Skip to main content
Springer Nature Link
Log in

How to lie without being (easily) convicted and the lengths of proofs in propositional calculus

  • Conference paper
  • First Online:
Computer Science Logic (CSL 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 933))

Included in the following conference series:

Abstract

We shall describe two general methods for proving lower bounds on the lengths of proofs in propositional calculus and give examples of such lower bounds. One of the methods is based on interactive proofs where one player is claiming that he has a falsifying assignment for a tautology and the second player is trying to convict him of a lie. The second method is based on boolean valuations. For the first method, a log n + log log n -O(logloglog n) lower bound is given on the lengths of interactive proofs of certain permutation tautologies.

Partially supported by US-Czechoslovak Science and Technology Program grant No. 93025

Partially supported by US-Czechoslovak Science and Technology Program grant No. 93025 and by NSF grant DMS92-05181

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ajtai, The complexity of the pigeonhole principle, in Proceedings of the 29-th Annual IEEE Symposium on Foundations of Computer Science, 1988, pp. 346–355.

    Google Scholar 

  2. P. Beame, R. Impagliazzo, J. Krajíček, T. Pitassi, P. Pudlák, and A. Woods, Exponential lower bounds for the pigeonhole principle, in Proceedings of the 24-th Annual ACM Symposium on Theory of Computing, 1992, pp. 200–220.

    Google Scholar 

  3. S. R. Buss, Bounded Arithmetic, Bibliopolis, 1986. Revision of 1985 Princeton University Ph.D. thesis.

    Google Scholar 

  4. S. R. Buss and et al., Weak formal systems and connections to computational complexity. Student-written Lecture Notes for a Topics Course at U.C. Berkeley, January–May 1988.

    Google Scholar 

  5. S. A. Cook, Feasibly constructive proofs and the propositional calculus, in Proceedings of the 7-th Annual ACM Symposium on Theory of Computing, 1975, pp. 83–97.

    Google Scholar 

  6. S. A. Cook and R. A. Reckhow, On the lengths of proofs in the propositional calculus, preliminary version, in Proceedings of the Sixth Annual ACM Symposium on the Theory of Computing, 1974, pp. 135–148.

    Google Scholar 

  7. —, The relative efficiency of propositional proof systems, Journal of Symbolic Logic, 44 (1979), pp. 36–50.

    Google Scholar 

  8. M. Dowd, Propositioned representation of arithmetic proofs, in Proceedings of the 10th ACM Symposium on Theory of Computing, 1978, pp. 246–252.

    Google Scholar 

  9. J. Krajíček, Bounded Arithmetic, Propositional Calculus and Complexity Theory, Cambridge University Press, To appear.

    Google Scholar 

  10. J. Krajíček, Lower bounds to the size of constant-depth Frege proofs. To appear in Journal of Symbolic Logic.

    Google Scholar 

  11. — Speed-up for propositional Frege systems via generalizations of proofs, Commentationes Mathematicae Universitatis Carolinae, 30 (1989), pp. 137–140.

    Google Scholar 

  12. -, On Frege and extended Frege proof systems. Typeset manuscript, 1993.

    Google Scholar 

  13. J. Krajíček and P. Pudlák, Propositional proof systems, the consistency of first-order theories and the complexity of computations, Journal of Symbolic Logic, 54 (1989), pp. 1063–1079.

    Google Scholar 

  14. —, Quantified propositional calculi and fragments of bounded arithmetic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 36 (1990), pp. 29–46.

    Google Scholar 

  15. V. P. Orevkov, On lower bounds on the lengths of proofs in propositional logic (russian), in Proc. of All Union Conference Metody matem. logiki v problemach iskusstvennogo intellekta i sistematicheskoje programmirovanie, Vilnius, vol. I, 1980, pp. 142–144.

    Google Scholar 

  16. R. Parikh, Some results on the lengths of proofs, Transactions of the American Mathematical Society, 177 (1973), pp. 29–36.

    Google Scholar 

  17. P. Pudlák, The lengths of proofs. To appear in Handbook of Proof Theory, ed. S. Buss.

    Google Scholar 

  18. S. Riis, Independence in Bounded Arithmetic, PhD thesis, Oxford University, 1993.

    Google Scholar 

  19. C. Shannon, On the synthesis of two-terminal switching circuits, Bell System Technical Journal, 28 (1949), pp. 59–98.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, Academy of Sciences of the Czech Republic, Prague

    Pavel Pudlák

  2. Department of Mathematics, University of California, San Diego

    Samuel R. Buss

Authors
  1. Pavel Pudlák
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Samuel R. Buss
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Leszek Pacholski Jerzy Tiuryn

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pudlák, P., Buss, S.R. (1995). How to lie without being (easily) convicted and the lengths of proofs in propositional calculus. In: Pacholski, L., Tiuryn, J. (eds) Computer Science Logic. CSL 1994. Lecture Notes in Computer Science, vol 933. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0022253

Download citation

  • DOI: https://doi.org/10.1007/BFb0022253

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60017-6

  • Online ISBN: 978-3-540-49404-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation