Active database systems enhance the functionality of traditional
  databases through the use of active rules or `triggers'.  
One of the principal
 questions for such systems 
 is that of {\em termination}
  -- is it possible for the rules to recursively activate one another indefinite
ly, given
  an initial triggering event.  
In this paper, we study the decidability of the termination problem, 
  our aim being to
  delimit the boundary between the decidable and the undecidable. 
We present
two families of rule languages, the {\em one literal} languages
  where each update is permitted to have just one atom in its body,
and the {\em unary} languages where 
  only unary relations may be updated, but
  higher arity relations may be accessed through views.  
Within each of these, we identify members close to the boundary
  of (un)decidability.
Our context is similar to the {\em while} query language
and the dynamics gives an interesting contrast to
 Datalog with negation;
our results shed insights on the power of triggers
as well as comparison of the termination problem to 
boundedness and query containment.