Cuts for circular proofs

Abstract

One of the authors introduced in [1] a calculus ofcircular proofs for studying the computability arising from thefollowing categorical operations: finite products and coproducts,initial algebras, final coalgebras. The calculus of[1] is cut-free; yet, even if sound and complete forprovability, it lacks an important property for the semantics ofproofs, namely fullness w.r.t. the class of natural categorical modelscalled μ-bicomplete category in [2].We fix, with this work, this problem by adding the cut rule to thecalculus. To this goal, we need to modifying the syntacticalconstraints on the cycles of proofs so to ensure soundness of thecalculus and at same time local termination of cut-elimination. Theenhanced proof system fully represents arrows of the intended model, afree μ-bicomplete category. We also describe a cut-eliminationprocedure as a model of computation arising from the above mentionedcategorical operations. The procedure constructs a cut-freeproof-tree with infinite branches out of a finite circular proof withcuts.[1] Luigi Santocanale. A calculus of circular proofs and its categorical semantics. In Mogens Nielsen and Uffe Engberg, editors, FoSSaCS, volume 2303 of Lecture Notes in Computer Science, pages 357–371. Springer, 2002.[2] Luigi Santocanale. μ-bicomplete categories and parity games. Theoretical Informatics and Applications, 36:195–227, September 2002.