Hardness of Ruling Out Short Proofs of Kolmogorov Randomness

TL;DR

This paper introduces the Feasible Chaitin Incompleteness (FCI) assumption, linking the hardness of ruling out short proofs of Kolmogorov randomness to broader complexity and unprovability issues, with wide-ranging implications.

Contribution

It formalizes a meta-complexity assumption connecting unprovability of randomness to proof complexity, offering a unified framework for several open problems in complexity theory and logic.

Findings

Random strings are hard to prove as random for any proof system.

Average-case hardness of proving randomness implies hardness of tautologies.

Certain natural languages are NP-intermediate with P/poly circuits despite not being in P.

Abstract

A meta-complexity assumption, Feasible Chaitin Incompleteness (FCI), asserts the hardness of ruling out length $t$ proofs that string $x$ is Kolmogorov random (e.g. $x{\in}R$), by analogy to Chaitin's result that proving $x{\in}R$ is typically impossible. By assertion, efficiently ruling out short proofs requires, impossibly, ruling out any proof. FCI has strong implications: (i) randomly chosen $x$ typically yields tautologies hard with high probability for any given proof system, densely witnessing its nonoptimality; (ii) average-case impossibility of proving $x{\in}R$ implies average-case hardness of proving tautologies and Feige's hypothesis; and (iii) a natural language is $\textbf{NP}$-intermediate -- the sparse complement of "$x{\in}R$ lacks a length $t$ proof" (where $R$'s complement is sparse) -- and has $\textbf{P/poly}$ circuits despite not being in $\textbf{P}$. FCI and its variants powerfully assert: (i) noncomputability facts translate to hardness conjectures; (ii) numerous open complexity questions have the expected answers (e.g. non-collapse of $\textbf{PH}$), so one overarching conjecture subsumes many questions; and (iii) an implicit mapping between certain unprovable and hard-to-prove sentences is an isomorphism. Further research could relate FCI to other open questions and hardness hypotheses; consider whether $R$ frustrates conditional program logic, implying FCI; and consider whether an extended isomorphism maps any true unprovable sentence to hard-to-prove sentences.