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The auxiliary carry flag AF watches for a 4-bit (nibble) carry, while the common carry flag CF watches for a carry-out from the MSB of the operand size.
Carry indicates the result isn't mathematically correct when interpreted as unsigned, overflow indicates the result isn't mathematically correct when interpreted as signed. So as examples for your 4-bit ALU: 1111 + 0001 = 0000 should set carry (15 + 1 = 0 is false) and clear overflow (-1 + 1 = 0 is true).
Carry Flag is a flag set when: a) two unsigned numbers were added and the result is larger than "capacity" of register where it is saved. Ex: we wanna add two 8 bit numbers and save result in 8 bit register.
In unsigned arithmetic, watch the carry flag to detect errors. In unsigned arithmetic, the overflow flag tells you nothing interesting. In signed arithmetic, watch the overflow flag to detect errors. In signed arithmetic, the carry flag tells you nothing interesting.
The rules for turning on the carry flag in binary/integer math are two:
The carry flag is set if the addition of two numbers causes a carry out of the most significant (leftmost) bits added. 1111 + 0001 = 0000 (carry flag is turned on)
The carry (borrow) flag is also set if the subtraction of two numbers requires a borrow into the most significant (leftmost) bits subtracted. 0000 - 0001 = 1111 (carry flag is turned on) Otherwise, the carry flag is turned off (zero).
In unsigned arithmetic, watch the carry flag to detect errors.
In signed arithmetic, the carry flag tells you nothing interesting.
The rules for turning on the overflow flag in binary/integer math are two:
If the sum of two numbers with the sign bits off yields a result number with the sign bit on, the "overflow" flag is turned on. 0100 + 0100 = 1000 (overflow flag is turned on)
If the sum of two numbers with the sign bits on yields a result number with the sign bit off, the "overflow" flag is turned on. 1000 + 1000 = 0000 (overflow flag is turned on)
Otherwise the "overflow" flag is turned off
Note that you only need to look at the sign bits (leftmost) of the three numbers to decide if the overflow flag is turned on or off.
If you are doing two's complement (signed) arithmetic, overflow flag on means the answer is wrong - you added two positive numbers and got a negative, or you added two negative numbers and got a positive.
If you are doing unsigned arithmetic, the overflow flag means nothing and should be ignored.
For more clarification please refer: http://teaching.idallen.com/dat2343/10f/notes/040_overflow.txt
Carry Flag is a flag set when:
a) two unsigned numbers were added and the result is larger than "capacity" of register where it is saved. Ex: we wanna add two 8 bit numbers and save result in 8 bit register. In your example: 255 + 9 = 264 which is more that 8 bit register can store. So the value "8" will be saved there (264 & 255 = 8) and CF flag will be set.
b) two unsigned numbers were subtracted and we subtracted the bigger one from the smaller one. Ex: 1-2 will give you 255 in result and CF flag will be set.
Auxiliary Flag is used as CF but when working with BCD. So AF will be set when we have overflow or underflow on in BCD calculations. For example: considering 8 bit ALU unit, Auxiliary flag is set when there is carry from 3rd bit to 4th bit i.e. carry from lower nibble to higher nibble. (Wiki link)
Overflow Flag is used as CF but when we work on signed numbers. Ex we wanna add two 8 bit signed numbers: 127 + 2. the result is 129 but it is too much for 8bit signed number, so OF will be set. Similar when the result is too small like -128 - 1 = -129 which is out of scope for 8 bit signed numbers.
You can read more about flags on wikipedia
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