Efficient detection of vacuity in ACTL formulas
Prepositional logic formulas containing implications can suffer from antecedent failure, in which the formula is true trivially because the pre-condition of the implication is not satisfiable. In other words, the post-condition of the implication does not affect the truth value of the formula. We call this a vacuous pass, and extend the definition of vacuity to cover other kinds of trivial passes in temporal logic. We define w-ACTL, a subset of CTL and show by construction that for every w-ACTL formula ϕ there is a formula w(ϕ), such that: both ϕ and w(ϕ) are true in some model M iff ϕ passes vacuously. A useful side-effect of w(ϕ) is that if false, any counter-example is also a non-trivial witness of the original formula ϕ.
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