Cuts for circular proofs

One of the authors introduced in [2] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products and coproducts, initial algebras, final coalgebras. The calculus of [2] is cut-free; yet, even if sound and complete for provability, it lacks an important property for the semantics of proofs, namely fullness w.r.t. the class of natural categorical models called μ-bicomplete category in [3]. We fix, with this work, this problem by adding the cut rule to the calculus. To this goal, we need to modify the syntactical constraints on the cycles of proofs so to ensure soundness of the calculus and at same time local termination of cut-elimination. The enhanced proof system fully represents arrows of the intended model, a free μ-bicomplete category. We also describe a cut-elimination procedure as a model of computation arising from the above mentioned categorical operations. The procedure constructs a cut-free proof-tree with infinite branches out of a finite circular proof with cuts.