  This study carries forward the line of enquiry that seeks to characterize
precisely which security policies are enforceable by runtime monitors. In this
regard, Basin et al.\ recently refined the structure that helps distinguish
between those actions that the monitor can potentially suppress or insert in
the execution, from those that the monitor can only observe. In this paper, we
generalize this model by organizing the universe of possible actions in a
lattice that naturally corresponds to the levels of monitor control. We then
delineate the set of properties that are enforceable under this paradigm and
relate our results to previous work in the field. Finally, we explore the set
of security policies that are enforceable if the monitor is given greater
latitude to alter the execution of its target, which allows us to reflect on
the capabilities of different types of monitors.
