Ruling Out Short Proofs of Unprovable Sentences is Hard

If no optimal propositional proof system exists, we (and independently Pudl'ak) 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.

💡 Research Summary

The paper investigates the computational difficulty of ruling out short proofs for sentences that are unprovable in a given formal system, under the assumption that no optimal propositional proof system exists. An optimal proof system would be one that can simulate any other proof system with only a polynomial overhead; its non‑existence is a widely considered plausible hypothesis in proof complexity. The authors (independently of Pudlák) show that this hypothesis implies a robust hardness result: for any sentence φ that is not provable, the statement “there is no proof of φ of length ≤ t” is itself hard to prove in any reasonable proof system.

The core construction is a uniform, computable reduction that maps an unprovable sentence φ to a family of sentences ψ_t that assert the non‑existence of a proof of φ of length t. Because φ is unprovable, ψ_t is true for every t, yet any proof system that could efficiently certify ψ_t would effectively give a short proof of the unprovable φ, contradicting the assumption. Hence ψ_t must require super‑polynomial (indeed, super‑linear) proof size in every proof system. This reduction translates non‑computability (the fact that φ cannot be decided) into proof‑complexity hardness.

To make the result concrete, the authors focus on Kolmogorov‑random strings. Let R be the set of strings x whose Kolmogorov complexity K(x) exceeds |x|+c for some constant c. It is well‑known that membership in R is not provable in any reasonable arithmetic theory: proving “x∈R” would amount to proving a lower bound on K(x), which is impossible because K is uncomputable. Consequently, for a random string x, the sentence “x∈R” is unprovable, and the derived family
 ψ_{x,t} := “there is no proof of length ≤ t that x∈R”
is hard for every t. The authors argue that the corresponding propositional encodings are tautologies that are dense in any enumeration of hard tautologies: any proof system that has at least one family of hard tautologies automatically has infinitely many such families derived from different random strings.

A striking corollary concerns the existence of natural NP‑intermediate languages. By redefining R so that its complement is sparse (e.g., by padding or by intersecting with a sparse set), the language
 L = {⟨x,1^t⟩ | there is no proof of length ≤ t that x∈R}
lies in NP (a nondeterministic machine can guess a proof of length ≤ t and verify it) but its complement is also sparse, making L neither NP‑complete nor in P under standard complexity assumptions. This provides a “natural” example of an NP‑intermediate language, unlike Ladner’s artificial constructions.

The paper also explores the implications for circuit complexity and the use of conditional reasoning (“if‑then” statements) in proofs. Since R appears random to any effective test, any predicate that tries to exploit a structural property of R behaves no better than random branching. In propositional terms, this means that NOT gates (which encode negation needed for case analysis) cannot be efficiently leveraged when reasoning about R; the proof must essentially enumerate possibilities, leading to exhaustive‑search‑type lower bounds. The authors conjecture that any computable predicate on R used in case‑based proofs cannot improve over random guessing, because R defeats any effective statistical test. This suggests a deeper limitation: the inability to efficiently cancel or simplify expressions involving R may be a source of hardness for both proof systems and Boolean circuits.

Overall, the paper establishes a novel bridge between non‑computability and proof‑complexity hardness. By showing that ruling out short proofs of unprovable statements is itself hard, it yields (1) a uniform method to generate dense families of hard tautologies, (2) a natural NP‑intermediate language based on Kolmogorov randomness, and (3) insights into why conditional logic and negation may be ineffective against random‑looking sets. These contributions enrich our understanding of why certain propositional formulas resist short proofs and open new avenues for studying the interplay between algorithmic randomness, proof complexity, and circuit lower bounds.