Toward a Characterization of Simulation Between Arithmetic Theories

2026-04-30

AI Summary

This work investigates when feasible arithmetic theories—such as those containing \( S^1_2 \)—can efficiently prove the bounded consistency of their extensions by true sentences, i.e., whether they can polynomially simulate such extensions. By integrating techniques from bounded arithmetic, interpretability theory, and the Busy Beaver function, we identify relative consistency strength as the decisive criterion governing the possibility of simulation. We further demonstrate that Busy Beaver assertions act as intrinsic barriers to simulation: if a base theory cannot polynomially simulate a given extension, then for all sufficiently large \( k \), it also fails to simulate the extension augmented with the assertion \( \mathrm{BB}(k) \). These findings offer a novel perspective on the interplay between proof complexity and simulability among formal 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 $Con_{\mathcal S{+}φ}(n)$ for a true sentence $φ$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S{+}φ$. The paper's two unconditional contributions constrain possible characterizations. First, for finitely axiomatized sequential $\mathcal S$, if $EA{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}φ}$, then $\mathcal S$ interprets $\mathcal S{+}φ$, implying ${\mathcal S}{\vdash^{n^{O(1)}}}Con_{\mathcal S}(p(n)){\rightarrow}Con_{\mathcal S{+}φ}(n)$ for some polynomial $p$, and hence ${\mathcal S}{\vdash^{n^{O(1)}}}Con_{\mathcal S{+}φ}(n)$. Second, if $\mathcal S$ fails to simulate $\mathcal S{+}φ$ for some true $φ$, then for all sufficiently large $k$ it also fails for $φ_{BB}(k)$ asserting the exact value of the $k$-state Busy Beaver function. Informally, any argument showing that $\mathcal S$ fails to simulate some $\mathcal S{+}φ$ also yields unprovable $φ_{BB}(k)$ witnessing the same obstruction. These results suggest that relative consistency strength is a serious candidate for governing when simulation is possible, while leaving open whether it is the correct criterion. The paper's central conjectural proposal is that the above sufficient condition is also necessary: if $EA{\not\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}φ}$, then for every constant $c{>}0$, ${\mathcal S}{\not\vdash^{n^c}}Con_{\mathcal S{+}φ}(n)$. Under this proposal, hardness follows in canonical cases where $φ$ is $Con_{\mathcal S}$ or a Kolmogorov-randomness axiom. The latter yields further conjectural consequences and extensions.

Problem

• bounded consistency
• proof simulation
• arithmetic theories
• relative consistency
• polynomial-time provability

Innovation

• bounded consistency
• proof simulation
• arithmetic theories
• Busy Beaver function
• relative consistency strength