Vacuity Checking in the Modal Mu-Calculus

Vacuity arises when a logical formula is trivially true in a given model due, for example, to antecedent failure. Beer et al. have recently introduced a logic-independent notion of vacuity and shown that certain logics, i.e., those with polarity, admit an eÆcient decision procedure for vacuity detection. We show that the modal mu-calculus, a very expressive temporal logic, is a logic with polarity and hence the results of Beer et al. are applicable. We also extend the de nition of vacuity to achieve a new notion of redundancy in logical formulas. Redundancy captures several forms of antecedent failure that escape traditional vacuity analysis, including vacuous actions in temporal modalities and unnecessarily strong temporal operators. Furthermore, we have implemented an eÆcient redundancy checker for the modal mu-calculus in the context of the XMC model checker. Our checker generates diagnostic information in the form of all maximal subformulas that are redundant and exploits the fact that XMC can cache intermediate results in memo tables between model-checking runs. We have applied our redundancy checker to a number of previously published case studies, and found instances of redundancy that have gone unnoticed till now. These ndings provide compelling evidence of the importance of redundancy detection in the design process.