🤖 AI Summary
Circular proofs have remained unexplored in implicit computational complexity (ICC), leaving a gap in foundational proof-theoretic characterizations of complexity classes.
Method: We introduce circular reasoning into the ICC framework by constructing a circular proof system grounded in the Bellantoni–Cook safe-normal function algebra, integrating both inductive and coinductive reasoning, and imposing proof-theoretic constraints tailored to specific complexity classes.
Contribution/Results: We establish the first tight correspondence between circular proofs and classical complexity classes: specifically, we characterize the class P of polynomial-time computable functions and the class ELEMENTARY of elementary functions. Moreover, we propose a recursion-theoretic approach to implicit complexity via circular proofs—offering a novel, proof-theoretically grounded pathway for analyzing computational complexity and opening new directions for applying circular reasoning in complexity theory.
📝 Abstract
Circular (or cyclic) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have been proposed. However, little is known about the complexity theoretic aspects of circular proofs, which exhibit sophisticated loop structures atypical of more common ‘recursion schemes’. This paper attempts to bridge the gap between circular proofs and implicit computational complexity (ICC). Namely we introduce a circular proof system based on Bellantoni and Cook’s famous safe-normal function algebra, and we identify proof theoretical constraints, inspired by ICC, to characterise the polynomial-time and elementary computable functions. Along the way we introduce new recursion theoretic implicit characterisations of these classes that may be of interest in their own right.
