A Proposed Characterization of p-Simulation Between Theories

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 $\mathcal{S}{+}ϕ$, then $\mathcal{S}$ $p$-simulates $\mathcal{S}{+}ϕ$ (Jeřábek in Pudlák17 proved simulation), since the interpretation maps an $\mathcal{S}{+}ϕ$-proof whose lines are all theorems into an $\mathcal{S}$-proof; (ii)~$\mathcal{S}$ proves ``$\mathcal{S}$ efficiently interprets $\mathcal{S}{+}ϕ$'' iff $\mathcal{S}$ proves ``$\mathcal{S}$ $p$-simulates $\mathcal{S}{+}ϕ$'' (if so, $\mathcal{S}$ already proves the $Π_1$ theorems of $\mathcal{S}{+}ϕ$). 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.