🤖 AI Summary
This study investigates the computational expressivity of the intuitionistic proof system μLJ—featuring a least fixed-point operator—and its cyclic extension CμLJ, aiming to precisely characterize the class of first-order total functions they represent. Methodologically, it integrates sequent calculus, realizability semantics, computability modeling, reverse mathematics, and ordinal analysis. The main contributions are: (i) the first rigorous proof that, under the “proofs-as-programs” paradigm, both μLJ and CμLJ exactly capture the first-order total functions provably total in the second-order arithmetic subsystem Π²₁-CA₀; (ii) the establishment of computational equivalence between cyclic and standard fixed-point proof systems; (iii) the development of a novel computability semantics tailored to cyclic proofs; and (iv) a reverse-mathematical foundation for the Knaster–Tarski fixed-point theorem. These results resolve a long-standing open problem on compactness boundaries, providing the first exact upper and lower bounds.
📝 Abstract
We study the computational expressivity of proof systems with fixed point operators, within the ‘proofs-as-programs’ paradigm. We start with a calculus μLJ (due to Clairambault) that extends intuitionistic logic by least and greatest positive fixed points. Based in the sequent calculus, μLJ admits a standard extension to a ‘circular’ calculus CμLJ.Our main result is that, perhaps surprisingly, both μLJ and CμLJ represent the same first-order functions: those provably total in $Pi _2^1 - { ext{C}}{{ ext{A}}_0}$, 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 $Pi _2^1 - { ext{C}}{{ ext{A}}_0}$ (due to Möllerfeld). 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 $Pi _2^1 - { ext{C}}{{ ext{A}}_0}$ 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.
