The following two statements are really "obviously false", but are still open:
EXPNPsubseteq depth-2-TC0
EXPNPsubseteq depth-2-AC0[6]
Just as a reminder:
EXPNP is exponential time plus an oracle for NP. It contains NEXP (nondeterministic exponential time), EXP, and NP.
By "depth-2-TC0" I mean the class of polynomial-size, depth-two circuit families where each gate is an arbitrary threshold function -- i.e., if it has m Boolean inputs x1,dots,xm, it is defined by reals a1,dots,an,theta and has output 1 iff sumaixigeqtheta.
By depth-2-AC0[6] I mean the class of polynomial-size, depth-two circuit families where each gate is a "standard" MOD6 gate: if it has m Boolean inputs x1,dots,xm, it has output 1 iff sumxineq0 mod 6.
Expanding on the second open problem: Given AsubseteqmathbbZ6, define an A-MOD6 gate to be one which outputs 1 iff sumxiinA mod 6.
The most embarrassing open problem in circuit complexity may be the following: Show that for all possible subsets A, the AND function requires superpolynomial-size depth-2 circuits of A-MOD6 gates.
[PS: Thanks to Arkadev Chattopadhyay for explaining some of these MOD6 problems to me.]
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