We show that tree-like (Gentzen’s calculus) PK where all cut formulas have depth at most a constant d does not simulate cut-free PK. Generally, we exhibit a family of sequents that have polynomial size cut-free proofs but requires superpolynomial tree-like proofs even when the cut rule is allowed on a class of cut-formulas that satisfies some plausible hardness assumption. This gives (in some cases, conditional) negative answers to several questions from a recent work of Maciel and Pitassi (LICS 2006). Our technique is inspired by the technique from Maciel and Pitassi. While the sequents used in earlier work are derived from the Pigeonhole principle, here we generalize Statman’s sequents. This gives the desired separation, and at the same time provides stronger results in some cases. 1

In this paper, we develop a quantified propositional proof systems that corresponds to logarithmic-space reasoning. We begin by defining a class ΣCNF(2) of quantified formulas that can be evaluated in log space. Then our new proof system GL ∗ is defined as G ∗ 1 with cuts restricted to ΣCNF(2) formulas and no cut formula that is not quantifier free contains a free variable that does not appear in the final formula. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of V L into a family of tautologies that have polynomial-size GL ∗ proofs. V L is a theory of bounded arithmetic that is known to correspond to logarithmic-space reasoning. To do the translation, we find an appropriate axiomatization of V L, and put V L proofs into a new normal form. To show that GL ∗ is not too strong, we prove the soundness of GL ∗ in such a way that it can be formalized in V L. This is done by giving a logarithmic-space algorithm that witnesses GL ∗ proofs. 1