Syntactic cut-elimination for common knowledge

We first look at an existing infinitary sequent system for common knowledge for which there is no known syntactic cut-elimination procedure and also no known non-trivial bound on the proof-depth. We then present another infinitary sequent system based on nested sequents that are essentially trees and with inference rules that apply deeply inside these trees. Thus we call this system “deep” while we call the former system “shallow”. In contrast to the shallow system, the deep system allows one to give a straightforward syntactic cut-elimination procedure. Since both systems can be embedded into each other, this also yields a syntactic cut-elimination procedure for the shallow system. For both systems we thus obtain an upper bound of φ20 on the depth of proofs, where φ is the Veblen function.