🧮 Unified Addition–Subtraction Circuit (4-bit)

A unified adder-subtractor circuit performs both binary addition and subtraction using a single chain of full adders. Instead of building separate units for each operation, it uses polarity control to transform subtraction into addition through conditional inversion and a $+1$ bias.

🧮 Core Insight

Subtraction in binary is defined as:

This is called two’s complement subtraction, where:

• $\sim B$ is the bitwise NOT of $B$ (1’s complement)
• Adding $1$ completes the transformation into two’s complement

⚙️ Circuit Architecture

The unified circuit uses:

• XOR gates to conditionally invert each bit of $B$
• A control signal Add/Sub to select operation mode
• The same control signal as the initial carry-in to add the $+1$ for subtraction

🔧 XOR as Conditional Inverter

Each XOR gate computes:

Where:

• $B_i$ is the $i$-th bit of operand $B$
• $S$ is the Add/Sub control signal

🔧 Carry-in as $+1$ Bias

The same control signal $S$ is fed into the least significant full adder as the initial carry-in:

This adds the $+1$ needed for two’s complement subtraction.

🧠 Why It Works

When $S = 1$, the circuit computes:

When $S = 0$, it computes:

So the same adder chain handles both operations with no structural changes—just a polarity shift.

🧮 Symbolic Flow

          x₃ x₂ x₁ x₀     ← Operand A
           │  │  │  │
           ▼  ▼  ▼  ▼
          FA FA FA FA     ← Full Adders
           ▲  ▲  ▲  ▲
           │  │  │  │
     y₃ y₂ y₁ y₀          ← Operand B
      \  \  \  \  
      XOR XOR XOR XOR     ← Conditional Inversion
       /   /   /   /
     Add/Sub (S)          ← Control signal
           │
         Carry-in         ← Also feeds FA₀

✅ Advantages

🧠 Philosophical Insight

Subtraction is not a separate operation—it’s addition under inverted polarity.

This design reflects the deep symmetry of binary arithmetic, where polarity control unifies seemingly distinct operations.