Proof.  Similar to the proof of Corollary    of Theorem 

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We can draw the following conclusions from the theorems and corollaries just presented Of the one step learning operators those that are guaranteed to be SMLs for any property are  delete   spec and  deletevspec  which implies that  delete-action is also an SML for any property We need never be concerned with the first step in a two step operator   It is guaranteed to be an SML  because  delete Or   spec is always the first step Next consider theorems that are needed to address the two step operators   Although we found results for the one step operators that were general enough to address property we were unable to do likewise for the two step operators  Our results for the two step op erators determine whether these operators are necessarily SMLs for Invariance or Response properties in particular Future work will consider other property classes The theorems are quite intuitive The first theorem distinguishes those learning operators that will satisfy Invariance properties from those that will not Theorem 3 A machine learning operator is guaranteed to be an SML with respect to any Invariance property - if and only if t G and t L are both true (which, for our two-step operators, implies that the operator satisfies the accessibility condition).

t Proof. Suppose ↑ G and ↑ L are both true  Let Invariance property Assume
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is the context of it must be the case that is true in every c state of y  If accessibility of atoms is not increased  i e  ↑ G and ↑
L then it must be the case that every c state of every string x
' where is also a c state of some string in         Therefore  for every string x
' it must be the case that is true in every c state of x In other words moving transition conditions around in an FSA without increasing accessibility will not alter the truth of an Invariance property which holds in every c state of every string in the language of the FSA
Suppose ↑ G or ↑ L   Increasing accessibility of atoms implies the possibility of introducing a c state in some string x where '
that was not in any string of This can cause violation of an Invariance property as in the counterexample in the proof of Corollary     Knowing that      is true in every c state of every string of provides no guarantee that is true in every c state of every string of
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Since we already have results to cover the one step operators  we need only consider the two step operators Corollary 5  The machine learning operator schemas  delete+gen,  spec+gen, and   stay are guaranteed to be SMLs with respect to any Invariance property    because for all of these operators ↑ G and ↑ L.

Corollary 6 The machine learning operator schemas  delete+add,  spec+add,  move, and change are not necessarily SMLs with respect to any Invariance property because for all of these operators ↑ G.