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I used to joke that since most homework/exam problems are in NP, we should just treat them as PCPs and check a constant number of bits. I stopped joking about it because a more junior TA appeared to take it seriously.
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I think that spot-grading makes a lot of sense, but I've definitely always needed to check a super-constant-in-submission-length number of places (and also not randomly sampled). Apparently I am not able to capture the full power of the PCP theorem. Kudos to those who can!