Cyclic Implicit Complexity

TL;DR

This paper explores the complexity of circular proofs by introducing a system based on Bellantoni and Cook's algebra, linking circular proof structures to polynomial-time and elementary computability classes.

Contribution

It introduces a circular proof system inspired by ICC principles, characterizing complexity classes through proof-theoretic constraints.

Findings

Characterizes polynomial-time functions using circular proofs.

Provides new recursion-theoretic implicit characterizations of complexity classes.

Bridges circular proof systems with implicit computational complexity.

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.