LOWER BOUNDS FOR MODAL LOGICS.

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 ψ_i must have a number of proof-lines exponential in terms of the size of ψ_i. 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. [ABSTRACT FROM AUTHOR]