Skip to main content
SpringerLink
Log in

The relative complexity of resolution and cut-free Gentzen systems

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

Resolution and a cut-free Gentzen system are compared with respect to their efficiency as proof systems for biconditionals in the classical propositional calculus. The relative efficiency is shown to depend strongly on the proof format adopted. If proofs are represented as trees, then resolution can simulate the cut-free system efficiently, but an efficient reverse simulation is not possible. If proofs are represented as sequences, then efficient simulations are possible in both directions. The examples used to show this also show that regular cut-free Gentzen systems cannot simulate unrestricted cut-free Gentzen systems efficiently.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S.A. Cook and R.A. Reckhow, The relative efficiency of propositional proof systems, J. Symbolic Logic 44 (1979) 36–50.

    Google Scholar 

  2. Z. Galil, On the complexity of regular resolution and the Davis-Putnam procedure, Theor. Comp. Sci. 4 (1977) 23–46.

    Google Scholar 

  3. J. Krajíček and G. Takeuti, On induction-free provability, Preprint.

  4. F.J. Pelletier, Seventy-five problems for testing automatic theorem provers, J. Auto. Reasoning 2 (1986) 191–216.

    Google Scholar 

  5. N.A. Shanin, G.V. Davydov, S.Yu. Maslov, G.E. Mints, V.P. Orevkov and A.O. Slisenko, An algorithm for a machine search of a natural logical deduction in a propositional calculus, Izdat. Nauka, Moscow (1965), reprinted in [6].

    Google Scholar 

  6. J. Siekmann and G. Wrightson (eds.),Automation of Reasoning (Springer, Berlin, 1983).

    Google Scholar 

  7. R.M. Smullyan,First-order Logic (Springer, New York, 1968).

    Google Scholar 

  8. G.S. Tseitin, On the complexity of derivation in propositional calculus,Studies in Constructive Mathematics and Mathematical Logic, part 2 (1968) pp. 115–125, reprinted in [6].

  9. A. Urquhart, Hard examples for resolution, J. ACM 34 (1987) 209–219.

    Google Scholar 

  10. A. Urquhart, The complexity of Gentzen systems for propositional logic, Theor. Comp. Sci. 66 (1989) 1–11.

    Google Scholar 

  11. H. Wang, Toward mechanical mathematics, IBM J. Res. Develop. 4 (1960) 2–22.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Philosophy and Computer Science, University of Toronto, M5S 1A4, Toronto, Ontario, Canada

    Alasdair Urquhart

Authors
  1. Alasdair Urquhart
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Urquhart, A. The relative complexity of resolution and cut-free Gentzen systems. Ann Math Artif Intell 6, 157–168 (1992). https://doi.org/10.1007/BF01531026

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01531026

Keywords

Search

Navigation