## The complexity of analytic tableaux.(English)Zbl 1109.03066

The paper settles the relative proof complexity of three variants of analytic tableaux, and tree resolution. Tree resolution was held for a long time to have exponential speed-up over analytic tableaux by a result of S. Cook and R. Reckhow [“On the lengths of proofs in the propositional calculus”, Proc. 6th ann. ACM Symp. Theory Comput., Seattle 1974, 135–148 (1974; Zbl 0375.02004)], until F. Massacci [Theor. Comput. Sci. 243, No. 1–2, 477–487 (2000; Zbl 0947.68130)] pointed out a gap in the proof, and constructed a quasi-polynomial tableaux refutation of their formulas.
In the present paper, the authors investigate the complexity of three variants of the tableau proof system: clausal tableaux, binary tableaux, and generalized clausal tableaux. It is shown that generalized clausal tableaux have an exponential speed-up over binary tableaux, and quasi-polynomially simulate tree resolution. The latter bound is tight, as the paper also provides a quasi-polynomial separation of tree resolution and generalized clausal tableaux. Together with the results of Cook and Reckhow, and Massacci (which imply an exponential speed-up of binary tableaux over clausal tableaux), this fully resolves the relative complexity of all the systems considered in the paper.

### MSC:

 03F20 Complexity of proofs

### Keywords:

proof complexity; analytic tableaux; tree resolution

### Citations:

Zbl 0375.02004; Zbl 0947.68130
Full Text:

### References:

 [1] Fibonacci Quarterly 11 pp 429– (1973) [2] First-order logic (1968) · Zbl 0172.28901 [3] DOI: 10.2178/bsl/1203350879 · Zbl 1133.03037 [4] DOI: 10.1093/jigpal/2.2.205 · Zbl 0810.03008 [5] An exponential example for analytic tableaux (1973) [6] 13th European Conference on Artificial Intelligence pp 408– (1998) [7] Mathematics for the analysis of algorithms (1990) [8] The relative efficiency of propositional proof systems 44 pp 36– (1979) · Zbl 0408.03044 [9] Proceedings of the Sixth Annual ACM Symposium on Theory of Computing 6 pp 15– (1974) [10] DOI: 10.1016/S0304-3975(00)00148-1 · Zbl 0947.68130
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