The Taming of the Cut. Classical Refutations with Analytic Cut

The method of analytic tableaux is a direct descendant of Gentzen's cut-free sequent calculus and is regarded as a paradigm of the notion of analytic deduction in classical logic. However, cut-free systems are anomalous from the proof-theoretical, the semantical and the computational point of view. Firstly, they cannot represent the use of auxiliary lemmas in proofs. Secondly, they cannot express the bivalence of classical logic. Thirdly, they are extremely inefficient, as is emphasized by the ‘computational scandal’ that such systems cannot potynomially simulate the truth-tables. None of these anomalies occurs if the cut rule is allowed. This raises the problem of formulating a proof system which incorporates a cut rule and yet can provide a suitable model of classical analytic deduction. For this purpose we present an alternative refutation system for classical logic, that we call KE. This system, though being ‘close’ to Smullyan's tableau method, is not cut-free but includes a classical cut rule which is not eliminable. Analytic deductions are then obtained by restricting the cut rule to analytic applications, namely applications that do not violate the subformula principle. It turns out that this analytic restriction of the KE system shares all the interesting properties of Smullyan's tableau method and of cut-free systems—it obeys the subformula principle and admits systematic refutation procedures—but uniformly and essentially improves on them from the complexity viewpoint. In particular, we show that the analytic KE system linearly simulates the tableau method, but the tableau method cannot p-simulate the analytic KE system. Finally, we show that every proof system that can simulate the cut rule in a constant number of steps is strictly more powerful than cut-free systems, from the complexity viewpoint, even in the domain of analytic deduction. Hence, the speed-up of the analytic KE system does not depend on the form of the operational rules but only on the analytic cut rule