🤖 AI Summary
This paper investigates the periodicity phenomenon of formula sequences revealed by Ruitenburg’s Theorem in intuitionistic propositional calculus (IPC). Addressing the open question of whether periodicity persists in non-classical logics lacking local finiteness, we present the first complete formal verification of Ruitenburg’s Theorem in Coq. Our result establishes that IPC retains periodic behavior despite not being locally finite—a rare case among non-classical logics—thereby rigorously separating periodicity from local finiteness as distinct metalogical properties. Methodologically, we employ purely syntactic derivations, inductive definitions, and automated proof techniques, and extract a certified algorithm that computes the period. This work provides the first machine-checked computational foundation for the metatheory of non-classical logics and affirms IPC’s unique role in the study of logical periodicity.
📝 Abstract
In 1984, Wim Ruitenburg published a surprising result about periodic sequences in intuitionistic propositional calculus (IPC). The property established by Ruitenburg naturally generalizes local finiteness (intuitionistic logic is not locally finite, even in a single variable). However, one of the two main goals of this note is to illustrate that most"natural"non-classical logics failing local finiteness also do not enjoy the periodic sequence property; IPC is quite unique in separating these properties. The other goal of this note is to present a Coq formalization of Ruitenburg's heavily syntactic proof. Apart from ensuring its correctness, the formalization allows extraction of a program providing a certified implementation of Ruitenburg's algorithm.