A proof complexity conjecture and the Incompleteness theorem

Abstract

Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function gT that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of gT intersects all infinite NP sets (i.e. whether it is a proof complexity generator hard for all proof systems).A propositional version of the construction shows that at least one of the following three statements is true:1. there is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm),2. E ⊆ P/poly, 3. there exists function h that stretches all inputs by one bit, is computable in sub-exponential time and its range Rng(h) intersects all infinite NP sets.