Logic for Computer Scientists/Modal Logic/Translation Method

Translation Method
There are several methods aiming at a translation of propositional modal logics into first order predicate logics. The idea is, to transform the semantic conditions for the reachability into logical formulae: One rule for the definition of the semantic was: $$ w \models \Box A   \;\;  \mbox{ iff } \mbox{for all } v \in W, (w,v)\not\in R \mbox{ or } v\models A $$ This can be compiled into a formula by substituting the modal formula $$\square P$$ by the first oder formula $$ \forall y R(x,y) \to P'(y)$$. Hence we can eliminate all modal operators by introducing the first order translations. The result of such a translation is a classical first order formula, which can be processed by the methods we have seen before.

For a modal formula $$F$$ we define its translation $$F^*$$: As a result, we have
 * $$P^* = P(x)$$, if $$P$$ is a propositional constant
 * $$(\lnot F)^* = \lnot (F)$$
 * $$(F \land G)^* = (F^* \land G^*)$$
 * $$(\square F)^* = (\forall y (R(x,y) \to F^*(x/y)))$$, where $$y$$ is a new variable not occurring in $$F^*$$ and $$ F^*(x/y)$$ is the result of replacing  all free occurrences of $$x$$ in $$F^*$$ by $$y$$.

Theorem 1
F is a valid modal formal in $$K$$ iff $$F^*$$ is a valid first order formula.

Together with the observation that validity in modal logic $$K$$ (like in many others) is decidable, we hence have a sublogic of first order classical predicate logic which is decidable! Modal logic can be seen as a fragment of 2-variable first-order logic $$FO^2$$.