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In [[modal logic]], the '''modal depth''' of a formula is the deepest nesting of [[modal operator]]s (commonly <math>\Box</math> and <math>\Diamond</math>). Modal formulas without modal operators have a modal depth of zero.


== Definition ==


Modal depth can be defined as follows. Let <math>MD(\phi)</math> be a [[Function (mathematics)|function]] that computes the modal depth for a modal formula <math>\phi</math>:
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:<math>MD(p) = 0</math>, where <math>p</math> is an [[atomic formula]].
:<math>MD(\top) = 0</math>
:<math>MD(\bot) = 0</math>
:<math>MD(\neg \varphi) = MD(\varphi)</math>
:<math>MD(\varphi \wedge \psi) = max(MD(\varphi), MD(\psi))</math>
:<math>MD(\varphi \vee \psi) = max(MD(\varphi), MD(\psi))</math>
:<math>MD(\varphi \rightarrow \psi) = max(MD(\varphi), MD(\psi))</math>
:<math>MD(\Box \varphi) = 1 + MD(\varphi)</math>
:<math>MD(\Diamond \varphi) = 1 + MD(\varphi)</math>
 
== Example ==
 
The following computation gives the modal depth of <math>\Box ( \Box p \rightarrow p )</math>:
 
:<math>MD(\Box ( \Box p \rightarrow p )) =</math>
:<math>1 + MD( \Box p \rightarrow p) =</math>
:<math>1 + max(MD(\Box p), MD(p)) =</math>
:<math>1 + max(1 + MD(p), 0) =</math>
:<math>1 + max(1 + 0, 0) =</math>
:<math>1 + 1 =</math>
: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 <math>M, w \models \Diamond \Diamond \varphi</math>, one needs to check whether there exists an accessible world <math>v</math> for which <math>M, v \models \Diamond \varphi</math>. If that is the case, one needs to check whether there is also a world <math>u</math> such that <math>M, u \models \varphi</math> and <math>u</math> is accessible from <math>v</math>. We have made two steps from the world <math>w</math> (from <math>w</math> to <math>v</math> and from <math>v</math> to <math>u</math>) 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., <math>\Box \varphi</math> holds for all <math>\varphi</math> in a world <math>w</math> when <math>\forall v \in W \ (w, v) \not \in R</math>, where <math>W</math> is the set of worlds and <math>R</math> is the accessibility relation). To check whether <math>M, w \models \Box \Box \varphi</math>, 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 <math>w</math>; the formula now trivially holds by the previous observation about the validity of formulas with a box as outer operator.
 
[[Category:Modal logic]]

Latest revision as of 20:29, 30 November 2014


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