I need a general method for solving systems of logical equations like:
$$
\begin{equation*}
 \begin{cases}
  c_{0} = a_{0} \land b_{0}\\\\ 
  c_{1} = a_{0} \land b_{1} ⊕ a_{1} \land b_{0}\\\\ 
  c_{2} = a_{0} \land b_{2} ⊕ a_{1} \land b_{1} ⊕ a_{2} \land b_{0}\\\\ 
  c_{3} = a_{1} \land b_{2} ⊕ a_{2} \land b_{1}\\\\ 
  c_{4} = a_{2} \land b_{2} 
 \end{cases}
\end{equation*}
$$
Where c is known and a and b are unknown variables. 
This system is a system of logical nonlinear equations, I want to know if it is possible to find a general solution for such a system. The number of unknowns is 1 more than the number of equations. Solutions will be symmetric (a and b can be swapped).