We give an exponential lower bound on number of proof-lines in the proof system K of modal logic, i.e., we give an example of K-tautologies ψ1,ψ2,… s.t. every K-proof of ψi must have a number of proof-lines exponential in terms of the size of ψi. The result extends, for the same sequence of K-tautologies, to the systems K4, Gödel—Löb’s logic, S and S4. We also determine some speed-up relations between different systems of modal logic on formulas of modal-depth one.
"Lower bounds for modal logics." J. Symbolic Logic 72 (3) 941 - 958, September 2007. https://doi.org/10.2178/jsl/1191333849