, Operator spec could be considered a special case of  change that
, for some consists only of step (1) and the additional two conditions  spec   K   1   2    = (  K   1   2 0 (i.e., spec delete), and
(since otherwise  spec has no effect).

Operator Schema 5 (ogen) Let   be an FSA with Boolean algebra   , and let  gen
'. Then we can define  gen K ' , where gen   K   1  3  
K
K 
for some
. Operator gen could be considered a special case of change that consists only of step (2) and the two additional conditions that ad) and (  K  1  3     ) = 0 (because otherwise   adds redundancy) prior to  gen .

learning operators are defined from these four primitives  Below  change in Next the operator hierarchy of Figure are two subtrees The right subtree consists of one step operators and the left subtree consists of two step operators We define the two one step operators just below   change first   since we just defined the primitive operators below them Operator Schema 6 (Odeletevspec) This operator consists of applying either of the primilive operators  delete or  spec Operator Schema 7 (Oaddvgen) This operator consists of applying either of the primitive operators  add OT  gen It is relevant at this point to introduce two more operators that are not in the hierarchy of Figure They are not in the hierarchy because they are merely minor variants of delete Vspec and add Vgen and they do not belong strictly below our most general operator change These operators are introduced here because they are very useful and also because they are guaranteed to preserve completeness of FSAs  In other words  if the FSA is complete prior to applying these operators then it will be complete after applying them Recall from Section that each FSA state is associated with a set of allowable actions that may be taken from that state These operators delete or add an action from the set of allowable actions from a state Operator Schema 8 (Odelete-action) Delete an allowable action from a state  1 by one or more applications of operator  deletevspec.  Each application may be to a different outgoing edge from  1 Operator Schema 9 (Oadd-action) Add an allowable action from a state  1 by one or more applications of operator  addvgen  Each application may be to a different outgoing edge from  1 To understand why  delete-action consists of one or more applications of  deleteVspec consider the following example In Figure    deleting F collect as an allowable action from F s COLLECTING state results in F deliver being the only allowable action from that state Furthermore this results in the edge COLLECTING COLLECTING being deleted and the edge  COLLECTING  DELIVERING  being specialized  The reasoning is similar for why add-action is one or more applications of addvgen The remaining operators which are all of the operators on the left subtree of change in Figure consist of two steps  the first to remove condition   from edge   1   2   and the