verb forms - Use of -ing after while

In the following two sentences, there is -ing after while:

1. 
   

   While exploring the paths of program, we established a natural partitioning of paths on-the-fly based on program dependencies - such that only one path in a partition is explored.

   
   


2. 
   

   Any method which covers various possible behaviours of a given program while avoiding path enumeration, can be extremely useful for software testing.

   
   

Can someone help to elaborate why the use of -ing after while is correct (in term of grammar rule)?

Answer

These are examples of reduced adverbial clauses,in which the subject and BE are deleted.

While we were exploring the paths of the program, we established ...
Any method which covers ... while it is avoiding path enumeration ...

This reduction is only permitted under two conditions:

1. 
   

   The verb of the clause must be in a progressive form, or rewritable as a progressive form.

   
   
   

   
   

   ^✲While explored the paths of the program ...
   ^✲... while avoids path enumeration ...

   
   

   
   


2. 
   

   The subject of the clause must be the same as that of the main clause which it modifies, or a pronoun which refers to it.

   
   

   
   

   ^✲While exploring the paths of the program, a natural partitioning was established ...

   
   

   
   
   

   This implies that it was the partitioning which was exploring. The technical term is dangling modifier—‘dangling’ because the clause ‘hangs loose’, not firmly attached to an appropriate subject. Note that the unreduced clause is OK, because the differing subjects are distinguished:

   
   

   
   

   ^OKWhile we were exploring ..., a natural partition was established ...

   
   

   
   

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^✲ marks an utterance as unacceptable