Uniform Interpolation via Nested Sequents

A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g., nested sequents, hypersequents, and labelled sequents). In this paper, we turn uniform interpolation, which is stronger than interpolation. We develop a constructive method proving via sequents apply it reprove the property $\mathsf{K}$, $\mathsf{D}$, $\mathsf{T}$. then use know-how same hypersequents obtain first direct proof $\mathsf{S5}$ cut-free sequent-like calculus. While our proof-theoretic, definition also uses semantic notions, including bisimulation modulo an atomic proposition.