ALU — operations, flags (zero, carry, overflow, negative)

4.1.6 · Coding › Computer Architecture (Deep)

1. ALU actually hai kya

2. Two's complement — neenv (derive karo!)

First principles se derivation. $n$ bits mein hmare paas $2^{n}$ patterns hain. Hum chahte hain ki $- x$ satisfy kare $x + (- x) \equiv 0 (mod 2^{n})$ . Toh: $- x \equiv 2^{n} - x (mod 2^{n})$ Ab $2^{n} - x = all ones (2^{n} - 1) - x + 1 = (\sim x) + 1$ .

Yahi famous rule hai: invert all bits and add 1.

3. Har flag ko first principles se derive karna

3.1 Zero flag — Z

3.2 Negative flag — N

3.3 Carry flag — C (unsigned story)

Ek $n$ -bit adder ko full-adders ki chain socho, har ek carry pass karta hai. Most significant full adder ka carry out $c_{o u t}$ hai.

3.4 Overflow flag — V (signed story)

Derivation. Signed overflow tab hota hai jab sign bit mein carry in aur sign bit se carry out alag ho: $V = c_{n} \oplus c_{n - 1}$ Equivalent mein, sign bits ke terms mein ( $a$ , $b$ = operands ke sign, $s$ = result ka sign), addition ke liye: $V = (a \land b \land \overline{s}) \lor (\overline{a} \land \overline{b} \land s)$ Padho ise: overflow tabhi hota hai jab dono operands ka sign same ho aur result ka sign flip ho jaye. (pos+pos→neg, ya neg+neg→pos). Opposite sign ke numbers jodne par aap kabhi overflow nahi kar sakte.

Figure — ALU — operations, flags (zero, carry, overflow, negative)

4. Worked examples (8-bit, $n = 8$ , range unsigned $0..255$ , signed $- 128..127$ )

5. Logic & shift operations (flags alag hote hain!)

Op	R	Z	N	C	V
ADD	$A + B$	result=0	MSB	carry-out	$c_{n} \oplus c_{n - 1}$
SUB	$A - B$	result=0	MSB	NOT borrow	sign rule
AND/OR/XOR	logic	result=0	MSB	0	0
LSL (shift left)	$A ≪ 1$	result=0	MSB	bit shifted out	—

6. Common mistakes (Steel-man + fix)

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek 8-slot counter jo sirf 0–255 tak ke numbers rakh sakta hai. ALU ek robot hai jo in slots mein add aur subtract karta hai. Har sum ke baad yeh choti choti flags uthata hai: ek "yeh zero hai!" flag, ek "top slot jal raha hai (negative lag raha hai)" flag, ek "answer fit nahi hua, upar se spill ho gaya" carry flag, aur ek special "agar hum maan lein ki aadhe numbers negative hain, toh sign galat aaya" overflow flag. Wahi robot subtraction second number ko ulta karke aur ek jodkar karta hai — jaise peeche jaana matlab ghumna aur aage chalna.

Connections

Two's Complement Representation
Full Adder and Ripple-Carry Adder
Condition Codes and Conditional Branches
Status / Flags Register (PSR / EFLAGS)
Signed vs Unsigned Comparison Instructions
Combinational vs Sequential Logic

Flashcards

ALU result ke alawa aur kya output karta hai?

Condition flags ka ek set (Z, C, N, V).

Zero flag aur uska hardware define karo.

Z=1 tabhi jab result bilkul zero ho; saare result bits par NOR se implement hota hai.

Negative flag kis ke barabar hota hai?

Result ka MSB (sign bit),

r_{n - 1}

Carry flag kaunsa range sawaal answer karta hai?

Kya UNSIGNED operation n-bit range se zyada ho gayi (carry/borrow out of MSB).

Overflow flag kaunsa range sawaal answer karta hai?

Kya SIGNED operation

[- 2^{n - 1}, 2^{n - 1} - 1]

se bahar gayi.

Carries ke terms mein overflow derive karo.

V = c_{n} \oplus c_{n - 1}

(sign bit mein carry XOR sign bit se carry out).

ALU adder use karke subtract kaise karta hai?

A - B = A + (\sim B) + 1

, carry-in = 1 feed karke.

\sim x + 1 = - x

kyun hai?

Kyunki

- x \equiv 2^{n} - x = (2^{n} - 1 - x) + 1 =\sim x + 1 (mod 2^{n})

Carry with no overflow ka example do.

255+1 (8-bit): C=1, V=0, result 0 (sahi signed -1+1).

Overflow with no carry ka example do.

127+1 (8-bit): C=0, V=1, result -128 (signed wrap).

Signed addition kabhi overflow NAHI kar sakta?

Jab operands ke signs opposite hon.

Subtraction par C=0 ka matlab kya hai (ARM/x86)?

Borrow hua, yaani A < B (unsigned).

Logic ops (AND/OR/XOR) typically kaun se flags set karte hain?

Result se Z aur N; C aur V clear ho jaate hain.

Left shift par carry kahaan se aata hai?

MSB se shift out hone wala bit carry flag ban jaata hai.