A Calculus of Circular Proofs and Its Categorical Semantics

We present a calculus of "circular proofs": the graph underlying a proof is not a finite tree but instead it is allowed to contain a certain amount of cycles. The main challenge in developing a theory for the calculus is to define the semantics of proofs, since the usual method by induction on the structure is not available. We solve this problem by associating to each proof a system of equations - defining relations among undetermined arrows of an arbitrary category with finite products and coproducts as well as constructible initial algebras and final coalgebras - and by proving that this system admits always a unique solution.

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