  Techniques for the efficient successive under-approximation of the greatest
fixpoint in TCTL formulas can be useful in fast refutation of inevitability
properties and vacuity checking. We first give an integrated algorithmic
framework for both under and over-approximate model-checking. We design the
{\em NZF (Non-Zeno Fairness) predicate}, with a greatest fixpoint formulation,
as a unified framework for the evaluation of formulas like
$\exists\pfrr\eta_1$, $\exists\pfrr\pevt\eta_1$, and $\exists\pevt\pfrr\eta_1$.
We then prove the correctness of a new formulation for the characterization of
the NZF predicate based on zone search and the least fixpoint evaluation. The
new formulation then leads to the design of an evaluation algorithm, with the
capability of successive under-approximation, for $\exists\pfrr\eta_1$,
$\exists\pfrr\pevt\eta_1$, and $\exists\pevt\pfrr\eta_1$. We then present
techniques to efficiently search for the zones and to speed up the
under-approximate evaluation of those three formulas. Our experiments show that
the techniques have significantly enhanced the verification performance against
several benchmarks over exact model-checking.
