Oracle query’s required The variables $a,b,c \in \{0,1\}$, thus $a^k, b^k, c^k \in \{0,1\}$

I want to pass a query to an oracle that returns the coefficients of each term $(1,a,b,c,ab,ac,bc,abc)$ in the expansion of products such as this one $(1-a+ab)(1-b+bc)(b-bc)$. There could be more variables and more brackets to expand.

Do I require a single FP query to do this or something more?



edit:

Input: $(1-a+ab)(1-b+bc)$

Expand: $a b^2 c - a b^2 - a b c + 2 a b - a + b c - b + 1$

Apply property of idempotence:  $a b c - a b - a b c + 2 a b - a + b c - b + 1$

Simplify: $1 - a - b + ab +bc$

Extract coefficients: \begin{matrix} 1 &amp; 1 \\ a &amp; -1 \\ b &amp; -1 \\ c &amp; 0 \\ ab &amp; 1 \\ ac &amp; 0 \\ bc &amp; 1 \\ abc &amp; 0 \end{matrix}

Question: What is the 'weakest' oracle capable of extracting the coefficients above from the input?
Your problem is #P-hard. Indeed, given a #SAT instance with variables $x_i$ and clauses $C_j$, let $\kappa_{i,b}$ be the product of the clauses $C_j$ satisfied by the truth assignment $x_i=b$, and consider
$$ P = \prod_i (\kappa_{i,0} + \kappa_{i,1}). $$
The coefficient of $\prod_j C_j$ in $P$ is the number of satisfying assignments.

In the other direction, the special case where the input formula is $\Pi\Sigma\Pi$ (that is, the product of polynomials) is #P-complete. Suppose that we are interested in the coefficient of some monomial $m$ in $\prod_k P_k$, where the $P_k$ are polynomials. Substitute zero in all variables not appearing in $m$. Now guess one term from each $P_k$, and accept if the terms together cover $m$.