Skip to main content
SpringerLink
Log in

Superposition Decides the First-Order Logic Fragment Over Ground Theories

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

The hierarchic superposition calculus over a theory T, called SUP(T), enables sound reasoning on the hierarchic combination of a theory T with full first-order logic, FOL(T). If a FOL(T) clause set enjoys a sufficient completeness criterion, the calculus is even complete. Clause sets over the ground fragment of FOL(T) are not sufficiently complete, in general. In this paper we show that any clause set over the ground FOL(T) fragment can be transformed into a sufficiently complete one, and prove that SUP(T) terminates on the transformed clause set, hence constitutes a decision procedure provided the existential fragment of the theory T is decidable. Thanks to the hierarchic design of SUP(T), the decidability result can be extended beyond the ground case. We show SUP(T) is a decision procedure for the non-ground FOL fragment plus a theory T, if every non-constant function symbol from the underlying FOL signature ranges into the sort of the theory T, and every term of the theory sort is ground. Examples for T are in particular decidable fragments of arithmetic.

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. Althaus, E., Kruglov, E., Weidenbach, C.: Superposition modulo linear arithmetic SUP(LA). In: Ghilardi, S., Sebastiani, R. (eds.) Frontiers of Combining Systems: 7th International Symposium, FroCoS 2009, volume 5749 of Lecture Notes in Artificial Intelligence, pp. 84–99, Trento, Italy, September , Springer, Berlin (2009)

  2. Armando A., Bonacina M.P., Ranise S., Schulz S.: New results on rewrite-based satisfiability procedures. ACM Trans. Comput. Log. 10(1), 129–179 (2009)

    Article  MathSciNet  Google Scholar 

  3. Armando A., Ranise S., Rusinowitch M.: A rewriting approach to satisfiability procedures. Inform. Comput. 183(2), 140–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  5. Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. J. Logic Comput. 4(3):217–247, 1994. Revised version of Max-Planck-Institut für Informatik technical report, MPI-I-91-208 (1991)

  6. Bachmair, L., Ganzinger, H., Waldmann, U.: Superposition with simplification as a decision procedure for the monadic class with equality. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) Computational Logic and Proof Theory, Third Kurt Gödel Colloquium, volume 713 of LNCS, pp. 83–96. Springer, August (1993)

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Baumgartner P., Fuchs A., Tinelli C.: ME(LIA)—model evolution with linear integer arithmetic constraints. In: LPAR 2008, volume 5330 of LNCS, pp. 258–273. Springer (2008)

  9. Bonacina M.P., Lynch C., de Moura L.M.: On deciding satisfiability by theorem proving with speculative inferences. J. Autom. Reason. 47(2), 161–189 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dershowitz N.: Orderings for term-rewriting systems. Theor. Comput. Sci. 17, 279–301 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dershowitz N., Manna Z.: Proving termination with multiset orderings. Commun. ACM. 22(8), 465–476 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  12. Eggers, A., Kruglov, E., Scheibler, K., Kupferschmid, S., Teige, T., Weidenbach, C.: SUP(NLA)—combining superposition and non-linear arithmetic. In: Sofronie-Stokkermans, V., Tinelli, C. (eds.) Frontiers of Combining Systems, 8th International Symposium, FroCos 2011, Lecture Notes in Computer Science. Springer, Berlin (2011)

  13. Fietzke, A., Hermanns, H., Weidenbach C.: Superposition-based analysis of first-order probabilistic timed automata. In: Proceedings of the 17th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR’10, pp. 302–316, Berlin, Heidelberg, Springer Verlag (2010)

  14. Fietzke, A., Kruglov, E.,Weidenbach C.: Automatic generation of inductive invariants by SUP(LA). Research Report MPI-I-2012-RG1-002, Max-Planck Institute for Informatics, Saarbrücken, Germany, March (2012)

  15. Fietzke, A., Kruglov, E., Weidenbach C.: Automatic generation of invariants for circular derivations in SUP(LA). In: Proceedings of the 18th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, volume 7180 of Lecture Notes in Computer Science, pp. 197–211. Springer, Berlin, March (2012)

  16. Fietzke, A., Weidenbach, C.: Superposition as a decision procedure for timed automata. In: Ratschan, S. (ed.) Fourth International Conference on Mathematical Aspects of Computer and Information Sciences (MACIS 2011), Beijing, China, 2011. Internal Conference Proceedings (2011)

  17. Ganzinger H., Sofronie-Stokkermans V., Waldmann U.: Modular proof systems for partial functions with Evans equality. Inform. Comput. 204(10), 1453–1492 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hillenbrand, T., Weidenbach, C.: Superposition for finite domains. Research Report MPI-I-2007-RG1-002, Max-Planck Institute for Informatics, Saarbruecken, Germany, April (2007)

  19. Hustadt U., Schmidt R.A., Georgieva L.: A survey of decidable first-order fragments and description logics. J. Relat. Methods Comput. Sci. 1, 251–276 (2004)

    Google Scholar 

  20. Ihlemann, C., Jacobs, S., Sofronie-Stokkermans V.: On local reasoning in verification. In: Ramakrishnan, C.R., Rehof, J. (eds.) Proceedings of TACAS 2008, volume 4963 of LNCS, pp. 265–281. Springer, Berlin (2008)

  21. Jacquemard, F., Meyer, C., Weidenbach C.: Unification in extensions of shallow equational theories. In: Nipkow, T. (ed.) Rewriting Techniques and Applications, 9th International Conference, RTA-98, volume 1379 of LNCS, pp. 76–90. Springer, Berlin (1998)

  22. Korovin, K., Voronkov A.: Integrating linear arithmetic into superposition calculus. In Duparc, J., Henzinger, T.A. (eds.) CSL 2007, volume 4646 of LNCS, pp. 223–237. Springer, Berlin (2007)

  23. Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving sat and sat modulo theories: from an abstract davis–putnam–logemann–loveland procedure to dpll(t). J. ACM. 53, pp. 937–977, November (2006)

  24. Sebastiani R.: Lazy satisability modulo theories. JSAT 3(3–4), 141–224 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Suchanek, F.M., Kasneci,G., Weikum G.: Yago: a core of semantic knowledge. In: Proceedings of the 16th international conference on World Wide Web, WWW ’07, pp. 697–706, New York, NY, USA, ACM (2007)

  26. Suda, M., Weidenbach, C., Wischnewski P.: On the saturation of yago. In: Proceedings of the 5th international conference on Automated Reasoning, IJCAR’10, pp. 441–456, Berlin, Heidelberg, Springer, Berlin (2010)

  27. Waldmann, U.: Superposition and chaining for totally ordered divisible abelian groups. In Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001, volume 2083 of LNAI, pp. 226–241. Springer, Berlin (2001)

  28. Weidenbach, C.: Combining superposition, sorts and splitting. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol 2, chapter 27, pp. 1965–2012. Elsevier, Amsterdam (2001)

  29. Weidenbach, C., Wischnewski, P.: Satisfiability checking and query answering for large ontologies. In: Fontaine, P., Schmidt, R., Schulz, S. (eds.) PAAR-2012: IJCAR’12 Workshop on Practical Aspects of Automated Reasoning, pp. 163–177 (2012)

Download references

Author information

Authors and Affiliations

  1. Universität des Saarlandes, Max-Planck-Institut für Informatik, Campus E1 4, 66123, Saarbrücken, Germany

    Evgeny Kruglov & Christoph Weidenbach

Authors
  1. Evgeny Kruglov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Weidenbach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgeny Kruglov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kruglov, E., Weidenbach, C. Superposition Decides the First-Order Logic Fragment Over Ground Theories. Math.Comput.Sci. 6, 427–456 (2012). https://doi.org/10.1007/s11786-012-0135-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-012-0135-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation