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.

We prove that the Cutting Plane proof system based on Gomory-Chvátal cuts polynomially simulates the lift-and-project system with integer coecients written in unary. The restriction on coefficients can be omitted when using Krajícek's cut-free Gentzen-style extension of both systems. We also prove that Tseitin tautologies have short proofs in this extension (of any of these systems and with any coefficients).