# Efficient detection of vacuity in ACTL formulas

- First Online:

DOI: 10.1007/3-540-63166-6_28

- Cite this paper as:
- Beer I., Ben-David S., Eisner C., Rodeh Y. (1997) Efficient detection of vacuity in ACTL formulas. In: Grumberg O. (eds) Computer Aided Verification. CAV 1997. Lecture Notes in Computer Science, vol 1254. Springer, Berlin, Heidelberg

## Abstract

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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