Modal depth
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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.
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. \operatorname{MD}(p) = 0 , where p is an atomic formula. \operatorname{MD}(\top) = 0 \operatorname{MD}(\bot) = 0 \operatorname{MD}(\neg \varphi) = \operatorname{MD}(\varphi) \operatorname{MD}(\varphi \wedge \psi) = \max(\operatorname{MD}(\varphi), \operatorname{MD}(\psi)) \operatorname{MD}(\varphi \vee \psi) = \max(\operatorname{MD}(\varphi), \operatorname{MD}(\psi)) \operatorname{MD}(\varphi \rightarrow \psi) = \max(\operatorname{MD}(\varphi), \operatorname{MD}(\psi)) \operatorname{MD}(\Box \varphi) = 1 + \operatorname{MD}(\varphi) \operatorname{MD}(\Diamond \varphi) = 1 + \operatorname{MD}(\varphi) = 1 + \operatorname{MD}( \Box p \rightarrow p) = 1 + \max(\operatorname{MD}(\Box p), \operatorname{MD}(p)) = 1 + \max(1 + \operatorname{MD}(p), 0) = 1 + \max(1 + 0, 0) = 1 + 1 = 2
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