Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-09T08:33:13.412Z Has data issue: false hasContentIssue false
  • English
  • Français

The complexity of resolution refinements

Published online by Cambridge University Press:  12 March 2014

Joshua Buresh-Oppenheim  and
Toniann Pitassi
Joshua Buresh-Oppenheim
Affiliation:
School of Computing Science, Simon Fraser University, Burnaby, BC, Canada. E-mail: [email protected]
Toniann Pitassi
Affiliation:
Computer Science Department, University of Toronto, Toronto, ON, Canada. E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Resolution is the most widely studied approach to propositional theorem proving. In developing efficient resolution-based algorithms, dozens of variants and refinements of resolution have been studied from both the empirical and analytic sides. The most prominent of these refinements are: DP (ordered), DLL (tree), semantic, negative, linear and regular resolution. In this paper, we characterize and study these six refinements of resolution. We give a nearly complete characterization of the relative complexities of all six refinements. While many of the important separations and simulations were already known, many new ones are presented in this paper; in particular, we give the first separation of semantic resolution from general resolution. As a special case, we obtain the first exponential separation of negative resolution from general resolution. We also attempt to present a unifying framework for studying all of these refinements.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1336 - 1352
Copyright
Copyright © Association for Symbolic Logic 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Alekhnovich, Michael, Johannsen, Jan, Pitassi, Toniann, and Urquhart, Alasdair, An exponential separation between regular and general resolution, Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC-02), ACM Press, New York, 2002, pp. 448–456.Google Scholar
[2]Ben-Sasson, E. and Wigderson, A., Short proofs are narrow – resolution made simple, Proceedings of the thirty-first annual ACM Symposium on Theory of Computing, Atlanta, GA, 1999, pp. 517–526.CrossRefGoogle Scholar
[3]Ben-Sasson, Eli, Size space tradeoffs for resolution, Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC-02), ACM Press, New York, 2002, pp. 457–464.Google Scholar
[4]Bonet, M.L., Esteban, J. L., Galesi, N., and Johannsen, J., On the relative complexity of resolution refinements and cutting planes proof systems, SIAM Journal on Computing, vol. 30 (2000), no. 5, pp. 1462–1484.CrossRefGoogle Scholar
[5]Bonet, M.L. and Galesi, N., A study of proof search algorithms for resolution and polynomial calculus, Proceedings of 40th FOCS, 1999, pp. 422–432.Google Scholar
[6]Buresh-Oppenheim, J., Clegg, M., Impagliazzo, R., and Pitassi, T., Homogenization and the polynomial calculus, 27th International Colloquium on Automata, Languages and Programming, 2000, pp. 926–937.CrossRefGoogle Scholar
[7]Cook, Stephen A., An observation on time-storage trade off, Conference record of fifth annual ACM Symposium on Theory of Computing, Austin, TX, 1973, pp. 29–33.Google Scholar
[8]Davis, M., Logemann, G., and Loveland, D., A machine program for theorem proving, Communications of the ACM, vol. 5 (1962), pp. 394–397.CrossRefGoogle Scholar
[9]Davis, M. and Putnam, H., A computing procedure for quantification theory, Communications of the ACM, vol. 7 (1960), pp. 201–215.Google Scholar
[10]Dunker, Ulf, Zur Effizienz der Beweissuche in der Logikverarbeitung, Ph.D. thesis, Universität Paderborn, 1997.Google Scholar
[11]Goerdt, A., Unrestricted resolution versus n-resolution, Theoretical Computer Science, vol. 93 (1992) , pp. 159–167.CrossRefGoogle Scholar
[12]Goerdt, A., Regular resolution versus unrestricted resolution, SIAM Journal on Computing, vol. 22 (1993) , no. 4, pp. 661–683.CrossRefGoogle Scholar
[13]Goerdt, Andreas, Davis-Putnam resolution versus unrestricted resolution, Annals of Mathematics and Artificial Intelligence, vol. 6 (1992), pp. 169–184.CrossRefGoogle Scholar
[14]Haken, A., The intractability of resolution, Theoretical Computer Science, vol. 39 (1985), pp. 297–305.CrossRefGoogle Scholar
[15]Johannsen, Jan, Exponential incomparability of tree-like and ordered resolution, 2001.Google Scholar
[16]Kowalski, R. and Kuehner, D., Linear resolution with selection function, Artificial Intelligence, vol. 2, pp. 227–260.CrossRefGoogle Scholar
[17]Kozen, Dexter, Lower bounds for natural proof systems, 18th Annual Symposium on Foundations of Computer Science, IEEE, 1977, pp. 254–266.Google Scholar
[18]Krishnamurthy, B., Short proofs for tricky formulas, Acta Informatica, vol. 22 (1985), pp. 253–275.CrossRefGoogle Scholar
[19]Paul, Wolfgang J., Tarjan, Robert E., and Celoni, J. R., Space bounds for a game on graphs, Mathematical Systems Theory, vol. 10 (1977), no. 3, pp. 239–251, Correction, Wolfgang J. Paul, Robert E. Tarjan, and J. R. Celoni, Space bounds for a game on graphs, Mathematical Systems Theory vol. 11 (1977), no. 1, p 85.CrossRefGoogle Scholar
[20]Raz, R. and McKenzie, P., Separation of the monotone NC hierarchy, Proceedings of 37th IEEE Foundations of Computer Science, 1997, pp. 234–243.Google Scholar
[21]Robinson, J. A., A machine oriented logic based on the resolution principle, Journal of the ACM, vol. 12 (1965), no. 1, pp. 23–41.CrossRefGoogle Scholar
[22]Tseitin, G. S., On the complexity of derivation in the propositional calculus, Studies in constructive mathematics and mathematical logic, part II (Slisenko, A. O., editor), 1968, pp. 115–125.Google Scholar
[23]Urquhart, A., The complexity of propositional proofs, The Bulletin of Symbolic Logic, vol. 1 (1995), no. 4, pp. 425–467.CrossRefGoogle Scholar
[24]Vellino, Andre, The complexity of automated reasoning, Ph.D. thesis, University of Toronto, 1989.Google Scholar