Uniform proof complexity
We define the notion of the uniform reduct of a propositional proof system as the set of those bounded formulas in the language of Peano Arithmetic which have polynomial size proofs under the Paris-Wilkie-translation. With respect to the arithmetic complexity of uniform reducts, we show that uniform reducts are Pi(1)(0)-hard and obviously in Sigma(2)(0). We also show under certain 2 regularity conditions that each uniform reduct is closed under bounded generalisation; that in the case the language includes a symbol for exponentiation, a uniform reduct is closed under modus ponens if and only if it already contains all true bounded formulas; and that each uniform reduct contains all true Pi(1)(b) (alpha)-formulas.
length of proofs propositional calculus translations bounded arithmetic