Looking for a new approach to mathematical proofs? This paper introduces an abstract category-theoretical framework for cyclic proof systems, offering a uniform treatment different from conventional derivation trees. Cyclic proof systems use finite graphs for derivations, relying on a soundness condition like the global trace condition (GTC). The work extends Brotherston’s approach using activation algebras for a more natural formalisation of trace conditions. By accounting for trace information composition, novel results are derived, including a Ramsey-style trace condition. Furthermore, the connection between trace and automata theory is explored, proving that verifying the GTC for abstract cyclic proofs with certain trace conditions is PSPACE-complete. This offers new insights into the complexity and structure of cyclic proofs.
This article is published in Mathematical Structures in Computer Science. It fits the scope of the journal as it deals with abstract mathematical structures and their applications to computer science, particularly in the context of proof theory and computational complexity.