I have already begun to paint the wall.
By the time my mum arrived, I hadn't begun to cook yet.
They will have begun to play before we arrive.
The present perfect tense is used to talk about something that happened in the past, but the exact time it happened is not important.
The past perfect is a verb tense used to indicate actions that were completed before some point in the past.
The future prerfect tense shows that an action will have been completed at some point in the future.