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The complexity of analytic tableaux

Published online by Cambridge University Press:  12 March 2014

Noriko H. Arai ,
Toniann Pitassi  and
Alasdair Urquhart
Noriko H. Arai
Affiliation:
Mathematical Informatics Research, Foundations of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi Chiyoda-KU, Tokyo 101-8430, Japan.E-mail:[email protected]
Toniann Pitassi
Affiliation:
Department of Computer Science, University of Toronto, Toronto, Ontario M5S 3G4, Canada.E-mail:[email protected]
Alasdair Urquhart
Affiliation:
Departments of Philosophy and Computer Science, University of TorontoToronto, Ontario M5S 1A1, Canada.E-mail:[email protected]
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Abstract

The method of analytic tableaux is employed in many introductory texts and has also been used quite extensively as a basis for automated theorem proving. In this paper, we discuss the complexity of the system as a method for refuting contradictory sets of clauses, and resolve several open questions. We discuss the three forms of analytic tableaux: clausal tableaux, generalized clausal tableaux, and binary tableaux. We resolve the relative complexity of these three forms of tableaux proofs and also resolve the relative complexity of analytic tableaux versus resolution. We show that there is a quasi-polynomial simulation of tree resolution by analytic tableaux; this simulation is close to optimal, since we give a matching lower bound that is tight to within a polynomial.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 777 - 790
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Aho, A. V. and Sloane, N. J. A., Some doubly exponential sequences, Fibonacci Quarterly, vol. 11 (1973), no. 4, pp. 429–437.Google Scholar
[2]Cook, Stephen A., An exponential example for analytic tableaux, manuscript, 1973.Google Scholar
[3]Cook, Stephen A. and Reckhow, Robert A., On the lengths of proofs in the propositional calculus, Proceedings of the Sixth Annual ACM Symposium on Theory of Computing, 1974, See also corrections for above in SIGACT News, vol. 6 (1974), pp. 15–22.Google Scholar
[4]Cook, Stephen A. and Reckhow, Robert A., The relative efficiency of propositional proof systems, this Journal, vol. 44 (1979), pp. 36–50.Google Scholar
[5]Greene, Daniel H. and Knuth, Donald E., Mathematics for the analysis of algorithms, third ed., Birkhäuser, 1990.CrossRefGoogle Scholar
[6]Massacci, Fabio, Cook and Reckhow are wrong: Subexponential tableau proofs for their family of formulae, 13th European Conference on Artificial Intelligence (Pradé, Henri, editor), Morgan Kaufmann, 1998, pp. 408–409.Google Scholar
[7]Massacci, Fabio, The proof complexity of analytic and clausal tableaux, Theoretical Computer Science, vol. 243 (2000), pp. 477–487.CrossRefGoogle Scholar
[8]Murray, Neil V. and Rosenthal, Erik, On the computational intractability of analytic tableau methods, Bulletin of the IGPL, vol. 2 (1994), no. 2, pp. 205–228.CrossRefGoogle Scholar
[9]Smullyan, Raymond M., First-order logic, Springer-Verlag, New York, 1968, reprinted by Dover, New York, 1995.CrossRefGoogle Scholar
[10]Urquhart, Alasdair, The complexity of propositional proofs, The Bulletin of Symbolic Logic, vol. 1 (1995), pp. 425–467.CrossRefGoogle Scholar