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A Sound Framework for δ-Rule Variants in Free-Variable Semantic Tableaux

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Abstract

We propose a generic sound δ-rule, based on a quite general method for the construction of Skolem terms, which can be used as a common framework for proving the soundness of known variants of the δ-rule, and we compare their relative effectiveness. Attempts to instantiate some of the δ-rules present in the literature within our framework allowed us to pinpoint unsoundness problems for two of them. In both cases, we propose revised versions that are proved sound by reducing them within our framework.

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References

  1. Ahrendt, W., Beckert, B., Hähnle, R., Menzel, W., Reif, W., Schellhorn, G., Schmitt, P.H.: Integrating automated and interactive theorem proving. In: Bibel, W., Schmitt, P.H. (eds.) Automated Deduction – A Basis for Applications: Systems and Implementation Techniques of Applied Logic Series, vol. 2, No. 9, Chapt. 4, pp. 97–116. Kluwer, Dordrecht (1998)

    Google Scholar 

  2. Baaz, M., Fermüller, C.G.: Non-elementary speedups between different versions of tableaux. In: Baumgartner, P., Hähnle, R., Posegga, J. (eds.) Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX’95. Lecture Notes in Computer Science, vol. 918, pp. 217–230. Springer, Berlin Heidelberg New York (1995)

    Google Scholar 

  3. 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.) Third Kurt Gödel Colloquium. Lecture Notes in Computer Science, vol. 713, pp. 108–119. Springer, Berlin Heidelberg New York (1993)

    Google Scholar 

  4. Cantone, D., Nicolosi Asmundo, M.: A further and effective liberalization of the δ-rule in free variable semantic tableaux. In: Caferra, R., Salzer, G. (eds.) Automated Deduction in Classical and Non-Classical Logics. Lecture Notes in Computer Science, vol. 1761, pp. 109–125. Springer, Berlin Heidelberg New York (2000)

    Google Scholar 

  5. Cantone, D., Nicolosi Asmundo, M: A sound framework for δ-rule variants in free variable semantic tableaux. In: Letz, R. (ed.) Proc. of the Fifth International Workshop on First-Order Theorem Proving (FTP 2005), pp. 51–69, Research Report 13/2005, Universität Koblenz-Landau, Institut für Informatik (2005)

  6. Cantone, D., Nicolosi Asmundo, M., Omodeo, E.G.: Global skolemization with grouped quantifiers. In: Falaschi, M., Navarro, M., Policriti, A. (eds.) Proc. of 1997 Joint Conf. on Declarative Programming, APPIA-GULP-PRODE’97, pp. 405–414. Grado, Italy (1997)

  7. D’Agostino, M., Gabbay, D.M, Hähnle, R., Posegga, J. (eds.): Handbook of Tableau Methods. Kluwer, Dordrecht (1999)

    Google Scholar 

  8. Davis, M., Fechter, R.: A free variable version of the first-order predicate calculus. J. Log. Comput. 1(4), 431–451 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics, pp. 243–320. MIT, Cambridge, MA (1990)

    Google Scholar 

  10. Fitting, M.C.: First-Order Logic and Automated Theorem Proving, Graduate Texts in Computer Science, 2nd ed. Springer, Berlin Heidelberg New York (1996)

    Google Scholar 

  11. Giese, M., Ahrendt, W.: Hilbert’s ε-term in automated theorem proving. In: Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX ’99. Lecture Notes in Computer Science, vol. 1617, pp. 171–185. Springer, Berlin Heidelberg New York (1999)

    Google Scholar 

  12. Hähnle, R.: Tableaux and related methods. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning (in 2 volumes), Chapt. 3, pp. 100–178. MIT, Cambridge, MA (2001)

    Google Scholar 

  13. Hähnle, R., Schmitt, P.H.: The liberalized δ-rule in free variable semantic tableaux. J. Autom. Reason. 13(2), 211–221 (1994)

    Article  MATH  Google Scholar 

  14. Letz, R.: First-order tableau methods. In: D’Agostino, M., Gabbay, D.M., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 125–196. Kluwer, Dordrecht (1999)

    Google Scholar 

  15. Letz, R., Stenz, G.: Model elimination and connection tableau procedures. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning (in 2 volumes), Chapt. 28, pp. 2015–2114. MIT, Cambridge, MA (2001)

    Google Scholar 

  16. Smullyan, R.M.: First-order Logic. Springer, Berlin Heidelberg New York (1968)

    MATH  Google Scholar 

  17. Sutcliffe, G., Suttner, G.: The CADE-15 ATP system competition. J. Autom. Reason. 23(1), 1–23 (1999)

    Article  MathSciNet  Google Scholar 

  18. Thion, V., Cerrito, S., Mayer, M.C.: A general theorem prover for quantified modal logics. In: Automated Reasoning with Analytic Tableaux and Related Methods, International Conference, TABLEAUX 2002. Lecture Notes in Computer Science, vol. 2381, pp. 266–280. Springer, Berlin Heidelberg New York (2002)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale A. Doria 6, I-95125, Catania, Italy

    Domenico Cantone & Marianna Nicolosi-Asmundo

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  1. Domenico Cantone
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  2. Marianna Nicolosi-Asmundo
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Correspondence to Domenico Cantone.

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This paper is based on results preliminarily presented in [5].

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Cantone, D., Nicolosi-Asmundo, M. A Sound Framework for δ-Rule Variants in Free-Variable Semantic Tableaux. J Autom Reasoning 38, 31–56 (2007). https://doi.org/10.1007/s10817-006-9045-y

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  • DOI: https://doi.org/10.1007/s10817-006-9045-y

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