We present in a uniform manner simple two-sorted theories corresponding to each of eight complexity classes between AC and P. We present simple translations between these theories and systems of the quanti ed propositional calculus.

Abstract We present a general method for introducing finitely axiomatizable "minimal " second-order theories for various subclasses of P. We show that our theory VTC 0 for the complexity class TC 0 is RSUV isomorphic to the first-order theory \Delta b

Let H be a proof system for the quantified propositional calculus (QPC). We j -witnessing problem for H to be: given a prenex S j -formula A, an H-proof of A, and a truth assignment to the free variables in A, find a witness for the outermost existential quantifiers in A. We point out that the S witnessing problems for the systems G 1 and G 1 are complete for polynomial time and PLS (polynomial local search), respectively. We introduce

We discuss the Paris-Wilkie translation from bounded arithmeticproofs to bounded depth propositional proofs in both relativized and non-relativized forms. We describe normal forms for proofs in boundedarithmetic, and a definition of \Sigma 0-depth for PK-proofs that makes the translation from bounded arithmetic to propositional logic particularlytransparent. Using this, we give new proofs of the witnessing theorems for S12and T 12; namely, new proofs that the \Sigma b1-definable functions of S12are polynomial time computable and that the \Sigma b1-definable functions of T 12 are in Polynomial Local Search (PLS). Both proofs generalize to \Sigma

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.