may is ambiguous with
can in the sense of
be allowed only in contexts where permission might be implied.
I may go now. I have permission to go now vs. It's possible that I'll go now.
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may not is even more likely to be ambiguous in the same situations. (This one is so ambiguous to my ear that I never use it.)
I may not go now. I don't have permission to go now. vs. It's possible that I won't go now.
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may have and may not have are not ambiguous in this way, as permission isn't given for past actions.
He may have left. It's possible that he left. [No alternative involving permission.]
I may not have understood. It's possible that I did not understand. [No alternative involving permission.]
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might, might not, might have, and might not have are ambiguous only in subordinate clauses, as in reports.
I might go; I might not go; I might have gone; I might not have gone. [All involve possibility, not permission.]
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All of the permission uses of might shown below are less used in modern English, especially the negatives, but you may encounter them on occasion.
He said I might go. He said that I had his permission to go. vs. He reported the possibility that I would go.
He said I might not go.
He said that I did not have his permission to go. (not as likely in
modern English) vs. He reported the possibility that I would not
go.
He said I might have gone.
He said that I would have had his permission to go. vs. He
reported the possibility that I had gone. (most likely
interpretation) vs. He gave his opinion that I should have
gone (as a mild reprimand), although I didn't. This last is nearly equivalent to He said it wouldn't have hurt me to go. (more used than its negative companion below)
He said I might not have gone.
He said that I would not have had his permission to go. (not as likely
in modern English) vs. He reported the possibility that I had not
gone. (most likely interpretation) vs. He gave his opinion
I should not have gone (as a mild reprimand), although I did.
This last is nearly equivalent to He said it wouldn't have hurt me to stay away.
CJ