciNet  Google Scholar 

  22. Robinson, A., Voronkov, A. (eds.): Handbook of Automated Reasoning. North-Holland (2001)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Grenoble (LIG, CNRS), France

    Abdelkader Kersani & Nicolas Peltier

Authors
  1. Abdelkader Kersani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolas Peltier
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Verona, Strada Le Grazie 15, 37134, Verona, Italy

    Maria Paola Bonacina

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kersani, A., Peltier, N. (2013). Completeness and Decidability Results for First-Order Clauses with Indices. In: Bonacina, M.P. (eds) Automated Deduction – CADE-24. CADE 2013. Lecture Notes in Computer Science(), vol 7898. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38574-2_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-38574-2_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38573-5

  • Online ISBN: 978-3-642-38574-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation