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Refutational theorem proving for hierarchic first-order theories

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

We extend previous results on theorem proving for first-order clauses with equality to hierarchic first-order theories. Semantically such theories are confined to conservative extensions of the base models. It is shown that superposition together with variable abstraction and constraint refutation is refutationally complete for theories that are sufficiently complete with respect to simple instances. For the proof we introduce a concept of approximation between theorem proving systems, which makes it possible to reduce the problem to the known case of (flat) first-order theories. These results allow the modular combination of a superposition-based theorem prover with an arbitrary refutational prover for the primitive base theory, whose axiomatic representation in some logic may remain hidden. Furthermore they can be used to eliminate existentially quantified predicate symbols from certain second-order formulae.

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References

  1. Avenhaus, J., Becker, K.: Conditional rewriting modulo a built-in algebra. SEKI Report SR-92-11, Fachbereich Informatik, Universität Kaiserslautern (1992)

  2. Bachmair, L., Ganzinger, H.: On restrictions of ordered paramodulation with simplification. In: Stickel, M. E. (ed.) 10th International Conference on Automated Deduction. Lecture Notes in Artificial Intelligence, Vol.449, pp. 427–441. Berlin, Heidelberg, New York: Springer 1990

    Google Scholar 

  3. Bachmair, L., Ganzinger, H.: Perfect model semantics for logic programs with equality. In: Furukawa, K. (ed.) Logic Programming, Proceedings of the Eighth International Conference, pp. 645–659. The MIT Press 1991

  4. Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. Tech. Rep. MPI-I-91-208, Max-Planck-Institut für Informatik, Saarbrücken (1991). Revised version to appear in J. Logic Comput.

    Google Scholar 

  5. Bergstra, J. A., Broy, M., Tucker, J. V., Wirsing, M.: On the power of algebraic specifications. In: Gruska, J., Chytil, M. (eds.) Mathematical Foundations of Computer Science, 10th Symposium. Lecture Notes in Computer Science, Vol.118, pp. 193–204. Berlin, Heidelberg, New York: Springer 1981

    Google Scholar 

  6. Bürckert, H.-J.: A resolution principle for clauses with constraints. In: Stickel, M. E. (ed.) 10th International Conference on Automated Deduction. Lecture Notes in Artificial Intelligence, Vol.449, pp. 178–192. Berlin, Heidelberg, New York: Springer 1990

    Google Scholar 

  7. Bürckert, H.-J.: A Resolution Principle for a Logic with Restricted Quantifiers. Lecture Notes in Artificial Intelligence, Vol.568. Berlin, Heidelberg, New York: Springer 1991

    Google Scholar 

  8. Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, Vol. B: Formal Models and Semantics, pp. 244–320. Amsterdam, New York, Oxford, Tokyo: Elsevier Science Publishers B. V. 1990

    Google Scholar 

  9. Gabbay, D., Ohlbach, H. J.: Quantifier elimination in second order predicate logic. South African Computer Journal7, 35–43 (1992). Also appeared in 3rd International Conference on Principles of Knowledge Representation and Reasoning, pp. 425–435. 1992

    Google Scholar 

  10. Jaffar, J., Lassez, J.-L.: Constraint logic programming. In: Fourteenth Annual ACM Symposium on Principles of Programming Languages, pp. 111–119 (1987)

  11. Kirchner, H.: Proofs in parameterized specifications. In: Book, R. V. (ed.) Rewriting Techniques and Applications, 4th International Conference. Lecture Notes in Computer Science, Vol.448, pp. 174–187. Berlin, Heidelberg, New York: Springer 1991

    Google Scholar 

  12. Kreisel, G., Krivine, J.-L.: Elements of Mathematical Logic. Amsterdam: North-Holland 1967

    Google Scholar 

  13. Nieuwenhuis, R.: First-order completion techniques. Technical report, Universidad Politécnica de Cataluña, Dept. Lenguajes y Sistemas Informáticos (1991)

  14. Stickel, M. E.: Automated deduction by theory resolution. J. Autom. Reasoning1, 333–355 (1985)

    Google Scholar 

  15. Van Benthem, J.: Correspondence theory. In: Gabbay, D. M., Guenthner, F. (eds.) Handbook of Philosophical Logic, Vol. II: Extensions of Classical Logic, pp. 167–247. Dordrecht, Boston, Lancaster: D. Reidel 1984

    Google Scholar 

  16. Wirsing, M.: Algebraic specification. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, pp. 675–788. Amsterdam, New York, Oxford, Tokyo: Elsevier Science Publishers B. V. 1990

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, SUNY at Stony Brook, 11794, Stony Brook, NY, USA

    Leo Bachmair

  2. Max-Planck-Institut für Informatik, Im Stadtwald, D-66123, Saarbrücken, Germany

    Harald Ganzinger & Uwe Waldmann

Authors
  1. Leo Bachmair
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  2. Harald Ganzinger
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  3. Uwe Waldmann
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Bachmair, L., Ganzinger, H. & Waldmann, U. Refutational theorem proving for hierarchic first-order theories. AAECC 5, 193–212 (1994). https://doi.org/10.1007/BF01190829

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  • DOI: https://doi.org/10.1007/BF01190829

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