"While" can be omitted in a time clause when the adverb clause is reduced to an adverb phrase. "While" in this case means "during the same time that" + subject + verb: The complete "while" clause would contain the same subject as the main clause, which is omitted in the reduced clause, the adverb phrase,
First, let's show some examples of "while" as an adverb phrase, reduced from an adverb clause of time:
1) While walking down the street, I found a hundred-dollar-bill on the sidewalk.
2) While talking on the phone, Bettina was also attending to her email.
3) While watching a late movie on TV, John fell asleep.
In all of the above sentences, "while" means "during the same time." "While" can be omitted in these sentences:
1a) Walking down the street, I found a hundred-dollar bill on the sidewalk.
2a) Talking on the phone, Bettina was also attending to her email.
3a) Watching a late movie on TV, John fell asleep. _______
In sentences 1a) and 2a) above, the "while" phrase can not come at the end of the sentence. If it did, it would be a misplaced modifier, and the sentence would be totally incorrect:
I found a hundred-dollar bill on the sidewalk walking down the street.
Bettina was also attending to her email talking on the phone.
However, since there is no other possible meaning attached to an intransitive verb with no object to modify, the third sentence could re-place the adverb phrase at the end of the sentence:
John fell asleep watching a late movie on TV. _______
"While" can not be reduced to an adverb phrase in clauses of contrast:
While John likes fish and chicken, he doesn't like red meat.
While liking fish and chicken, he doesn't like red meat.
Liking fish and chicken, he doesn't like red meat. (This sentence is illogical; it gives the opposite meaning. )
While Henry works in the city, he lives in the country.
While working in the city, he lives in the country.
Working in the city, he lives in the country _______
So, we could say that "while" can not be omitted from a reduced adverb clause if 1) the reduced clause comes at the end of the sentence after an object; 2) the reduced clause is meant to express contrast.
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