The correct answer is B̅.

Let's analyze the given truth table of the logic circuit step-by-step to find the Boolean expression for the output $Y$:

The inputs are $A$ and $B$, and the output is $Y$.
From the given truth table:
1. When $A=0$ and $B=0$, the output $Y=1$.
2. When $A=0$ and $B=1$, the output $Y=0$.
3. When $A=1$ and $B=0$, the output $Y=1$.
4. When $A=1$ and $B=1$, the output $Y=0$.

Let us observe the relationship between the input variables and the output $Y$:
- When $B=0$ (rows 1 and 3), the output $Y$ is always $1$, regardless of whether $A$ is $0$ or $1$.
- When $B=1$ (rows 2 and 4), the output $Y$ is always $0$, regardless of whether $A$ is $0$ or $1$.

Thus, the output $Y$ does not depend on the input $A$ at all. It is always the logical negation (complement) of the input $B$.
Therefore, the Boolean expression for the output is:
$Y=\overline{B}$
which is represented as B̅.