Reasoning in circles

Abstract: Circular proofs, introduced by Daniyar Shamkanov, are proofs in which assumptions are allowed that are not axioms but do appear at least twice along a branch. Shamkanov has shown that a formula belongs to the provability logic GL exactly if it has a circular proof in the modal logic K4. Shamkanov uses Tait style proof systems and infinitary proofs. In this paper we prove the same result but then for sequent calculi and without the detour via infinitary systems. We also obtain a mild generalisation of the result, implying that its intuitionistic analogue holds as well.