true What actually happens is that deletions reduce the number of strings in the language If the original language satisfies the property then so does is the smaller language  Theorem formalizes this Theorem 1  Let  ' be an FSA with Boolean algebra   .  Let    be identical to  ', but with additional edges, i.e.,
is defined as
-
, where Then Proof. The language may be enlarged by the addition of new edges that have newly learned transition conditions  On the other hand  because every accepting run remains an accepting
' implies x round to the content of the content to the content of and we are never reducing the
,
size of the language  Therefore
□
The results about the machine learning operator schemas  delete and  add follow as corol laries Corollary 1   delete is an SML with respect to any property   .

' By Then x x Define delete Proof. Assume implies x
,
Theorem Therefore x x
' implies x We conclude that ie delete 口 To be consistent with Theorem in Corollary only but not in the rest of the paper we use Corollary 2  add is not necessarily an SML for any property, including Invariance and Response properties.

Proof. Assume '
Then x x
' implies x By Theorem Then we cannot be certain that          ie  that  add  '
For instance a counterexample for Invariance property □   occurs if we add an accessible edge with transition condition Now we consider a priori results for  spec and  gen  Again  we begin with a relevant theorem for operator schema Theorem 2 Let  ' be an FSA with Boolean algebra   , and let
-
be defined as x    where 3
', such that K 
,
, 
K    1     3 l
. Then K 1 3 Proof. Similar to the proof of Theorem
□
Corollary 3 spec is an SML for any property.

Proof.  Similar to the proof of Corollary    of Theorem 口 Corollary 4  gen is not necessarily an SML for any property, including Invariance and Response properties.