Modal depth

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

Definition[edit]

Modal depth can be defined as follows. Let be a function that computes the modal depth for a modal formula :

, where is an atomic formula.

Example[edit]

The following computation gives the modal depth of :

2

Modal depth and semantics[edit]

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 , one needs to check whether there exists an accessible world for which . If that is the case, one needs to check whether there is also a world such that and is accessible from . We have made two steps from the world (from to and from to ) 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., holds for all in a world when , where is the set of worlds and is the accessibility relation). To check whether , 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.