All proper normal extensions of S5--square have the polynomial size model property
Maarten Marx, Nick Bezhanishvili

Abstract:
All proper normal extensions of S5--square have the polynomial size 
model property
Maarten Marx, Nick Bezhanishvili

It is shown that all proper normal extensions of the bi-modal system
$S5^2$ have the poly-size model property. In fact, every normal proper
extension $L$ of $S5^2$ is complete with respect to a class of finite
frames $F_L$.  To each such class 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 $L$-satisfiable formula $\phi$ is
satisfiable on an $L$-frame whose universe is bounded by $P(|\phi|)$,
for $|\phi|$ the number of subformulas of $\phi$.  It is shown that
this bound is optimal.
                
Keyword(s): cylindric algebras, products of modal logic