THE QUANTUM INTELLIGENCER

Efficient Interpretability and p-Simulation: A Logical Framework for Proof Complexity and Cryptographic Hardness In Plain English: This paper tackles a deep question in computer science: how can one mathematical system quickly check or reproduce proofs from another system? The authors suggest a rule: if one system can easily understand and represent a slightly more powerful version of itself, then it can also quickly replicate its proofs. They show this idea connects to bigger questions, like whether truly secure encryption is possible or whether certain hard math problems can never be solved quickly. If their rule holds in broader cases, it could help prove some of the most important unsolved problems in computing. Summary: The paper proposes a characterization of p-simulation between axiomatic theories in the context of propositional proof complexity. It argues that if a computably enumerable (c.e.) theory $\mathcal{S}$ can efficiently interpret its extension $\mathcal{S}+\phi$, then $\mathcal{S}$ p-simulates $\mathcal{S}+\phi$ as a proof system for tautologies. This strengthens a result by Jeřábek (in Pudlák, 2017), who showed simulation under interpretability, by establishing the stronger p-simulation under the condition of efficient interpretability. Furthermore, the paper proves that $\mathcal{S}$ can prove the statement “$\mathcal{S}$ efficiently interprets $\mathcal{S}+\phi$” if and only if it can prove “$\mathcal{S}$ p-simulates $\mathcal{S}+\phi$”, and that in such cases, $\mathcal{S}$ already proves all $\Pi_1$ theorems of $\mathcal{S}+\phi$. This equivalence provides a logical criterion for when extending a theory does not yield proof-theoretic advantages in terms of efficiency. The paper then explores the potential of this framework to resolve major open problems in complexity theory. It formulates conjectures that go beyond the nonexistence of optimal proof systems, suggesting that such principles could imply Feige’s Hypothesis (regarding the hardness of refuting random 3SAT instances), the existence of one-way functions, and strong circuit lower bounds—cornerstones of modern cryptography and complexity. Thus, the work aims to unify logical metatheoretic methods with foundational questions in computational hardness. Key Points: - The paper introduces a proposed characterization linking efficient interpretability between theories to p-simulation in proof complexity. - If a c.e. theory $\mathcal{S}$ efficiently interprets $\mathcal{S}+\phi$, then $\mathcal{S}$ p-simulates $\mathcal{S}+\phi$. - This improves upon Jeřábek’s earlier result (in Pudlák, 2017), which established simulation but not necessarily p-simulation. - $\mathcal{S}$ proves efficient interpretability of $\mathcal{S}+\phi$ iff it proves p-simulation of $\mathcal{S}+\phi$. - When this holds, $\mathcal{S}$ already proves all $\Pi_1$ theorems of $\mathcal{S}+\phi$. - The framework is extended to conjectures that, if validated, would imply Feige’s Hypothesis, one-way functions, and circuit lower bounds. - The work suggests logical methods may help resolve central open problems in computational complexity. Notable Quotes: - "This paper proposes a characterization of when one axiomatic theory, as a proof system for tautologies, $p$-simulates another..." - "...$\mathcal{S}$ proves ‘$\mathcal{S}$ efficiently interprets $\mathcal{S}{+}\phi$’ iff $\mathcal{S}$ proves ‘$\mathcal{S}$ $p$-simulates $\mathcal{S}{+}\phi$’..." - "To explore whether this framework conceivably resolves other open questions, the paper formulates conjectures stronger than ‘no optimal proof system exists’ that imply Feige's Hypothesis, the existence of one-way functions, and circuit lower bounds." Data Points: - Reference to prior work: Jeřábek in Pudlák (2017) established simulation under interpretability. - The paper focuses on c.e. (computably enumerable) theories and extensions by a single sentence $\phi$. - The key logical level involved is $\Pi_1$ theorems—statements of the form “for all x, P(x)” where P is computable. - No specific numerical data or experimental results are presented, as the work is theoretical. Controversial Claims: - The paper suggests that its proposed logical framework could imply the existence of one-way functions—a foundational assumption in cryptography—which is a highly non-trivial and unproven claim in complexity theory. - It formulates conjectures stronger than the nonexistence of optimal proof systems, which itself is a major open problem, and claims these could imply circuit lower bounds, a result that would represent a breakthrough in computational complexity. - The equivalence between proving efficient interpretability and p-simulation within the base theory $\mathcal{S}$ may depend on subtle assumptions about the representability of proof transformations and the strength of $\mathcal{S}$, which could be challenged in weaker systems. Technical Terms: - p-simulation, axiomatic theory, proof system, tautologies, computably enumerable (c.e.) theory, efficient interpretability, $\Pi_1$ theorems, propositional proof complexity, optimal proof system, Feige’s Hypothesis, one-way functions, circuit lower bounds, simulation, interpretability, extension by $\phi$, logical metatheory, computational complexity —Ada H. Pemberley Dispatch from The Prepared E0