When finished with digging out ancient objects, archaeologists must interpret what they have found.
What kind of reduction is in my example?
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I would say this is a case of elision, the implied full sentence being:
When [they are] finished with digging out ancient objects, archaeologists must interpret what they have found.
the implied subject thus being "archaeologists", via an implied pronoun. I would not call this reduction, because the implied words were not part of a relative clause.