delete+gen and  spec+gen  Note that  gen does not increase global accessibility  ↑ G   but it can increase local accessibility ↑ L   Is ↑ L a net effect due to the generalization step Because atoms are being transferred from one outgoing edge of some vertex 1 to another outgoing edge of 1 with these two operators by definition the local accessibility of those atoms from an initial state will not be increased as a net effect In other words the atoms are accessible from an initial state if and only if  1 is  and these two learning operators do not increase the accessibility of  1  Furthermore  by definition   κ   1   3 prior to learning  so the accessibility of   3 is not increased   We conclude that ↑ L is a net effect A similar line of reasoning explains why operator stay will not increase local accessibility Operator stey cannot increase global accessibility even if it adds an edge because the only edge that this operator could add is    1   1   In conclusion  all three operators that satisfy the accessibility condition have a net effect of       increasing accessibility  ↑ G and ↑ L    On the other hand  because operators  delete+add and  spec+add have  add as their second step they can increase accessibility Results from lower in the hierarchy of Figure are inherited up the tree For example because  delete+ add can increase global accessibility  move can as well  The following is a summary of the relevant results we have so far about how the two step learning operators can change accessibility To avoid overwhelming the reader  we present only those results necessary for understanding this paper

$$\uparrow\mathrm{\boldmath~G~}\uparrow\mathrm{\boldmath~L~}$$
$$s p e c+g e\,n$$
$$\mathbf{u}\;\mathbf{u}$$
Let $e\in\mathbb{T}^{q}\mathbb{E}^{n}$. 
stay   delete+gen   spec+gen  ↑ G ↑ L

$delet+ad\qquad spec+ad\qquad move\quad change\quad\uparrow$ G. 
Before concluding this section  we briefly consider a different partition of the learning operators than that reflected in the taxonomy of Figure     This different partition is neces sary for understanding the a priori proofs about the preservation of Response properties  in For this partition  we wish to distinguish those operators that can introduce at Section least one new string with an infinitely repeating substring eg abcdedede where the ellipsis represents infinite repetition of d followed by e  into the FSA language versus those that cannot Any operator that can add atoms to the transition condition for an edge in a cycle  add an edge to an existing cycle  or add an edge to create a new cycle belongs to the first class the class that can add such substrings Thus this first class includes our operators that can create new cycles  e g    stay because it can add a new edge    1   1 as well as our operators that can generalize the transition condition along some edge of a cycle delete+gen because it can generalize e g k i i The operators are divided between these two classes as follows add  gen  addVgen  add-action  stay  delete+gen  spec+gen  delete+add  spec+add  move change delete   spec   deleteVspec   delete-action It is important to note that of the two step operators are in the first class At this point we have defined a set of useful operators via their operator schemas that one could apply to an FSA plan for adaptation  With these operators  it is possible to improve the effectiveness of a plan  and to adapt it to handle previously unforeseen external