Modal depth

In modal logic, the modal depth of a formula is the deepest nesting of modal operators (commonly $$\Box$$ and $$\Diamond$$). Modal formulas without modal operators have a modal depth of zero.

Definition
Modal depth can be defined as follows. Let $$MD(\phi)$$ be a function that computes the modal depth for a modal formula $$\phi$$:


 * $$MD(p) = 0$$, where $$p$$ is an atomic formula.
 * $$MD(\top) = 0$$
 * $$MD(\bot) = 0$$
 * $$MD(\neg \varphi) = MD(\varphi)$$
 * $$MD(\varphi \wedge \psi) = max(MD(\varphi), MD(\psi))$$
 * $$MD(\varphi \vee \psi) = max(MD(\varphi), MD(\psi))$$
 * $$MD(\varphi \rightarrow \psi) = max(MD(\varphi), MD(\psi))$$
 * $$MD(\Box \varphi) = 1 + MD(\varphi)$$
 * $$MD(\Diamond \varphi) = 1 + MD(\varphi)$$

Example
The following computation gives the modal depth of $$\Box ( \Box p \rightarrow p )$$:


 * $$MD(\Box ( \Box p \rightarrow p )) =$$
 * $$1 + MD( \Box p \rightarrow p) =$$
 * $$1 + max(MD(\Box p), MD(p)) =$$
 * $$1 + max(1 + MD(p), 0) =$$
 * $$1 + max(1 + 0, 0) =$$
 * $$1 + 1 =:2$$

Modal depth and semantics
The modal depth of a formula indicates 'how far' one needs to look in a Kripke model when checking the validity of the formula. For each modal operator, one needs to transition from a world in the model to a world that is accessible through the accessibility relation. The modal depth indicates the longest 'chain' of transitions from a world to the next that is needed to verify the validity of a formula.

For example, to check whether $$M, w \models \Diamond \Diamond \varphi$$, one needs to check whether there exists an accessible world $$v$$ for which $$M, v \models \Diamond \varphi$$. If that is the case, one needs to check whether there is also a world $$u$$ such that $$M, u \models \varphi$$ and $$u$$ is accessible from $$v$$. We have made two steps from the world $$w$$ (from $$w$$ to $$v$$ and from $$v$$ to $$u$$) in the model to determine whether the formula holds; this is, by definition, the modal depth of that formula.

The modal depth is an upper bound (inclusive) on the number of transitions as for boxes, a modal formula is also true whenever a world has no accessible worlds (i.e., $$\Box \varphi$$ holds for all $$\varphi$$ in a world $$w$$ when $$\forall v \in W \ (w, v) \not \in R$$, where $$W$$ is the set of worlds and $$R$$ is the accessibility relation). To check whether $$M, w \models \Box \Box \varphi$$, it may be needed to take two steps in the model but it could be less, depending on the structure of the model. Suppose no worlds are accessible in $$w$$; the formula now trivially holds by the previous observation about the validity of formulas with a box as outer operator.