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Refined Complexity Analysis of Cut Elimination

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Computer Science Logic (CSL 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2803))

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Abstract

In [1,2] Zhang shows how the complexity of cut elimination depends on the nesting of quantifiers in cut formulas. By studying the role of contractions we can refine that analysis and show how the complexity depends on a combination of contractions and quantifier nesting. With the refined analysis the upper bound on cut elimination coincides with Statman’s lower bound. Every non-elementary growth example must display a combination of nesting of quantifiers and contractions similar to Statman’s lower bound example. The upper and lower bounds on cut elimination immediately translate into bounds on Herbrand’s theorem. Finally we discuss the role of quantifier alternations and show an elementary upper bound for the ∀ − ∧-case (resp. ∃ − ∨-case).

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References

  1. Zhang, W.: Cut elimination and automatic proof procedures. Theoretical Computer Science 91, 265–284 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Zhang, W.: Depth of proofs, depth of cut-formulas and complexity of cut formulas. Theoretical Computer Science 129, 193–206 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Schwichtenberg, H.: Proof Theory: Some Applications of Cut-Elimination. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 868–894. North-Holland Publishing Company, Amsterdam (1977)

    Google Scholar 

  4. Buss, S.R.: An Introduction to Proof Theory. In: Buss, S.R. (ed.) Handbook of Proof Theory, pp. 2–78. Elsevier Science B.V, Amsterdam (1998)

    Google Scholar 

  5. Luckhardt, H.: Komplexität der Beweistransformation bei Schnittelimination. Handwritten notes (1991)

    Google Scholar 

  6. Kohlenbach, U.: Notes to [5]. Handwritten notes (1995)

    Google Scholar 

  7. Girard, J.Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Blass, A.: A game semantics for linear logic. Annals of Pure and Applied Logic 56, 183–220 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ketonen, J., Weyrauch, R.: A Decidable Fragment of Predicate Calculus. Theoretical Computer Science 32, 297–307 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Bellin, G., Ketonen, J.: A Decision Procedure Revisited: Notes on Direct Logic, Linear Logic and its Implementation. Theoretical Computer Science 95, 115–142 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  11. Statman, R.: Lower Bounds on Herbrand’s Theorem. Proc. of the Amer. Math. Soc. 75, 104–107 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  12. Orevkov, V.P.: Complexity of Proofs and Their Transformations in Axiomatic Theories. Translations of Mathematical Monographs, vol. 128. American Mathematical Society, Providence (1993)

    MATH  Google Scholar 

  13. Pudlak, P.: The Length of Proofs. In: Buss, S.R. (ed.) Handbook of Proof Theory, pp. 548–637. Elsevier Science B.V, Amsterdam (1998)

    Google Scholar 

  14. Baaz, M., Leitsch, A.: On Skolemizations and Proof Complexity. Fundamenta Informatica 20/4, 353–379 (1994)

    MathSciNet  Google Scholar 

  15. Gerhardy, P.: Improved Complexity Analysis of Cut Elimination and Herbrand’s Theorem. Master’s thesis (2003)

    Google Scholar 

  16. Carbone, A.: Asymptotic Cyclic Expansion and Bridge Groups of Formal Proofs. Journal of Algebra 242, 109–145 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Baaz, M., Leitsch, A.: Cut Normal Forms and Proof Complexity. Annals of Pure and Applied Logic 97, 127–177 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. BRICS Department of Computer Science, University of Aarhus Ny Munkegade, DK-8000, Aarhus C, Denmark

    Philipp Gerhardy

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  1. Philipp Gerhardy
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Editors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040, Wien, Austria

    Matthias Baaz

  2. Members of EACSL Jury for the Ackermann Award,  

    Johann A. Makowsky

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Gerhardy, P. (2003). Refined Complexity Analysis of Cut Elimination. In: Baaz, M., Makowsky, J.A. (eds) Computer Science Logic. CSL 2003. Lecture Notes in Computer Science, vol 2803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45220-1_19

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  • DOI: https://doi.org/10.1007/978-3-540-45220-1_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40801-7

  • Online ISBN: 978-3-540-45220-1

  • eBook Packages: Springer Book Archive

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