We present two constructions of models of bounded arithmetic, both in the form of a generalization of the ultrapower construction, that yield nonelementary extensions but do not introduce new lengths. As an application we show, assuming the existence of a one-way permutation g hard against polynomial-size circuits, that strictR 1 2 (g) is weaker than R 1 2 (g). In particular, if such a permutation can be defined by a term in the language of R 1 2 , then strictR 1 2 is weaker than R 1 2 .

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We show that for every integer k ≥ 2, the Res(k) propositional proof system does not have the weak feasible disjunction property. Next, we generalize a recent result of Atserias and Müller [3] to Res(k). We show that if NP is not included in P (resp. QP, SUBEXP) then for every integer k ≥ 1, Res(k) is not automatable in polynomial (resp. quasi-polynomial, subexponential) time.

We consider three relatively strong families of subsystems of AC 0 [2]-Frege proof systems, i.e. propositional proof systems using constantdepth formulas with an additional parity connective, for which exponential lower bounds on proof size are known. In order of increasing strength, the subsystems are: (i) constant-depth proof systems with parity axioms and the (ii) treelike and (iii) daglike versions of systems introduced by Krajíček which we call PK c d (⊕). In a PK c d (⊕)-proof, lines are disjunctions (cedents) in which all disjuncts have depth at most d, parities can only appear as the outermost connectives of disjuncts, and all but c disjuncts contain no parity connective at all. We prove that treelike PK O(1) O(1) (⊕) is quasipolynomially but not polynomially equivalent to constant-depth systems with parity axioms. We also verify that the technique for separating parity axioms from parity connectives due to Impagliazzo and Segerlind can be adapted to give