A boolean algebra expression can be converted into an idempotent algebra using $$\bar a \equiv 1-a, \qquad a \vee b \equiv a+b -ab, \qquad a \wedge b \equiv a \otimes b$$
where $\otimes$ is the idempotent product (no powers). For example, $$(a+b)\otimes(a-b) = a -ab +ab - b = a-b.$$
The CNF formula
$$\phi = (a\vee b) \; (b \vee c)(b \vee \bar c)(\bar b \vee \bar c) \; (a \vee c)(\bar a \vee \bar c)$$
can be converted into what I would call the idempotent expression $$\phi = (a + b - ab)\otimes (b-bc) \otimes (a+c-2ac).$$
This expression expands to give $\phi = ab - abc$. I would like an algorithm that, given a CNF formula as input, outputs the term with the lowest homogeneity. In this example, the oracle would return $ab$. (If there are multiple terms all with minimal homogeneity, the algorithm can return any one of them.)
Question 1: What is the complexity of this task? How high in the polynomial hierarchy is it?
Secondly, given a different idempotent expression $$\phi = ac+ad+bc+bd-abc-abd-2acd-2bcd + 2abcd,$$
I am interested in summing over the terms with equal homogeneity. By letting all variables be $\epsilon$ we get $$\phi = 4\epsilon ^2 - 6\epsilon^3 + 2\epsilon^4.$$ This yields a homogeneity vector of $[0,0,4,-6,2]$.
Question 2: What is the complexity of computing the homogeneity vector, given an idempotent expression as input? How high in the polynomial hierarchy is it?