Formalization of the Resolution Calculus for First-Order Logic

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9807)

Abstract

A formalization in Isabelle/HOL of the resolution calculus for first-order logic is presented. Its soundness and completeness are formally proven using the substitution lemma, semantic trees, Herbrand’s theorem, and the lifting lemma. In contrast to previous formalizations of resolution, it considers first-order logic with full first-order terms, instead of the propositional case.

Keywords

First-order logic Resolution Isabelle/HOL Herbrand’s theorem Soundness Completeness 

References

  1. 1.
    Ben-Ari, M.: Mathematical Logic for Computer Science, 3rd edn. Springer (2012)Google Scholar
  2. 2.
    Berghofer, S.: First-order logic according to Fitting. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/FOL-Fitting.shtml
  3. 3.
    Blanchette, J.C., Fleury, M., Schlichtkrull, A., Traytel, D.: IsaFoL: Isabelle Formalization of Logic. https://bitbucket.org/jasmin_blanchette/isafol
  4. 4.
    Blanchette, J.C., Traytel, D.: Formalization of Bachmair and Ganzinger’s “Resolution Theorem Proving”. https://bitbucket.org/jasmin_blanchette/isafol/src/master/Bachmair_Ganzinger/
  5. 5.
    Blanchette, J.C., Popescu, A., Traytel, D.: Unified classical logic completeness – A coinductive pearl. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 46–60. Springer, Heidelberg (2014)Google Scholar
  6. 6.
    Braselmann, P., Koepke, P.: Gödel completeness theorem. Formalized Math. 13(1), 49–53 (2005)Google Scholar
  7. 7.
    Braselmann, P., Koepke, P.: A sequent calculus for first-order logic. Formalized Math. 13(1), 33–39 (2005)Google Scholar
  8. 8.
    Chang, C.L., Lee, R.C.T.: Symbolic Logic and Mechanical Theorem Proving, 1st edn. Academic Press Inc., Orlando (1973)MATHGoogle Scholar
  9. 9.
    Fleury, M.: Formalisation of ground inference systems in a proof assistant. Master’s thesis, École normale supérieure Rennes (2015). http://www.mpi-inf.mpg.de/fileadmin/inf/rg1/Documents/fleury_master_thesis.pdf
  10. 10.
    Harrison, J.V.: Formalizing basic first order model theory. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 153–170. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Harrison, J.: Towards self-verification of HOL Light. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 177–191. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Illik, D.: Constructive completeness proofs and delimited control. Ph.D. thesis, École Polytechnique (2010)Google Scholar
  13. 13.
    Kumar, R., Arthan, R., Myreen, M.O., Owens, S.: Self-formalisation of higher-order logic – Semantics, soundness, and a verified implementation. J. Autom. Reason 56(3), 221–259 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Leitsch, A.: On different concepts of resolution. Math. Logic Q. 35(1), 71–77 (1989)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Leitsch, A.: The Resolution Calculus. Springer, Texts in theoretical computer science (1997)CrossRefMATHGoogle Scholar
  16. 16.
    Margetson, J., Ridge, T.: Completeness theorem. Archive of Formal Proofs, Formal proof development. http://isa-afp.org/entries/Completeness.shtml
  17. 17.
    Paulson, L.C.: A mechanised proof of Gödel’s incompleteness theorems using Nominal Isabelle. J. Autom. Reason. 55(1), 1–37 (2015)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Riazanov, A., Voronkov, A.: Vampire. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 292–296. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  19. 19.
    Ridge, T., Margetson, J.: A mechanically verified, sound and complete theorem prover for first order logic. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 294–309. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. ACM 12(1), 23–41 (1965)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Schlichtkrull, A.: Formalization of resolution calculus in Isabelle. Master’s thesis, Technical University of Denmark (2015). https://people.compute.dtu.dk/andschl/Thesis.pdf
  22. 22.
    Schlöder, J.J., Koepke, P.: The Gödel completeness theorem for uncountable languages. Formalized Math. 20(3), 199–203 (2012)MATHGoogle Scholar
  23. 23.
    Schulz, S.: System description: E 1.8. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 735–743. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Sternagel, C., Thiemann, R.: An Isabelle/HOL formalization of rewriting for certified termination analysis. http://cl-informatik.uibk.ac.at/software/ceta/
  25. 25.
    Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 140–145. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.DTU ComputeTechnical University of DenmarkKongens LyngbyDenmark

Personalised recommendations