After computer breaking down we could not restore the lost data.
Tell me please, is "breaking down" a verbal noun or is it like a complex gerund here? Is this sentence correct at all?
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Your sentence has two problems:
The singular common count noun "computer" requires a determiner.
"After" is a preposition whose object is "computer". That implies that the sentence's main clause ("we could not restore the lost data") occurs after the computer. That does not make sense.
Here are two options to fix this sentence:
This is correct, but the possessive noun with the gerund sounds a bit awkward.
Now "after" is a subordinating conjunction introducing an adverbial clause. This is probably the best way to write the sentence.