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Deriving Modular Programs from Short Proofs

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Automated Reasoning (IJCAR 2001)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 2083))

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Abstract

We present a polynomial translation of dag sequent proofs into tree sequent proofs for first-order classical and intuitionistic logic. The basic idea is to interpret a reference in a dag proof as a lemma application, which is then simulated using an application of the cut rule. The result of this translation is a tree proof with cuts, which are only applied in order to “factorize” identical subproofs. We illustrate a central application of the presented cut-based translation, that is automated extraction of modular programs from first-order intuitionistic proofs.

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References

  1. S. F. Allen, R. L. Constable et al. The NuPRL Open Logical Environment. In CADE-17, LNAI 1831, pp. 170–176, Springer, 2000.

    Google Scholar 

  2. J. L. Bates and R. L. Constable. Proofs as Programs. ACM Transactions on Programming Languages and Systems, 7(1):113–136, January 1985.

    Article  MATH  Google Scholar 

  3. R. L. Constable, S. F. Allen et al. Implementing Mathematics with the NuPRL Proof Development System. Prentice Hall, 1986.

    Google Scholar 

  4. H. B. Curry. Foundations of Mathematical Logic. Dover, Dover edition, 1977.

    Google Scholar 

  5. H. B. Curry, R. Feys, and W. Craig. Combinatory Logic, volume 1. North-Holland, Amsterdam, 1958.

    Google Scholar 

  6. A. Degtyarev and A. Voronkov. The Inverse Method. In J. A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, pp. 177–268, Elsevier Science Publishers, 2001.

    Google Scholar 

  7. E. Eder. Relative Complexities of First Order Calculi. Vieweg, 1992.

    Google Scholar 

  8. U. Egly. On Different Intuitionistic Calculi and Embeddings from INT to S4. Studia Logica, 2001. To appear

    Google Scholar 

  9. U. Egly and S. Schmitt. Intuitionistic Proof Transformations and their Application to Constructive Program Synthesis. In AISC’98, LNAI 1476, pp. 132–144, 1998.

    Google Scholar 

  10. U. Egly and S. Schmitt. On Intuitionistic Proof Transformations, Their Complexity and Application to Constructive Program Synthesis. Fundamenta Informaticae, 39:59–83, 1999.

    MathSciNet  MATH  Google Scholar 

  11. M. C. Fitting. Intuitionistic Logic, Model Theory and Forcing. Studies in logic and the foundations of mathematics. North-Holland, 1969.

    Google Scholar 

  12. G. Gentzen. Untersuchungen über das logische Schlieβen. Mathematische Zeitschrift, 39:176–210, 405-431, 1935.

    Article  MathSciNet  MATH  Google Scholar 

  13. W. Howard. The Formulas-as-Types Notion of Construction. In To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism., pp. 479–490, Academic Press 1980.

    Google Scholar 

  14. R. Letz. Polynomial Simulation of Sequent Systems by Tree Sequent Systems. Technical report, Institut für Informatik, TU München, 1992.

    Google Scholar 

  15. S. Maehara. Eine Darstellung der intuitionistischen Logik in der klassischen. Nagoya Mathematical Journal, 7:45–64, 1954.

    MathSciNet  MATH  Google Scholar 

  16. P. Martin-LÖf. Intuitionistic Type Theory, volume 1 of Studies in Proof Theory Lecture Notes. Bibliopolis, Napoli, 1984.

    Google Scholar 

  17. G. Mints. Resolution Strategies for the Intuitionistic Logic. In B. Mayoh, E. Tyugu, and J. Penyam, editors, Constraint Programming, Nato ASI Series, pp. 289–311. Springer Verlag, 1994.

    Google Scholar 

  18. S. Schmitt and C. Kreitz. On Transforming Intuitionistic Matrix Proofs into Standard-sequent Proofs. In 4th TABLEAUX Workshop, LNAI 918, pp. 106–121, 1995.

    Google Scholar 

  19. S. Schmitt, L. Lorigo, C. Kreitz, and A. Nogin. JProver: Integrating Connection-based Theorem Proving into Interactive Proof Assistants. In IJCAR, this volume, 2001.

    Google Scholar 

  20. W. W. Tait. Intensional Interpretation of Functionals of Finite Type. Journal of Symbolic Logic, 32(2):187–199, 1967.

    Article  MathSciNet  Google Scholar 

  21. A. S. Troelstra Marginalia on Sequent Calculi. Studia Logica, 62:291–303,1999.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. S. Troelstra and H. Schwichtenberg. Basic Proof Theory. Second Edition. Cambridge Univ. Press, 2000.

    Google Scholar 

  23. R. Vestergaard Revisting Kreisel: A Computational Anomaly in the Troelstra-Schwichtenberg G3i System. Technical report, 1999. http://iml.univ-mrs.fr:80/~vester/Writings/index.html

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Author information

Authors and Affiliations

  1. Institut für Informationssysteme 184/3, TU Wien, Favoritenstr. 9-11, A-1040, Wien, Austria

    Uwe Egly

  2. Department of Sciences and Engineering, Saint Louis University, Madrid Campus, Avd. del Valle 34, 28003, Madrid, Spain

    Stephan Schmitt

Authors
  1. Uwe Egly
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  2. Stephan Schmitt
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Editor information

Editors and Affiliations

  1. Automated Reasoning Project and Department of Computer Science, Australian National University, Canberra, ACT, 0200, Australia

    Rajeev Goré

  2. AG Theoretische Informatik und Logik, Institut für Computersprachen, Technische Universität Wien, Favoritenstr. 9, E185-2, 1040, Wien, Austria

    Alexander Leitsch

  3. Institut für Informatik, Technische Universität München, 80290, München, Germany

    Tobias Nipkow

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© 2001 Springer-Verlag Berlin Heidelberg

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Egly, U., Schmitt, S. (2001). Deriving Modular Programs from Short Proofs. In: Goré, R., Leitsch, A., Nipkow, T. (eds) Automated Reasoning. IJCAR 2001. Lecture Notes in Computer Science, vol 2083. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45744-5_47

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  • DOI: https://doi.org/10.1007/3-540-45744-5_47

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42254-9

  • Online ISBN: 978-3-540-45744-2

  • eBook Packages: Springer Book Archive

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