GATE CS 2003 Q46 — Computer Arithmetic

In the given ALU circuit, the inputs to the $n$-bit adder are:

• First input: $A$
• Second input: $B \oplus K$ (since $B$ is XORed with the control line $K$)
• Carry-in: $C_0$

The output of the adder is given by the expression:

Let us analyze the possible operations by setting the control lines $K$ and $C_0$:

1. Addition ($A + B$): By setting $K = 0$ and $C_0 = 0$, we get:
   $$S = A + (B \oplus 0) + 0 = A + B$$ This operation is possible.
2. Subtraction ($A - B$): In $2$’s complement representation, $A - B = A + \bar{B} + 1$. By setting $K = 1$ and $C_0 = 1$, we get:
   $$S = A + (B \oplus 1) + 1 = A + \bar{B} + 1 = A - B$$ This operation is possible.
3. Increment ($A + 1$): To perform $A + 1$, the second operand of the adder must be $0$ with $C_0 = 1$. Since the second operand is $B \oplus K$, its value depends on the external input $B$. We cannot force $B \oplus K = 0$ for all values of $B$ using only the control lines $K$ and $C_0$. Thus, $A + 1$ is not possible.

Therefore, we can perform $A + B$ and $A - B$, but not $A + 1$.