On the size of refutation Kripke models for some linear modal and tense logics

LetL be any modal or tense logic with the finite model property. For eachm, definer_L(m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr_L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr_L(m) for some linear modal and tense logicsL.