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A Tableau-Based Decision Procedure for a Fragment of Set Theory with Iterated Membership

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Abstract

MLSS is a decidable fragment of set theory involving the predicates membership and set equality and the operators union, intersection, set difference, and singleton. In this paper we extend MLSS with the iterated membership predicate, that is, with a predicate denoting the transitive closure of the membership relation. We call the resulting language MLSS+. We prove that MLSS+ is decidable by providing a decision procedure for it based on Smullyan semantic tableaux. As an application of our results, we show how our decision procedure can be used as a black box in order to allow an interactive theorem prover to verify some basic properties of the ordinal numbers.

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References

  1. Aczel, P.: Non-Well-Founded Sets, CSLI Publications, Stanford, CA, 1988.

    Google Scholar 

  2. Cantone, D.: A fast saturation strategy for set-theoretic tableaux, in D. Galmiche (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Comput. Sci. 1227, Springer, 1997, pp. 122–137.

  3. Cantone, D. and Cincotti, G.: The decision problem in graph theory with reachability related constructs, in P. Baumgartner and H. Zhang (eds.), Third International Workshop on First-Order Theorem Proving (FTP 2000), Technical Report 5/2000, Universität Koblenz-Landau, 2000, pp. 68–80.

  4. Cantone, D. and Ferro, A.: Techniques of computable set theory with applications to proof verification, Comm. Pure Appl. Math. 48(9–10) (1995), 1–45.

    Google Scholar 

  5. Cantone, D., Ferro, A. and Omodeo, E. G.: Computable Set Theory, Internat. Ser. Monographs Comput. Sci. 6, Clarendon Press, 1989.

  6. Cantone, D., Omodeo, E. G. and Policriti, A.: The automation of syllogistic, II: Optimization and complexity issues, J. Automated Reasoning 6(2) (1990), 173–187.

    Google Scholar 

  7. Cantone, D., Omodeo, E. G. and Policriti, A.: Set Theory for Computing: From Decision Procedures to Logic Programming with Sets, Monographs in Computer Science, Springer, 2001.

  8. Cantone, D. and Cannata, R. R.: Deciding set-theoretic formulae with the predicate Finite by a tableau calculus, Le Matematiche L(I) (1995), 99–118.

    Google Scholar 

  9. Cantone, D. and Zarba, C. G.: A tableau-based decision procedure for a fragment of set theory involving a restricted form of quantification, in N. V. Murray (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Comput. Sci. 1617, Springer, 1999, pp. 97–112.

  10. Cantone, D. and Zarba, C. G.: A new fast tableau-based decision procedure for an unquantified fragment of set theory, in R. Caferra and G. Salzer (eds.), Automated Deduction in Classical and Non-Classical Logics, Lecture Notes in Comput. Sci. 1761, Springer, 2000, pp. 127–137.

  11. Cantone, D. and Zarba, C. G.: A tableau calculus for integrating first-order reasoning with elementary set theory reasoning, in R. Dyckhoff (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Comput. Sci. 1847, Springer, 2000, pp. 143–159.

  12. Ferro, A., Omodeo, E. G. and Schwartz, J. T.: Decision procedures for elementary sublanguages of set theory, I: Multi-level syllogistic and some extensions, Comm. Pure Appl. Math. 33(5) (1980), 599–608.

    Google Scholar 

  13. Fitting, M. C.: First-Order Logic and Automated Theorem Proving, 2nd edn, Graduate Texts in Computer Science, Springer, 1996.

  14. Piazza, C. and Policriti, A.: Towards tableau-based decision procedures for non-well-founded fragments of set theory, in R. Dyckhoff (ed.), Automated Reasoning with Analytic Tableaux and Related Methods, Lecture Notes in Comput. Sci. 1847, Springer, 2000, pp. 368–382.

  15. Smullyan, R. M.: First-Order Logic, Springer, 1968.

  16. Zarba, C. G.: Combining sets with elements, in N. Dershowitz (ed.), Verification: Theory and Practice, Lecture Notes in Comput. Sci. 2772, Springer, 2004, pp. 762–782.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Catania, Viale Andrea Doria 6, 95125, Catania, Italy

    Domenico Cantone

  2. LORIA and INRIA-Lorraine, 615, rue du Jardin Botanique, BP 101, 54602, Villers-lès-Nancy Cedex, France

    Calogero G. Zarba

  3. European Commission ESTAT.C.6, Economic indicators of the Euro zone, B-1049, Bruxelles, Belgium

    Rosa Ruggeri Cannata

Authors
  1. Domenico Cantone
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  2. Calogero G. Zarba
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  3. Rosa Ruggeri Cannata
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Corresponding author

Correspondence to Domenico Cantone.

Additional information

This research was in part supported by murst grant prot. 2001017741 under the Italian project “Ragionamento su aggregati e numeri a supporto della programmazione e relative verifiche”.

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Cantone, D., Zarba, C.G. & Cannata, R.R. A Tableau-Based Decision Procedure for a Fragment of Set Theory with Iterated Membership. J Autom Reasoning 34, 49–72 (2005). https://doi.org/10.1007/s10817-004-8271-4

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  • DOI: https://doi.org/10.1007/s10817-004-8271-4

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