ECCC-Report TR23-047https://eccc.weizmann.ac.il/report/2023/047Comments and Revisions published for TR23-047en-usSun, 16 Apr 2023 15:20:26 +0300
Paper TR23-047
| Ruling Out Short Proofs of Unprovable Sentences is Hard |
Hunter Monroe
https://eccc.weizmann.ac.il/report/2023/047If no optimal propositional proof system exists, we (and independently Pudlák) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts about noncomputability into complexity theory. For instance, because proving string $x$ is Kolmogorov random ($x{\in}R$) is typically impossible, it is typically hard to prove "no length $t$ proof shows $x{\in}R$", or tautologies encoding this. Therefore, a proof system with one family of hard tautologies has these densely in an enumeration of families. The assumption also implies that a natural language is $\textbf{NP}$-intermediate: with $R$ redefined to have a sparse complement, the complement of the language $\{\langle x,1^t\rangle|$ no length $t$ proof exists of $x{\in}R\}$ is also sparse.
Efficiently ruling out length $t$ proofs of $x{\in}R$ might violate the constraint on using the fact of $x{\in}R$'s unprovability. We conjecture: any computable predicate on $R$ that might be used in if-then statements (or case-based proofs) does no better than branching at random, because $R$ appears random by any effective test. This constraint could also inhibit the usefulness in circuits and propositional proofs of NOT gates and cancellation---needed to encode if-then statements. If $R$ defeats if-then logic, exhaustive search is necessary.
Sun, 16 Apr 2023 15:20:26 +0300https://eccc.weizmann.ac.il/report/2023/047