Is solving the following system of boolean equations NP-hard? I reduced a problem I'm currently working on to the following system of boolean equations:
$$
X_i \iff
\begin{cases}
\bigvee_{B \in A_i} \bigwedge_{k \in B} X_k \\
true \\
false
\end{cases}
$$
Where $|B|&gt;=1$ for all $B \in A_i$ and each $X_i$ has exactly one of these defining equations.
I need to find $X_i$ that satisfies all these equations ($A_i$ are given). I'm wondering if this problem is NP-hard. I couldn't reduce the general boolean satisfiability problem to it yet.
Ideally, I would find an algorithm to solve this problem in polynomial time. If it's NP-hard I guess I have to hope that SAT solvers can solve it reasonably fast.
Thanks!

Here's an example:
$$
X_1 \iff true \\
X_2 \iff false \\
X_3 \iff X_4 \\
X_4 \iff X_1 \vee (X_2 \wedge X_3)
$$
[Now that the question's been clarified I'll post my previous comment as an answer.]
It's in $\mathsf{P}$. Start with unit propagation. Afterwards, what's left on the right-hand sides will be monotone, so will be satisfied by setting all remaining variables to True.
(If you want to count the number of solutions, on the other hand, that might be harder.)