# filtration

Let $M=(W,R,V)$ be a Kripke model for a modal logic $L$. Let $\Delta$ be a set of wff’s. Define a binary relation $\sim_{\Delta}$ on $W$:

 $w\sim_{\Delta}u\qquad\mbox{iff}\qquad\models_{w}A\mbox{ iff }\models_{u}A\mbox% { for any }A\in\Delta.$

Then $\sim_{\Delta}$ is an equivalence relation on $W$. Let $W^{\prime}$ be the set of equivalence classes of $\sim_{\Delta}$ on $W$. It is easy to see that if $\Delta$ is finite, so is $W^{\prime}$. Next, let

 $V^{\prime}(p):=\{[w]\in W^{\prime}\mid w\in V(p)\}.$

Then $V^{\prime}$ is a well-defined function. We call a binary relation $R^{\prime}$ on $W^{\prime}$ a filtration of $R$ if

• $wRu$ implies $[w]R^{\prime}[u]$

• $[w]R^{\prime}[u]$ implies that for any wff $A$ with $\square A\in\Delta$, if $\models_{w}\square A$, then $\models_{u}A$.

The triple $M^{\prime}:=(W^{\prime},R^{\prime},V^{\prime})$ is called a filtration of the model $M$.

###### Proposition 1.

(Filtration Lemma) Let $\Delta$ be a set of wff’s closed under the formation of subformulas: any subformula of any formula in $\Delta$ is again in $\Delta$. Then

 $M^{\prime}\models_{[w]}A\qquad\mbox{iff}\qquad M\models_{w}A.$
Title filtration Filtration1 2013-03-22 19:35:39 2013-03-22 19:35:39 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 03B45