Journal: Journal of Logic and Computation 2005, 15: 433-446
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^0_1-hard and obviously in \Sigma^0_2. We also show under certain 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^b_1(\alpha)-formulas.
Steve Cook made a comment
on Problem 2.
He showed that the existence of a proof system whose uniform reduct equals
the set of all true bounded formulas is equivalent to the existence of
an optimal proof system.
Cook's Comments (ps-file), in arXiv