  We formulate, in lattice-theoretic terms, two novel algorithms inspired by
Bradley's property directed reachability algorithm. For finding safe invariants
or counterexamples, the first algorithm exploits over-approximations of both
forward and backward transition relations, expressed abstractly by the notion
of adjoints. In the absence of adjoints, one can use the second algorithm,
which exploits lower sets and their principals. As a notable example of
application, we consider quantitative reachability problems for Markov Decision
Processes.
