The problem involves finding a simplified logical expression for the given digital circuit. Let's analyze the circuit step-by-step:

1. The circuit has three inputs: \(x\), \(y\), and \(z\).
2. The first gate is a NOR gate with inputs \(x\), \(y\), and \(z\). The output of a NOR gate is the negation of the OR operation: 
   \(\overline{x + y + z}\).
3. The output of the NOR gate feeds into an AND gate along with the output of a NOT gate.
4. The NOT gate receives \(x\) and outputs its negation: 
   \(\overline{x}\).
5. The first AND gate thus checks: 
   \(\overline{x + y + z} \cdot \overline{x}\).
6. There is a second AND gate that directly takes inputs \(y\) and \(z\). Its output is \(y \cdot z\).
7. The outputs from both AND gates are fed into an OR gate: 
   \((\overline{x + y + z} \cdot \overline{x}) + (y \cdot z)\).
8. The output of the OR gate and the NOT gate output are fed into a final AND gate. The expression simplifies to: \(\overline{x} \cdot \left( (\overline{x + y + z} \cdot \overline{x}) + (y \cdot z) \right)\).
9. Simplifying this logic further results in the simplified logical expression: \(0\).

The operation shows that under all possible input combinations, the output of the final AND gate will always be zero, hence the simplified logical expression for the output is \(0\).

Therefore, the correct option is: \(0\).