A proof complexity conjecture and the Incompleteness theorem

Abstract

Abstract Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ 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 $g_T$ intersects all infinite ${\mbox {NP}}$ 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. $E \not \subseteq P/poly$ . 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 ${\text {NP}}$ sets.