  This paper deals with model checking problems with respect to LTL properties
under fairness assumptions. We first present an efficient algorithm to deal
with a fragment of fairness assumptions and then extend the algorithm to handle
arbitrary %fairness assumptions ones. Notably, by making use of some syntactic
transformations, our algorithm avoids to construct corresponding B\"uchi
automata for the whole fairness assumptions, which can be very large in
practice. We implement our algorithm in NuSMV and consider a large selection of
formulas. Our experiments show that in many cases our approach exceeds the
automata-theoretic approach up to several orders of magnitude, in both time and
memory.
