Could you please resolve a confusion with Schaefer's theorem for me? Namely, why does it not imply many problems in P are NP-complete? For example, primality testing surely cannot be reduced to one of the six classes in the theorem, so why does that not imply it's NP-complete?
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Primality testing is not of the form $\text{SAT}(S)$ for a finite set $S$ of finite-arity relations over the Boolean domain. Thus, it's not covered by Shaefer's theorem, and Shaefer's theorem says nothing about the hardness of primality testing. See this answer: https://cs.stackexchange.com/a/43831/755.
See also Why does Schaefer's theorem not prove that P=NP? for a more detailed explanation.
