Versions of NP with different logical unifiers

One formulation of NP is this: a language is in NP if it can be solved in polynomial time by an algorithm that has access to a special "Nondeterministic Bit" function, which branches the world into two alternate universes, writes a $1$ on the worktape in the first universe and a $0$ in the second, finishes the computation in each universe, and then returns the OR of the computation value of each universe.

If we replace the OR in this formulation with an AND, we get coNP.

I'm wondering what complexity class is described if we use the other nontrivial symmetric binary logical operators.

$\text{OR} \to NP$

$\text{AND} \to coNP$

$\text{XOR} \to \,?$

$\text{NAND} \to \, ?$

$\text{NOR} \to \,?$

$\text{EQUALS} \to \, ?$

• If your "reduce" is a summation of the number of computations that accept, you get #P. – usul Feb 6 '15 at 16:47
• Be careful about how you define the model. For one thing, it's critical that the program is never allowed to read the computation value ("return value") of any computation. – D.W. Feb 7 '15 at 2:06

In the case of XOR or EQUALS, you get the class $\oplus$P, which includes the graph isomorphism problem.