  Designing algorithms capable of efficiently constructing minimal models of
CNFs is an important task in AI. This paper provides new results along this
research line and presents new algorithms for performing minimal model finding
and checking over positive propositional CNFs and model minimization over
propositional CNFs. An algorithmic schema, called the Generalized Elimination
Algorithm (GEA) is presented, that computes a minimal model of any positive
CNF. The schema generalizes the Elimination Algorithm (EA) [BP97], which
computes a minimal model of positive head-cycle-free (HCF) CNF theories. While
the EA always runs in polynomial time in the size of the input HCF CNF, the
complexity of the GEA depends on the complexity of the specific eliminating
operator invoked therein, which may in general turn out to be exponential.
Therefore, a specific eliminating operator is defined by which the GEA
computes, in polynomial time, a minimal model for a class of CNF that strictly
includes head-elementary-set-free (HEF) CNF theories [GLL06], which form, in
their turn, a strict superset of HCF theories. Furthermore, in order to deal
with the high complexity associated with recognizing HEF theories, an
"incomplete" variant of the GEA (called IGEA) is proposed: the resulting
schema, once instantiated with an appropriate elimination operator, always
constructs a model of the input CNF, which is guaranteed to be minimal if the
input theory is HEF. In the light of the above results, the main contribution
of this work is the enlargement of the tractability frontier for the minimal
model finding and checking and the model minimization problems.
