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Cyclic Implicit Complexity
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DOI：10.1145/3793666.png)
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Abstract
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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 article 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.
Keywords:
Cyclic proofs
implicit complexity
function algebras
safe recursion
higher-order complexity
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