💥 Integer Overflow

💥 Integer Overflow

🧠 Motivation

ℹ️

Why we care Integer overflow is a semantic mismatch between mathematical expectation and hardware constraints.

In modular arithmetic, overflow is benign. In fixed-width binary systems, it can flip signs, corrupt logic, or cause unintended wraparound.

🔣 What Is Integer Overflow?

Integer overflow occurs when the result of an arithmetic operation exceeds the maximum value representable within a fixed number of bits.

In unsigned systems: values wrap from 2ⁿ − 1 back to 0
In signed (2’s complement) systems: values wrap from +2ⁿ⁻¹ − 1 to −2ⁿ⁻¹

🧮 Example: 4-bit Signed Overflow

A	B	A + B	Expected	Actual	Interpretation
`0111` (+7)	`0001` (+1)	`1000`	+8	−8	Overflow occurred

MSB flipped → system interprets result as negative

🔁 Why It Happens

✅ Mathematical View

You expect: 7 + 1 = 8
In modular arithmetic: 8 mod 16 = 8 ✅ (still valid within modulus 2⁴ = 16)

❌ Hardware View

4-bit signed integers range from −8 to +7
1000 is interpreted as −8
Result is semantically incorrect, even if bitwise correct

⚠️ Overflow Detection Logic

🔧 Signed Addition Overflow

Occurs when:

Adding two positive numbers yields a negative result
Adding two negative numbers yields a positive result

🧠 Detection Formula

Overflow = C_in_MSB ⊕ C_out_MSB

Where:

C_in_MSB: carry into most significant bit
C_out_MSB: carry out from most significant bit

📊 Visual Anchor: 4-bit Signed Wraparound

+7 → 0111 +8 → 1000 ← interpreted as −8
+9 → 1001 ← interpreted as −7
...
+15 → 1111 ← interpreted as −1

Last updated on 2025-11-08

💡 Open-Drain Logic Output 💻 Floating-Point & Fixed-Point