# PP-2000-14: All proper normal extensions of S5--square have the polynomial size model property

PP-2000-14: Marx, Maarten and Bezhanishvili, Nick (2000) All proper normal extensions of S5--square have the polynomial size model property. [Report]

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