Isomorphisms Between Impossible and Hard Tasks

Abstract

If no efficient proof shows that an unprovable arithmetic sentence 'x is Kolmogorov random' ('x∈R') lacks a length t proof, an isomorphism associates for each x impossible and hard tasks: ruling out any proof and length t proofs respectively. This resembles Pudlák's feasible incompleteness. This possible isomorphism implies widely-believed complexity theoretic conjectures hold-in effect, translating theorems from noncomputability about proof speedup and average-case hardness directly to complexity.Formally, we conjecture: sentence "Peano arithmetic (PA) lacks any length t proof of 'x∈R'" lacks t O(1) length proofs in any consistent extension T of PA if and only if T cannot prove 'x∈R'. If so, tautologies encoding the sentence lack t O(1) length proofs in any proof system P for x∈R sufficiently long (relative to the description of a program enumerating theorems of a theory T proving 'P is sound'). R's density implies: TAUT / ∈AvgP, Feige's hypothesis holds, and, a new conjecture, P 's nonoptimality has dense witnesses. If the isomorphism holds for any Π 0 sentence, PH does not collapse, because the arithmetic hierarchy does not collapse. alent since P A 'x∈R' implies T 'x∈R' by the assumption that T extends PA.