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Journal of Automated Reasoning

, Volume 42, Issue 1, pp 99–122 | Cite as

First Order Stålmarck

Universal Lemmas Through Branch Merges
  • Magnus BjörkEmail author
Article

Abstract

We present a proof method with a novel way of introducing universal lemmas. The method is a first order extension of Stålmarck’s method, containing a branch-and-merge rule known as the dilemma rule. The dilemma rule creates two branches in a tableau-like way, but later recombines the two branches, keeping the common consequences. While the propositional version uses normal set intersection in the merges, the first order version searches for pairwise unifiable formulae in the two branches. Within branches, the system uses a special kind of variables that may not be substituted. At branch merges, these variables are replaced by universal variables, and in this way universal lemmas can be introduced. Relevant splitting formulae are found through failed unifications of variables in branches. This article presents the calculus and proof procedure, and shows soundness and completeness. Benchmarks of an implementation are also presented.

Keywords

Automated theorem proving First order logic Stålmarck’s method Universal lemmas Intersections 

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References

  1. 1.
    Andersson, G., Bjesse, P., Cook, B., Hanna, Z.: A proof engine approach to solving combinational design automation problems. In: Design Automation Conference (DAC), pp. 725–730, ACM, New York (2002)Google Scholar
  2. 2.
    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. Springer, Berlin (2003)Google Scholar
  3. 3.
    Baumgartner, P., Tinelli, C.: The model evolution calculus with equality. In: CADE. Lecture Notes in Computer Science, vol. 3632 pp. 392–408. Springer, Berlin (2005)Google Scholar
  4. 4.
    Baumgartner, P., Tinelli, C.: The model evolution calculus as a first-order DPLL method. Artif. Intell. 172(4–5), 591–632 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Beckert, B., Hähnle, R., Schmitt, P.H.: The even more liberalized δ-Rule in free variable semantic tableaux. In: Gottlob, G., Leitsch, A., Mundici, D. (eds.) Proceedings, 3rd Kurt Gödel Colloquium (KGC), Brno, Czech Republic, pp. 108–119. Springer, Berlin (1993)Google Scholar
  6. 6.
    Billon, J.-P.: The disconnection method: a confluent integration of unification in the analytic framework. In: TABLEAUX’96. LNAI, vol. 1071, pp. 110–126. Springer, Berlin (1996)Google Scholar
  7. 7.
    Björk, M.: Extending Stålmarck’s method to first order logic. In: Mayer, M.C., Pirri, F. (eds.) TABLEAUX 2003 Position Papers and Tutorials, pp. 23–36, Dipartimento di Informatica e Automazione, Università degli Studi di Roma Tre (2003)Google Scholar
  8. 8.
    Björk, M.: Adding equivalence classes to Stålmarck’s method in first order logic. In: IJCAR Doctoral Programme. http://CEUR-WS.org/Vol-106/02-bjork.ps: CEUR Workshop Proceedings, vol. 106 (2004)
  9. 9.
    Björk, M.: A first order extension of Stålmarck’s method. In: Sutcliffe, G., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning, pp. 276–291. Springer, Berlin (2005)CrossRefGoogle Scholar
  10. 10.
    Björk, M.: A first order extension of Stålmarcks method. Ph.D. thesis, Department of Computing Science, Chalmers University of Technology (2006)Google Scholar
  11. 11.
    Borälv, A.: The industrial success of verification tools based on Stålmarck’s method. In: Computer Aided Verification, CAV. Lecture Notes in Computer Science, vol. 1254. Springer, Berlin (1997)Google Scholar
  12. 12.
    Borälv, A.: Case study: formal verification of a computerized railway interlocking. Form. Asp. Comput. 10(4), 338–360 (1998)zbMATHCrossRefGoogle Scholar
  13. 13.
    Cook, B., Gonthier, G.: Using Stålmarck’s algorithm to prove inequalities. In: 7th International Conference on Formal Engineering Methods (ICFEM), pp. 330–344 (2005)Google Scholar
  14. 14.
    Davis, M.: The early history of automated deduction. In [25], chap. 1, pp. 3–15 (2001)Google Scholar
  15. 15.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5, 394–397 (1962)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. JACM 7(3), 201–215 (1960)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fitting, M.C.: First-Order Logic and Automated Theorem Proving, 2nd edn. Springer, New York (1996)zbMATHGoogle Scholar
  18. 18.
    Gilmore, P.C.: A proof method for quantification theory: its justification and realization. IBM J. Res. Develop. 4, 28–35 (1960)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Hähnle, R., Schmitt, P.H.: The liberalized δ-rule in free variable semantic tableaux. J. Autom. Reason. 13(2), 211–222 (1994)zbMATHCrossRefGoogle Scholar
  20. 20.
    Letz, R., Stenz, G.: Proof and model generation with disconnection tableaux. In: Proceedings of the 8th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning. LNAI, vol. 2250, pp. 142–156. Springer, New York (2001)CrossRefGoogle Scholar
  21. 21.
    Letz, R., Stenz, G.: Integration of equality reasoning into the disconnection calculus. In: TABLEAUX. LNCS, vol. 2381, pp. 176–190. Springer, Berlin (2002)Google Scholar
  22. 22.
    McCune, W.: OTTER 3.3 Reference Manual’. CoRR. http://arxiv.org/abs/cs.SC/0310056, cs.SC/0310056 (2003)
  23. 23.
    Mondadori, M.: Classical analytical deduction. Annali dell’ Università di Ferrara, Nuova Serie, sezione III, Filosofia, discussion paper, n. 1, Università degli Studi di Ferrara (1988)Google Scholar
  24. 24.
    Ramakrishnan, I.V., Sekar, R., Voronkov, A.: Term indexing. In [25], chap. 26, pp. 1853–1965 (2001)Google Scholar
  25. 25.
    Robinson, A., Voronkov, A. (eds.): Handbook of Automated Reasoning. Elsevier, Amsterdam (2001)zbMATHGoogle Scholar
  26. 26.
    Robinson, J.A.: A machine-oriented logic based on the resolution Principle. J. ACM 12(1), 23–41 (1965)zbMATHCrossRefGoogle Scholar
  27. 27.
    Sheeran, M., Stålmarck, G.: A tutorial on Stålmarck’s proof procedure for propositional logic. Form. Methods Syst. Des. 16(1), 23–58 (2000)CrossRefGoogle Scholar
  28. 28.
    Smullyan, R.M.: First-Order Logic, 2nd corrected edn. Dover Publications, New York. First published 1968 by Springer-Verlag (1995)Google Scholar
  29. 29.
    Stenz, G., Wolf, A.: E-SETHEO: an automated3 theorem prover—system abstract. In: Dyckhoff, R. (ed.) Proc. of the TABLEAUX’2000. LNAI, vol. 1847, pp. 436–440. Springer, New York (2000)Google Scholar
  30. 30.
    Sutcliffe, G.: The IJCAR-2004 automated theorem proving competition. AI Commun. 18(1), 33–40 (2005)MathSciNetGoogle Scholar
  31. 31.
    Sutcliffe, G., Suttner, C.: The TPTP problem library: CNF release v1.2.1. J. Autom. Reason. 21(2), 177–203 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Chalmers University of TechnologyGothenburgSweden

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