When restricted to proving \Sigma qi formulas, the quantified propositional proof system G*i is closely related to the \Sigma bitheorems of Buss's theory Si2. Namely, G*i has polynomial-size proofs of the translations of theorems of Si2, and Si2proves that G*i is sound. However, little is known about G*i when proving more complex formulas. In this paper, weprove a witnessing theorem for G*1 similar in style to theKPT witnessing theorem for T i2. This witnessing theorem is then used to show that Si2 proves G*1 is sound with respectto prenex \Sigma q i+1 formulas. Note that unless the polynomial hierarchy collapses Si 2 is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that G*1 is p-equivalent to a quantified ver-sion of extended-Frege. This is followed by a proof that Gi p-simulates G*i+1. We finish by proving that S2 can be axiomatized by S12 plus axioms stating that the cut-free versionof G * is sound. All together this shows that the connection between G*i and Si2 does not extend to more complex formulas.

We develop an arithmetical theory VNC¹∗ and its variant VNC¹∗, corresponding to “slightly nonuniform” NC¹. Our theories sit between VNC¹ and VL, and allow evaluation of log-depth bounded fan-in circuits under limited conditions. Propositional translations of ΣB 0 (LVNC 1)-formulas provable in VNC¹∗ admit L-uniform polynomial-size Frege proofs.

We show that the quantified propositional proof systems Gi are polynomially equiva-(or prenex lent to their restricted versions that require all cut formulas to be prenex Σ q i Π q i). Previously this was known only for the treelike systems G ∗ i.

We show that the quantified propositional proof systems Gi are polynomially equiva-(or prenex lent to their restricted versions that require all cut formulas to be prenex Σ q i Π q i). Previously this was known only for the treelike systems G ∗ i. 1

It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider tree-like Sequent Calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cut-free proofs and “lift ” it so it applies to proofs with constant-depth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. We show that these tree-like proofs with constant-depth cuts cannot polynomially simulate similar dag-like proofs, even when the dag-like proofs are cut-free. We present a new proof of the non-finite axiomatizability of the theory of bounded arithmetic I∆0(R). Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy G ∗ i of quantified propositional proof systems does not collapse.

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