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).

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). The challenge is not unsolvable and there is an example of a solution . However, it seems to me that there is a simpler solution.

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.

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).