Abstract

We show that every proper normal extension of the bi-modal system S52 has the poly-size model property. In fact, to every proper normal extension L of S52 corresponds a natural number b(L) - the bound of L. For every L, there exists a polynomial P(·) of degree b(L) + 1 such that every L-consistent formula ϕ is satisfiable on an L-frame whose universe is bounded by P(|ϕ|), where |ϕ| denotes the number of subformulas of ϕ. It is shown that this bound is optimal.