We study the computational expressivity of proof systems with fixed point operators, within the `proofs-as-programs' paradigm. We start with a calculus $\mu\mathsf{LJ}$ (due to Clairambault) that extends intuitionistic logic by least and greatest positive fixed points. Based in the sequent calculus, $\mu\mathsf{LJ}$ admits a standard extension to a `circular' calculus $\mathsf{C}\mu\mathsf{LJ}$. Our main result is that, perhaps surprisingly, both $\mu\mathsf{LJ}$ and $\mathsf{C}\mu\mathsf{LJ}$ represent the same first-order functions: those provably total in $\Pi^1_2$-$\mathsf{CA}_0$, a subsystem of second-order arithmetic beyond the `big five' of reverse mathematics and one of the strongest theories for which we have an ordinal analysis (due to Rathjen). This solves various questions in the literature on the computational strength of (circular) proof systems with fixed points. For the lower bound we give a realisability interpretation from an extension of Peano Arithmetic by fixed points that has been shown to be arithmetically equivalent to $\Pi^1_2$-$\mathsf{CA}_0$ (due to M\"ollerfeld). For the upper bound we construct a novel computability model in order to give a totality argument for circular proofs with fixed points. In fact we formalise this argument itself within $\Pi^1_2$-$\mathsf{CA}_0$ in order to obtain the tight bounds we are after. Along the way we develop some novel reverse mathematics for the Knaster-Tarski fixed point theorem.

翻译：我们在“证明即程序”范式下研究带不动点算子的证明系统的计算表达能力。我们从扩展直觉主义逻辑的带最小与最大正不动点的演算$\mu\mathsf{LJ}$（源于Clairambault）出发。基于相继式演算的$\mu\mathsf{LJ}$可标准扩展为“循环”演算$\mathsf{C}\mu\mathsf{LJ}$。我们的主要结果是，或许令人惊讶地，$\mu\mathsf{LJ}$和$\mathsf{C}\mu\mathsf{LJ}$都表示相同的一阶函数：即在$\Pi^1_2$-$\mathsf{CA}_0$中可证全的函数。$\Pi^1_2$-$\mathsf{CA}_0$是超越逆数学“五大系统”的二阶算术子系统，也是我们拥有序数分析（源于Rathjen）的最强理论之一。这解决了文献中关于带不动点的（循环）证明系统计算强度的若干问题。对于下界，我们给出了从Peano算术带不动点扩展的可实现性解释，该扩展已被证明算术等价于$\Pi^1_2$-$\mathsf{CA}_0$（源于Möllerfeld）。对于上界，我们构建了一个新颖的可计算性模型，以便为带不动点的循环证明给出全性论证。事实上，我们将此论证本身形式化于$\Pi^1_2$-$\mathsf{CA}_0$中，从而获得我们追求的紧界。在此过程中，我们针对Knaster-Tarski不动点定理发展了一些新颖的逆数学结果。