I came across this paper, which mentions an interesting variation on SAT:
We call a CNF formula F insensitive if every total assignment α satisfies the same number of clauses of F.
I hadn't come across this before. You can easily define the same property for DNF formulae as well.
Which brings me to the following question:
Given either a DNF or CNF formula, how hard is it to tell if the formula is insensitive?
It is clear that the decision problem "is this formula insensitive?" for both DNF and CNF is in co-NP, because two assignments satisfying different numbers of clauses serve as a "no" certificate. Its complement, which is "is this formula sensitive?" is in NP. But are these problems complete in their respective classes?
I suspect the answer is "yes," because most variants of SAT turn out to be NP-complete. But I don't know how to prove this, and I cannot find any information on "insensitive" formulas outside of this paper. Does any info on this exist?