A Proposed Characterization of p-Simulation Between Theories

TL;DR

This paper introduces a new characterization of p-simulation between theories, linking interpretability and proof simulation, and explores implications for open problems like proof systems and cryptographic assumptions.

Contribution

It provides a novel framework connecting interpretability and p-simulation, and formulates conjectures that imply major open problems in complexity theory.

Findings

Shows that efficient interpretation implies p-simulation between theories

Proves equivalence between interpretability and p-simulation under certain conditions

Formulates conjectures stronger than 'no optimal proof system' that imply key complexity hypotheses

Abstract

This paper proposes a characterization of when one axiomatic theory, as a proof system for tautologies, $p$-simulates another, by showing: (i)~if c.e. theory $\mathcal{S}$ efficiently interprets $\mathcal{S}{+}\phi$, then $\mathcal{S}$ $p$-simulates $\mathcal{S}{+}\phi$ (Je\v{r}\'abek in Pudl\'ak17 proved simulation), since the interpretation maps an $\mathcal{S}{+}\phi$-proof whose lines are all theorems into an $\mathcal{S}$-proof; (ii)~$\mathcal{S}$ proves ``$\mathcal{S}$ efficiently interprets $\mathcal{S}{+}\phi$'' iff $\mathcal{S}$ proves ``$\mathcal{S}$ $p$-simulates $\mathcal{S}{+}\phi$'' (if so, $\mathcal{S}$ already proves the $\Pi_1$ theorems of $\mathcal{S}{+}\phi$). 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.