Skip to main content

Detection of First Order Axiomatic Theories

  • Conference paper
  • 490 Accesses

Part of the Lecture Notes in Computer Science book series (LNAI,volume 8152)


Automated theorem provers for first-order logic with equality have become very powerful and useful, thanks to both advanced calculi — such as superposition and its refinements — and mature implementation techniques. Nevertheless, dealing with some axiomatic theories remains a challenge because it gives rise to a search space explosion. Most attempts to deal with this problem have focused on specific theories, like AC (associative commutative symbols) or ACU (AC with neutral element). Even detecting the presence of a theory in a problem is generally solved in an ad-hoc fashion. We present here a generic way of describing and recognizing axiomatic theories in clausal form first-order logic with equality. Subsequently, we show some use cases for it, including a redundancy criterion that can be applied to some equational theories, extending some work that has been done by Avenhaus, Hillenbrand and Löchner.


  • Theorem Prover
  • Equational Theory
  • Predicate Symbol
  • Horn Clause
  • Axiomatic Theory

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.

This is a preview of subscription content, access via your institution.

Buying options

USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
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


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley (1995)

    Google Scholar 

  2. Avenhaus, J., Hillenbrand, T., Löchner, B.: On using ground joinable equations in equational theorem proving. Journal of Symbolic Computation 36(1-2), 217–233 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Bachmair, L., Ganzinger, H.: Associative-commutative superposition. In: Dershowitz, N., Lindenstrauss, N. (eds.) CTRS 1994. LNCS, vol. 968, pp. 1–14. Springer, Heidelberg (1995)

    CrossRef  Google Scholar 

  4. Denzinger, J., Schulz, S.: Learning domain knowledge to improve theorem proving. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 62–76. Springer, Heidelberg (1996)

    CrossRef  Google Scholar 

  5. Ganzinger, H., Stuber, J.: Superposition with equivalence reasoning and delayed clause normal form transformation. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 335–349. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

  6. Hillenbrand, T., Jaeger, A., Löchner, B.: System description: Waldmeister – improvements in performance and ease of use. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 232–236. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  7. Korovin, K.: iProver – An Instantiation-Based Theorem Prover for First-Order Logic (System Description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 292–298. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  8. Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning. Elsevier, MIT Press (1999)

    Google Scholar 

  9. Riazanov, A., Voronkov, A.: Vampire 1.1 (System description). In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 376–380. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  10. Robinson, J.A.: A Machine-Oriented Logic Based on the Resolution Principle. J. ACM 12(1), 23–41 (1965)

    CrossRef  MATH  Google Scholar 

  11. Schulz, S.: E - a brainiac theorem prover. AI Commun. 15(2,3), 111–126 (2002)

    MATH  Google Scholar 

  12. Stuber, J.: Superposition theorem proving for abelian groups represented as integer modules. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 33–47. Springer, Heidelberg (1996)

    CrossRef  Google Scholar 

  13. Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure: The FOF and CNF Parts, v3.5.0. Journal of Automated Reasoning 43(4), 337–362 (2009)

    CrossRef  MATH  Google Scholar 

  14. Weidenbach, C., Schmidt, R.A., Hillenbrand, T., Rusev, R., Topic, D.: System Description: Spass Version 3.0. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 514–520. Springer, Heidelberg (2007)

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations


Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Burel, G., Cruanes, S. (2013). Detection of First Order Axiomatic Theories. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds) Frontiers of Combining Systems. FroCoS 2013. Lecture Notes in Computer Science(), vol 8152. Springer, Berlin, Heidelberg.

Download citation

  • DOI:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40884-7

  • Online ISBN: 978-3-642-40885-4

  • eBook Packages: Computer ScienceComputer Science (R0)