  We design a variation of the Karp-Miller algorithm to compute, in a forward
manner, a finite representation of the cover (i.e., the downward closure of the
reachability set) of a vector addition system with one zero-test. This
algorithm yields decision procedures for several problems for these systems,
open until now, such as place-boundedness or LTL model-checking. The proof
techniques to handle the zero-test are based on two new notions of cover: the
refined and the filtered cover. The refined cover is a hybrid between the
reachability set and the classical cover. It inherits properties of the
reachability set: equality of two refined covers is undecidable, even for usual
Vector Addition Systems (with no zero-test), but the refined cover of a Vector
Addition System is a recursive set. The second notion of cover, called the
filtered cover, is the central tool of our algorithms. It inherits properties
of the classical cover, and in particular, one can effectively compute a finite
representation of this set, even for Vector Addition Systems with one
zero-test.
