Characterizing p-Simulation Between Theories

Abstract: This paper characterizes 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 $\mathcal{S}{+}\phi$); and (iii)~no $\mathcal{S}$ $p$-simulates all theories. Result (iii) implies $\textbf{P}{\neq}\textbf{NP}{\neq}\textbf{coNP}$, using the nonrelativizing fact no c.e. theory interprets all c.e. theories'' (false for $\mathcal{S}$ with predicate for true sentences). To explore whether this framework resolves other open questions, the paper formulates conjectures stronger thanno optimal proof system exists'' that imply Feige's Hypothesis, the existence of one-way functions, and circuit lower bounds.