Toward a Characterization of Simulation Between Arithmetic Theories

Abstract

We study when a sound arithmetic theory $\mathcal S\supseteq S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S+\phi}(n)$ for a true sentence $\phi$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S+\phi$. The paper gives two unconditional constraints on possible characterizations. First, for finitely axiomatized sequential $\mathcal S$, if $EA\vdash Con_{\mathcal S}\rightarrow Con_{\mathcal S+\phi}$, then $\mathcal S$ interprets $\mathcal S+\phi$, implying $\mathcal S\vdash^{n^{O(1)}}Con_{\mathcal S}(p(n))\rightarrow Con_{\mathcal S+\phi}(n)$ for some polynomial $p$, and hence $\mathcal S\vdash^{n^{O(1)}}Con_{\mathcal S+\phi}(n)$. Second, if $\mathcal S$ fails to simulate $\mathcal S+\phi$ for some true $\phi$, then for all sufficiently large $k$ it also fails to simulate $S^1_2+\phi_{BB}(k)$, where $\phi_{BB}(k)$ asserts the exact value of the $k$-state Busy Beaver function. Thus any hard true extension yields a canonical Busy Beaver witness to nonsimulation. $\mathcal B$-certified simulation of a target $\mathcal U$ yields $\mathcal B{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal U}$, giving certification barriers rather than external lower bounds. The paper's central conjectural proposal is: for sound, finitely axiomatized sequential $\mathcal S$, if $EA\not\vdash Con_{\mathcal S}\rightarrow Con_{\mathcal S+\phi}$, then for every constant $c>0$, $\mathcal S\not\vdash^{n^c}Con_{\mathcal S+\phi}(n)$. Under this proposal, hardness follows when $\phi$ is $Con_{\mathcal S}$ or a Kolmogorov-randomness axiom. The latter yields further conjectural consequences and extensions.