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^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

Uniformity and Nonuniformity in Proof Complexity

This thesis is dedicated to the study of the relations between uniform and nonuniform proof complexity and computational complexity. Nonuniform proof complexity studies the lengths of proofs in various propositional proof systems such as Frege . Uniform proof complexity studies the provability strength of bounded arithmetic theories which use only concepts computable in specific computational complexity classes, e.g. the two-sorted bounded arithmetic theory VNC1 uses only concepts computable in NC1. We are interested in transferring concepts, tools, and results from computational complexity to proof complexity. We introduce the notion of proof complexity class which corresponds to the notion of computational complexity class. We show the possibility of developing a systematic framework for studying proof complexity classes associated with computational complexity classes. The framework is based on soundness statements for proof complexity classes and evaluation problems for circuit complexity classes. The soundness statements are universal for proof complexity classes as circuit evaluation problems are complete for computational complexity classes. We introduce the notion of io-typed theories to design theories corresponding to computational complexity classes which are not closed under composition. We use io-types to control the composition of provably total functions of theories. We design a new class of theories n^ε-ioV∞ (εPh.D