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Optimal axiomatizations for multiple-valued operators and quantifiers based on semi-lattices

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Automated Deduction — Cade-13 (CADE 1996)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1104))

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Abstract

We investigate the problem of finding optimal axiomatizations for operators and distribution quantifiers in finitely-valued first-order logics. We show that the problem can be viewed as the minimization of certain two-valued prepositional formulas. We outline a general procedure leading to optimized quantifier rules for the sequent calculus, for natural deduction and for clause formation. In the case of operators and quantifiers based on semi-lattices, rules with a minimal branching degree can be obtained by instantiating a schema, which can also be used for optimal tableaux with sets-as-signs.

Supported by FWF grant P10282-MAT.

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References

  1. M. Baaz and C. G. Fermüller. Resolution-based theorem proving for many-valued logics. J. Symbolic Computation, 19:353–391, 1995.

    Article  Google Scholar 

  2. M. Baaz, C. G. Fermüller, G. Salzer, and R. Zach. MUltlog 1.0: Towards an expert system for many-valued logics. In 13th Int. Conf. on Automated Deduction (CADE'96), LNCS (LNAI). Springer, 1996.

    Google Scholar 

  3. M. Baaz, C. G. Fermüller, and R. Zach. Systematic construction of natural deduction systems for many-valued logics. In Proc. 23rd International Symposium on Multiple-valued Logic, pages 208–215. IEEE Computer Society Press, Los Alamitos, May 24–27 1993.

    Google Scholar 

  4. W. A. Carnielli. Systematization of finite many-valued logics through the method of tableaux. J. Symbolic Logic, 52(2):473–493, 1987.

    Google Scholar 

  5. R. Hähnle. Automated Deduction in Multiple-valued Logics. Clarendon Press, Oxford, 1993.

    Google Scholar 

  6. R. Hähnle. Commodious axiomatization of quantifiers in multiple-valued logic. In Proc. 26th Int. Symp. on Multiple-Valued Logics, Santiago de Compostela, Spain. IEEE Press, Los Alamitos, May 1996.

    Google Scholar 

  7. McCluskey. Minimization of boolean functions. Bell Syst. Techn. J., 35:1417–1444, 1956.

    Google Scholar 

  8. A. Mostowski. On a generalization of quantifiers. Fund. Math., 44:12–36, 1957.

    Google Scholar 

  9. K. F. Nutz. Digitaltechnik, BASIC: Berechnung und Optimierung von Digitalschaltungen mit Hilfe von BASIC-Programmen. Oldenburg, 1984.

    Google Scholar 

  10. W. V. Quine. The problem of simplifying truth functions. American Mathematical Monthly, 59:521–531, 1952.

    Google Scholar 

  11. J. B. Rosser and A. R. Turquette. Many-Valued Logics. North-Holland, Amsterdam, 1952.

    Google Scholar 

  12. G. Rousseau. Sequents in many valued logic I. Fund. Math., 60:23–33, 1967.

    Google Scholar 

  13. N. Zabel. Nouvelles Techniques de Déduction Automatique en Logiques Polyvalentes Finies et Infinies du Premier Ordre. PhD thesis, Institut National Polytechnique de Grenoble, 1993.

    Google Scholar 

  14. R. Zach. Proof Theory of Finite-valued Logics. Diplomarbeit, Technische Universität Wien, 1993.

    Google Scholar 

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Author notes

  1. Gernot Salzer

    Present address: Technische Universität Wien, Karlsplatz 13, A-1040, Wien, Austria

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    1. Gernot Salzer
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    Correspondence to Gernot Salzer .

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    M. A. McRobbie J. K. Slaney

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    © 1996 Springer-Verlag Berlin Heidelberg

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    Salzer, G. (1996). Optimal axiomatizations for multiple-valued operators and quantifiers based on semi-lattices. In: McRobbie, M.A., Slaney, J.K. (eds) Automated Deduction — Cade-13. CADE 1996. Lecture Notes in Computer Science, vol 1104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61511-3_122

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    • DOI: https://doi.org/10.1007/3-540-61511-3_122

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    • Publisher Name: Springer, Berlin, Heidelberg

    • Print ISBN: 978-3-540-61511-8

    • Online ISBN: 978-3-540-68687-3

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