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Sharpened lower bounds for cut elimination

Published online by Cambridge University Press:  12 March 2014

Samuel R. Buss
Samuel R. Buss*
Affiliation:
Department of Mathematics, University of California, San Diego, La Jolla, Ca 92093-0112, USA, E-mail: sbuss@math.ucsd.edu
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Abstract

We present sharpened lower bounds on the size of cut free proofs for first-order logic. Prior lower bounds for eliminating cuts from a proof established superexponential lower bounds as a stack of exponentials, with the height of the stack proportional to the maximum depth d of the formulas in the original proof. Our results remove the constant of proportionality, giving an exponential stack of height equal to d − O(1). The proof method is based on more efficiently expressing the Gentzen–Solovay cut formulas as low depth formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 656 - 668
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Baaz, Matthias, and Leitsch, Alexander, On Skolemizations and proof complexity, Fundamenta Informaticae, vol. 20 (1994), pp. 353–379.Google Scholar
[2] Baaz, Matthias, and Leitsch, Alexander, Cut normal forms and proof complexity, Annals of Pure and Applied Logic, vol. 97 (1999), pp. 127–177.Google Scholar
[3] Baaz, Matthias, and Leitsch, Alexander, Fast cut-elimination by CERES, Proofs, categories and computations (Feferman, S. and Sieg, W., editors), College Publications, London, 2010, pp. 31–49.Google Scholar
[4] Beckmann, Arnold and Buss, Samuel R., Corrected upper bounds for free-cut elimination, Theoretical Computer Science, vol. 412 (2011), no. 39, pp. 5433–5445.Google Scholar
[5] Buss, Samuel R., An introduction to proof theory, Handbook of proof theory (Buss, S. R., editor). North-Holland, 1998, pp. 1–78.Google Scholar
[6] Buss, Samuel R., Cut elimination in situ, typeset manuscript, 2011.Google Scholar
[7] Buss, Samuel R. and Johnson, Alan, The quantifier complexity of polynomial-size iterated definitions in first-order logic, Mathematical Logic Quarterly, vol. 56 (2010), no. 6, pp. 573–590.Google Scholar
[8] Ferrante, Jeanne and Rackoff, Charles W., The computational complexity of logical theories, Lecture Notes in Mathematics, vol. 718, Springer Verlag, Berlin, 1979.CrossRefGoogle Scholar
[9] Gentzen, Gerhard, Beweisbarkeit und Unbeweisbarkeit von Anfangsfällen der transfiniten Induktion in der reinen Zahlentheorie, Mathematische Annalen, vol. 119 (1943), pp. 140–161, English translation in [10], pp. 287, 308.CrossRefGoogle Scholar
[10] Gentzen, Gerhard, Collected papers of Gerhard Gentzen, North-Holland, 1969, Edited by Szabo, M. E..Google Scholar
[11] Gerhardy, Philipp, Refined complexity analysis of cut elimination, Computer Science Logic 2003, Lecture Notes in Computer Science, vol. 2803, Springer Verlag, 2003, pp. 212–225.Google Scholar
[12] Gerhardy, Philipp, The role of quantifier alternations in cut elimination, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 2, pp. 165, 171.Google Scholar
[13] Girard, Jean-Yves, Proof theory and logical complexity, Humantities Press, 1987.Google Scholar
[14] Nelson, Edward, Predicative arithmetic, Princeton University Press, 1986.Google Scholar
[15] Orevkov, V. P., Lower bounds for lengthening of proofs after cut-elimination, Journal of Soviet Mathematics, vol. 20 (1982), pp. 2337–2350. Original Russian version in Zapiski Nauchnykh Seminarov (L.O.M.I.) Steklov , vol. 88 (1979), pp. 137–162.CrossRefGoogle Scholar
[16] Orevkov, V. P., Upper bound on the lengthening of proofs by cut elimination, Journal of Soviet Mathematics, vol. 34 (1986), pp. 1810–1819, Original Russian version in Zapiski Nauchnykh Seminarov [L.O.M.I.) Steklov , vol. 137 (1984), pp. 87–98.CrossRefGoogle Scholar
[17] Orevkov, V. P., Applications of cut elimination to obtain estimates of proof lengths, Soviet Mathematics Doklady, vol. 36 (1988), pp. 292–295, Original Russian version in Doklady Akademii Nauk , vol. 296 (1987), no. 3, pp. 539, 542.Google Scholar
[18] Paris, J. B. and Dimitracopoulos, C., A note on the undefinability of cuts, this Journal, vol. 48 (1983), no. 3, pp. 564, 569.Google Scholar
[19] Pudlák, Pavel, The lengths of proofs, Handbook of proof theory (Buss, S. R., editor), Elsevier, North-Holland, 1998, pp. 547–637.CrossRefGoogle Scholar
[20] Solovay, Robert M., Letter to P. Hajek, 08 1976.Google Scholar
[21] Statman, Richard, Lower bounds on Herbrand's theorem, Proceedings of the American Mathematical Society, vol. 75 (1979), no. 1, pp. 104–107.Google Scholar
[22] Statman, Richard, Speed-up by theories with infinite models, Proceedings of the American Mathematical Society, vol. 81 (1981), pp. 465–469.Google Scholar
[23] Troelstra, Anne S. and Schwichtenberg, Helmut, Basic proof theory, 2nd ed., Tracts in Theoretical Computer Science, vol. 43, Cambridge University Press, Cambridge, 2000.Google Scholar
[24] Yessenin-Volpin, A. S., The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics, Intuitionism and proof theory (Kino, A., Myhill, J., and Vesley, R. E., editors). North-Holland, 1970, pp. 1–45.Google Scholar
[25] Zhang, Wenhui, Cut elimination and automatic proof procedures, Theoretical Computer Science, vol. 91 (1991), pp. 265–284.Google Scholar
[26] Zhang, Wenhui, Depth of proofs, depth of cut-formulas, and complexity of cut formulas, Theoretical Computer Science, vol. 129 (1994), pp. 193–206.CrossRefGoogle Scholar
[27] Zhang, Wenhui, Structure of proofs and the complexity of cut elimination, Theoretical Computer Science, vol. 353 (2006), pp. 63–70.Google Scholar