In modal logic, the modal depth of a formula is the deepest nesting of modal operators (commonly
Modal depth can be defined as follows.
be a function that computes the modal depth for a modal formula
: The following computation gives the modal depth of
: 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.
, one needs to check whether there exists an accessible world
If that is the case, one needs to check whether there is also a world
We have made two steps from the world
) 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.,
is the set of worlds and
is the accessibility relation).
, 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
; the formula now trivially holds by the previous observation about the validity of formulas with a box as outer operator.