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CS 251 Fall 2019Principles of Programming LanguagesBen Woodλ CS 240 Spring 2020Foundations of Computer SystemsBen Wood
https://cs.wellesley.edu/~cs240/s20/
Logic for Arithmeticadders
Arithmetic Logic Unit
Logic for Arithmetic 1
Addition: 1-bit half adder ex
AB
Sum
Carry out
A B Carry out Sum
0 0
0 1
1 0
1 1
+A
BSum
Carry out
Logic for Arithmetic 2
Addition: 1-bit full adder
Carry in A B Carry
out Sum
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
ex+A
BSum
Carry in
Carry out
AB
Sum
Carry in
Carry out
Logic for Arithmetic 3
Addition: n-bit ripple-carry adder
+A0
B0Sum0
Carry in0
+An-1
Bn-1Sumn-1
Carry outn-1
+A1
B1Sum1
+A2
B2Sum2
…
There are faster, more complicated ways too…
AB
Sum
Carry in
Carry out
Logic for Arithmetic 4
ALU
Processor Components
Registers MemoryInstruction Fetch and Decode
1 324
Abstraction!
Logic for Arithmetic 5
Arithmetic Logic Unit (ALU)
Operand A
Operand B
Condition Codes(sign, overflow, carry-out, zero)
Result
Operation
Hardware unit for arithmetic and bitwise operations.
words
word
a few bits
a few bits
1
ALU
Logic for Arithmetic 6
1-bit ALU for bitwise operationsBuild an n-bit ALU from n 1-bit ALUs.Each bit i in the result is computed from the corresponding bit i in the two inputs.
MU
X
A
B
0
1
Operation
Result
Op A B Result
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
ex
Logic for Arithmetic 7
1-bit ALU
A
B
0
1
Operation
Result
2
2Carry in
+ Sum
Carry out
MUX
Logic for Arithmetic 8
A0
B0
0
1Result0
2
Carry in
+ Sum
MUX
An-1
Bn-1
0
1Resultn-1
2+ Sum
Carry out
MUX
A1
B1
0
1Result1
2+ Sum
MUX
Operation
… ....
2
+A0
B0
Sum0
Carry in
+An-1
Bn-1
Sumn-1
Carry out
+A1
B1
Sum1
+A2
B2
Sum2
…
n-bit ripple carry adder
n-bit ALU
Logic for Arithmetic 9
ALU conditionsExtra ALU outputs describing properties of result.Zero Flag:1 if result is 00...0 else 0Sign Flag:1 if result is negative else 0Carry Flag:1 if carry out else 0(Signed) Overflow Flag:1 if signed overflow else 0
Implement these.
A0
B0
0
1Result0
2
Carry in
+ Sum
MUX
An-1
Bn-1
0
1Resultn-1
2+ Sum
Carry out
MUX
A1
B1
0
1Result1
2+ Sum
MUX
Operation
… ....
2
ex
ex
Logic for Arithmetic 10
Add subtraction
Logic for Arithmetic 12
0
1Result0
2+
MUX
0
1Resultn-1
2+Carry out
MUX
0
1Result1
2+
MUX
Operation
… ....
B10
1
B00
1
Bn-10
1
....
2
How can we control ALU inputsor add minimal new logicto compute A-B?
A0
A1
An-1
A NAND B
A NOR B
A<B
A==B
How can we control ALU inputsor add minimal new logicto compute each?
Logic for Arithmetic 13
0
1Result0
2+
MUX
0
1Resultn-1
2+Carry out
MUX
0
1Result1
2+
MUX
Operation
… ....
A1 0
1
B10
1
A0 0
1
B00
1
Negate B
An-1 0
1
Bn-10
1
....
Invert A
....
2ex
Controlling the ALU
Logic for Arithmetic 17
ALU control lines Function
0000 AND0001 OR0010 add0110 subtract1100 NOR
Operand A
Operand B
Result
Control Lines
Condition Codes
Abstraction!
ALU
![Flag Control instructions CLC clear carry flag CF = 0 STC set carry flag CF= 1 CMC complement carry flag [CF] CF](https://img.dokumen.tips/doc/110x75/56649e925503460f94b97808/flag-control-instructions-clc-clear-carry-flag-cf-0-stc-set-carry-flag-cf.jpg)
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