Journal of Symbolic Logic

The complexity of analytic tableaux

Noriko H. Arai, Toniann Pitassi, and Alasdair Urquhart

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

J. Symbolic Logic, Volume 71, Issue 3 (2006), 777-790.

First available in Project Euclid: 4 August 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Arai, Noriko H.; Pitassi, Toniann; Urquhart, Alasdair. The complexity of analytic tableaux. J. Symbolic Logic 71 (2006), no. 3, 777--790. doi:10.2178/jsl/1154698576.

Export citation


  • A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, vol. 11 (1973), no. 4, pp. 429--437.
  • Stephen A. Cook, An exponential example for analytic tableaux, manuscript, 1973.
  • Stephen A. Cook and Robert A. Reckhow, 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.
  • --------, The relative efficiency of propositional proof systems, Journal of Symbolic Logic, vol. 44 (1979), pp. 36--50.
  • Daniel H. Greene and Donald E. Knuth Mathematics for the analysis of algorithms, third ed., Birkhäuser, 1990.,
  • Fabio Massacci, Cook and Reckhow are wrong: Subexponential tableau proofs for their family of formulae, 13th European Conference on Artificial Intelligence (Henri Pradé, editor), Morgan Kaufmann, 1998, pp. 408--409.
  • --------, The proof complexity of analytic and clausal tableaux, Theoretical Computer Science, vol. 243 (2000), pp. 477--487.
  • Neil V. Murray and Erik Rosenthal, On the computational intractability of analytic tableau methods, Bulletin of the IGPL, vol. 2 (1994), no. 2, pp. 205--228.
  • Raymond M. Smullyan First-order logic, Springer-Verlag, New York, 1968, reprinted by Dover, New York, 1995.,
  • Alasdair Urquhart, The complexity of propositional proofs, Bulletin of Symbolic Logic, vol. 1 (1995), pp. 425--467.