Hardness of Ruling Out Short Proofs of Kolmogorov Randomness
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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∈R), by analogy to Chaitin's result that proving x∈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∈R implies average-case hardness of proving tautologies and Feige's hypothesis; and (iii) a natural language is NP-intermediate – the sparse complement of "x∈R lacks a length t proof" (where R's complement is sparse) – and has P/poly circuits despite not being in 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 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.
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