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Introduction to Computer Science • Robert Sedgewick and Kevin Wayne • Copyright © 2005 • http://www.cs.Princeton.EDU/IntroCS
Combinational Circuits
2
Virtual machines
High-Level Language
Assembly Language
Operating System
Instruction SetArchitecture
Microarchitecture
Digital Logic Level 0
Level 1
Level 2
Level 3
Level 4
Level 5
Abstractions for computers
3
A simple microcomputer from the year of 2005
IR
DECODE
CONTROLAND
SEQUENCING
PC
ACC B
ALU
CLOCK
I/OD
EVICE
I/OD
EVICE
DATA BUS
CONTROL BUS
ADDRESS BUS
MEMORYI/OPORT
FLAG
4
Instruction set
OPCODE MNEMONIC OPCODE MNEMONIC 0 NOP A CMP 1 LDA B JG2 STA C JE 3 ADD D JL4 SUB5 IN6 OUT7 JMP8 JN9 HLT
OPCODE OPERAND
4 12
Introduction to Computer Science • Robert Sedgewick and Kevin Wayne • Copyright © 2005 • http://www.cs.Princeton.EDU/IntroCS
The Princeton TOY Machine
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What is TOY?
An imaginary machine similar to:ν Ancient computers. (PDP-8, world's first
commercially successful minicomputer. 1960s)
– 12-bit words– 2K words of memory– Used in Apollo project
7
What is TOY?
An imaginary machine similar to:ν Ancient computers.ν Today's microprocessors.
Pentium Celeron
8
What is TOY?
An imaginary machine similar to:ν Ancient computers.ν Today's microprocessors.
Why study TOY?ν Machine language programming.
– how do high-level programs relate to computer?– a favor of assembly programming
ν Computer architecture.– how is a computer put together?– how does it work?
ν Optimized for understandability, not cost or performance.
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Inside the Box
Switches. Input data and programs.
Lights. View data.
Memory.ν Stores data and programs.ν 256 "words." (16 bits each)ν Special word for stdin /
stdout.
Program counter (PC).ν An extra 8-bit register.ν Keeps track of next
instruction to be executed.
Registers.ν Fastest form of storage.ν Scratch space during
computation.ν 16 registers. (16 bits each)ν Register 0 is always 0.
Arithmetic-logic unit (ALU). Manipulate data stored in registers.
Standard input, standard output. Interact with outside world.
10
Instruction set architecture (level 2)
Machine contents at a particular place and time.ν Record of what program has done.ν Completely determines what machine will do.
0008 0005 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
8A00 8B01 1CAB 9C02 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
0000 0000 0000 0000 0000 0000 0000 0000
00:
08:
10:
18:
20:
28:
.
.
E8:
F0:
F8: 0000 0000 0000 0000 0000 0000 0000 0000
Main Memorypc10
nextinstruction
program
Registers
0000
R2
0000
R3
0000
R8
0000
R9
R0
0000
R1
0000
RA
0000
RB
0000
0000
R6
0000
R7
0000
RC
0000
RD
R4
0000
R5
0000
RE
0000
RF
0000
variables
data
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Instruction set architecture (level 2)
Program: Sequence of instructions.
16 instruction types:ν 16-bit word (interpreted one way).ν Changes contents of registers,
memory, andPC in specified, well-defined ways.
Data:ν 16-bit word (interpreted other way).
Program counter (PC):ν Stores memory address of "next
instruction."
0: haltInstructions
1: add2: subtract3: and4: xor5: shift left6: shift right7: load address8: load9: storeA: load indirectB: store indirectC: branch zeroD: branch positiveE: jump registerF: jump and link
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TOY Architecture (level 1)
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level 0
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Boolean Algebra
Based on symbolic logic, designed by George BooleBoolean variables take values as 0 or 1.Boolean expressions created from:
ν NOT, AND, OR
15
NOT
NOT
Digital gate diagram for NOT:
X X’
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AND
AND
Digital gate diagram for AND:
X⋅Y XY
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OR
OR
Digital gate diagram for OR:
X+Y
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Operator Precedence
Examples showing the order of operations: NOT > AND > OR
Use parentheses to avoid ambiguity
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Defining a function
Description: square of x minus 1Algebraic form : x2-1Enumeration:
::245154833201
f(x)x
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Defining a function
Description: number of days of the x-th month of a non-leap yearAlgebraic form: ?Enumeration:
311230113110309318317306315304313282311
f(x)x
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Truth Table
Truth table.ν Systematic method to describe Boolean function.ν One row for each possible input combination.ν N inputs ⇒ 2N rows.
AND truth table
0 0
0 1
1 0
1 1
0
0
0
1
x y x y
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Proving the equivalence of two functions
Prove that x2-1=(x+1)(x-1)
Using algebra: (you need to follow some rules)(x+1)(x-1) = x2+x-x-1= x2-1
Using enumeration:
:::2424515154883332001
x2-1(x+1)(x-1)x
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Important laws
x + 1 = 1x + 0 = xx + x = 1
x．1 = xx．0 = 0x．x = 0
DeMorgan Law
x．y = x + y
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Truth Tables (1 of 3)
A Boolean function has one or more Boolean inputs, and returns a single Boolean output.A truth table shows all the inputs and outputs of a Boolean function
Example: ¬X ∨ Y
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Truth Tables (2 of 3)
Example: X ∧ ¬Y
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Truth Tables (3 of 3)
When s=0, return x; otherwise, return y.Example: (Y ∧ S) ∨ (X ∧ ¬S)
muxX
Y
S
Z
Two-input multiplexer
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Truth Table for Functions of 2 Variables
Truth table.ν 16 Boolean functions of 2 variables.
ZERO
Truth table for all Boolean functions of 2 variables
y
0 0
0 1 0
1 0 0
1 1 0
0
0
1
0
0
1
0
0
x
0
0
1
1
AND
0
0
0
1
y
0
1
0
1
XOR
0
1
1
0
OR
0
1
1
1
x
0
NOR
Truth table for all Boolean functions of 2 variables
y
0 1
0 1 0
1 0 0
1 1 0
y'
1
0
1
0
x'
1
1
0
0
1
0
1
1
EQ
1
0
0
1
1
1
0
1
NAND
1
1
1
0
ONE
1
1
1
1
x
0
every 4-bit value represents one
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Truth Table for Functions of 3 Variables
Truth table.ν 16 Boolean functions of 2 variables.ν 256 Boolean functions of 3 variables.ν 2^(2^n) Boolean functions of n variables!
AND
some functions of 3 variables
z
0 0
0 1 0
1 0 0
1 1 0
y
0
x
0
0
0
0
0
0 1
1 0
1 1
01
1
1
1
0
0
0
1
OR
0
1
1
1
1
1
1
1
MAJ
0
0
0
1
0
1
1
1
ODD
0
1
1
0
1
0
0
1
every 4-bit value represents one
every 8-bit value represents one
every 2n-bit value represents one
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Sum-of-Products
Sum-of-products. Systematic procedure for representing a Boolean function using AND, OR, NOT.
ν Form AND term for each 1 in Boolean function.ν OR terms together.
x'yz
expressing MAJ using sum-of-products
z xyz' xyzxy'zMAJyx
0
0
0
1
0
1
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
x'yz + xy'z + xyz' + xyz
0
0
0
1
0
1
1
1
proves that { AND, OR, NOT }are universal
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Universality of AND, OR, NOT
Fact. Any Boolean function can be expressed using AND, OR, NOT.
ν { AND, OR, NOT } are universal.ν Ex: XOR(x,y) = xy' + x'y.
Exercise. Show {AND, NOT}, {OR, NOT}, {NAND}, {NOR} are universal.Hint. DeMorgan's law: (x'y')' = x + y.
x'
Expressing XOR Using AND, OR, NOT
y
0 1
1 1
0 0
1 0
x'y
0
1
0
0
x'y + xy'
0
1
1
0
xy'
0
0
1
0
y'
1
0
1
0
x XOR y
0
1
1
0
x
0
0
1
1
NOT xx'x AND yx yx OR y
MeaningNotation
x + y
31
Computer Architecture
This lecture. Boolean circuits.
Ahead. Putting it all together and building a TOY machine.
same issues apply to modern day microprocessors, just a matter of scale and speed.
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Digital Circuits
What is a digital system?ν Analog: signals vary continuously.ν Digital: signals are 0 or 1.
Why digital systems?ν Accuracy and reliability.ν Staggeringly fast and cheap.
Basic abstractions.ν On, off.ν Wire: propagates on/off value.ν Switch: controls propagation of on/off values
through wires.
0.0V0.5V
2.8V3.3V
0 1 0
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Wires
Wires.ν On (1): connected to power.ν Off (0): not connected to power.ν If a wire is connected to a wire that is on, that
wire is also on.ν Typical drawing convention: "flow" from top, left
to bottom, right.
0
powerconnection
1
1
1
34
Controlled switch. [relay implementation]ν 3 connections: input, output, control.ν Magnetic force pulls on a contact that cuts
electrical flow.ν Control wire affects output wire, but output does
not affect control; establishes forward flow of information over time.
Controlled Switch
X
X
35
Circuit Anatomy
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Logic Gates: Fundamental Building Blocks
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NOT
38
NOT
1
0
0
1
39
OR
40
Series relays = NOR
0 10 0
101 1
0
0
0
1
41
NOR
42
AND
43
AND
44
Logic Gates: Fundamental Building Blocks
45
What about parallel relays?
0
11
0
0
1
=NAND
46
Can we implement AND/OR using parallel relays?
Now we know how to implement AND,OR and NOT. We can just use them as black boxes without knowing how they were implemented. Principle of information hiding.
47
Multiway Gates
Multiway gates.ν OR: 1 if any input is 1; 0 otherwise.ν AND: 1 if all inputs are 1; 0 otherwise.ν Generalized: negate some inputs.
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Multiway Gates
Multiway gates.ν OR: 1 if any input is 1; 0 otherwise.ν AND: 1 if all inputs are 1; 0 otherwise.ν Generalized: negate some inputs.
49
Translate Boolean Formula to Boolean Circuit
Sum-of-products. XOR.
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Translate Boolean Formula to Boolean Circuit
Sum-of-products. XOR.
51
Translate Boolean Formula to Boolean Circuit
Sum-of-products. XOR.
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Translate Boolean Formula to Boolean Circuit
Sum-of-products. Majority.
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Translate Boolean Formula to Boolean Circuit
Sum-of-products. Majority.
54
Translate Boolean Formula to Boolean Circuit
Sum-of-products. Majority.
55
Simplification Using Boolean Algebra
Every function can be written as sum-of-product
Many possible circuits for each Boolean function.ν Sum-of-products not necessarily optimal in:
– number of switches (space)– depth of circuit (time)
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Boolean expression simplification
Karnaugh map
57
Example
xy z
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Simplification Using Boolean Algebra
Many possible circuits for each Boolean function.ν Sum-of-products not necessarily optimal in:
– number of switches (space)– depth of circuit (time)
MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz = xy + yz + xz.
59
Layers of Abstraction
Layers of abstraction.ν Build a circuit from wires and switches.
[implementation]ν Define a circuit by its inputs and outputs. [API]ν To control complexity, encapsulate circuits.
[ADT]
60
Layers of Abstraction
Layers of abstraction.ν Build a circuit from wires and switches.
[implementation]ν Define a circuit by its inputs and outputs. [API]ν To control complexity, encapsulate circuits.
[ADT]
61
ODD Parity Circuit
ODD(x, y, z).ν 1 if odd number of inputs are 1. ν 0 otherwise.
x'y'z
Expressing ODD using sum-of-products
z xy'z' xyzx'yz'ODDyx
0
1
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
x'y'z + x'yz' + xy'z' + xyz
0
1
1
0
1
0
0
1
62
ODD Parity Circuit
ODD(x, y, z).ν 1 if odd number of inputs are 1. ν 0 otherwise.
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ODD Parity Circuit
ODD(x, y, z).ν 1 if odd number of inputs are 1. ν 0 otherwise.
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Expressing a Boolean Function Using AND, OR, NOT
Ingredients.ν AND gates.ν OR gates.ν NOT gates.ν Wire.
Instructions.ν Step 1: represent input and output signals with
Boolean variables.ν Step 2: construct truth table to carry out
computation.ν Step 3: derive (simplified) Boolean expression using
sum-of products.ν Step 4: transform Boolean expression into circuit.
65
Let's Make an Adder Circuit
Goal. x + y = z for 4-bit integers.ν We build 4-bit adder: 9 inputs, 4 outputs.ν Same idea scales to 128-bit adder.ν Key computer component.
842 7
753 9+
606 6
111 0
66
Let's Make an Adder Circuit
Step 1. Represent input and output in binary.
x1x2x3 x0
y1y2y3 y0+
z1z2z3 z0
100 0
110 1+
001 1
011 0
67
Let's Make an Adder Circuit
Goal. x + y = z for 4-bit integers.
Step 2. [first attempt]ν Build truth table.
Q. Why is this a bad idea?A. 128-bit adder: 2256+1 rows >> # electrons in universe!
x1x2x3 x0
y1y2y3 y0+
z1z2z3 z0
4-Bit Adder Truth Table
y2y30
0
0
0
1
.
1
0
0
0
0
0
.
1
x0x10
0
0
0
0
.
1
0
0
0
0
0
.
1
x2x30
0
0
0
0
.
1
0
0
0
0
0
.
1
y0y10
1
0
1
0
.
1
0
0
1
1
0
.
1
z2z30
0
0
0
1
.
1
0
0
0
0
0
.
1
z0z10
1
0
1
0
.
1
0
0
1
1
0
.
1
28+1 = 512 rows!
c00
0
0
0
0
.
1
cincout
68
1-bit half adder
x y s c
ADDxy
cs
We add numbers one bit at a time.
69
1-bit full adder
x y s
ADD
x
Cout
Cin
y
s
CoutCin
70
8-bit adder
71
Let's Make an Adder Circuit
Goal. x + y = z for 4-bit integers.
Step 2. [do one bit at a time]ν Build truth table for carry bit.ν Build truth table for summand bit.
x1x2x3 x0
y1y2y3 y0+
z1z2z3 z0
c1c2c3 c0 = 0
Carry Bit
ci ci+1yixi
0
0
0
1
0
1
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
Summand Bit
ci ziyixi
0
1
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
cout
72
Let's Make an Adder Circuit
Goal. x + y = z for 4-bit integers.
Step 3.ν Derive (simplified) Boolean expression.
MAJ
0
0
0
1
0
1
1
1
ODD
0
1
1
0
1
0
0
1
ci ci+1yixi
0
0
0
1
0
1
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
ci ziyixi
0
1
1
0
1
0
0
1
0
1
0
1
0
1
0
1
0
0
1
1
0
0
1
1
0
0
0
0
1
1
1
1
Carry Bit Summand Bit
73
Let's Make an Adder Circuit
Goal. x + y = z for 4-bit integers.
Step 4.ν Transform Boolean expression into circuit.ν Chain together 1-bit adders.
74
Adder: Interface
75
Adder: Component Level View
76
Adder: Switch Level View
77
Subtractor
Subtractor circuit: z = x – y.ν One approach: design like adder circuitν Better idea: reuse adder circuit
– 2’s complement: to negate an integer, flip bits, then add 1
78
Shifter
4-bit Shifter
SHIFT
s0 s1 s2 s3
z0 z1 z2 z3
x0
x1
x2
x3
Only one of them will be on at a time.
79
Shifter
x0000s3
x1x000s2
x2x1x00s1
x3x2x1x0s0
z3z2z1z0
z0 = s0‧x0 + s1‧0 + s2‧0 + s3‧0z1 = s0‧x1 + s1‧x0 + s2‧0 + s3‧0z2 = s0‧x2 + s1‧x1 + s2‧x0 + s3‧0z3 = s0‧x3 + s1‧x2 + s2‧x1 + s3‧x0
80
Shifter
z0 = s0‧x0 + s1‧0 + s2‧0 + s3‧0z1 = s0‧x1 + s1‧x0 + s2‧0 + s3‧0z2 = s0‧x2 + s1‧x1 + s2‧x0 + s3‧0z3 = s0‧x3 + s1‧x2 + s2‧x1 + s3‧x0
81
N-bit Decoder
N-bit decoderν N address inputs, 2N data outputsν Addresses output bit is 1;
all others are 0
82
N-bit Decoder
N-bit decoderν N address inputs, 2N data outputsν Addresses output bit is 1;
all others are 0
83
2-Bit Decoder Controlling 4-Bit Shifter
Ex. Put in a binary amount to shift.r0r1
84
Arithmetic Logic Unit
Arithmetic logic unit (ALU). Computes all operations in parallel.
ν Add and subtract.ν Xor.ν And.ν Shift left or right.
Q. How to select desired answer?
85
1 Hot OR
1 hot OR.ν All devices compute their answer;
we pick one.ν Exactly one select line is on.ν Implies exactly one output line is
relevant.
adder
xor
shifter
x．1 = xx．0 = 0
x + 0 = x
86
Bus
16-bit busν Bundle of 16 wiresν Memory transfer
Register transfer
8-bit busν Bundle of 8 wiresν TOY memory address
4-bit busν Bundle of 4 wiresν TOY register address
87
Bitwise AND, XOR, NOT
Bitwise logical operationsν Inputs x and y: n bits eachν Output z: n bitsν Apply logical operation to each corresponding pair
of bits
88
ALU
TOY ALUν Big combinational logic ν 16-bit busν Add, subtract, and, xor, shift left, shift right,
copy input 2
89
Device Interface Using Buses
Device. Processes a word at a time.Input bus. Wires on top.Output bus. Wires on bottom.Control. Individual wires on side.
16-bit words for TOY memory
90
ALU
Arithmetic logic unit.ν Add and subtract.ν Xor.ν And.ν Shift left or right.
Arithmetic logic unit.ν Computes all operations in parallel.ν Uses 1-hot OR to pick each bit answer.
91
Summary
Lessons for software design apply to hardware design!
ν Interface describes behavior of circuit.ν Implementation gives details of how to build it.
Layers of abstraction apply with a vengeance!ν On/off.ν Controlled switch. [relay, transistor]ν Gates. [AND, OR, NOT]ν Boolean circuit. [MAJ, ODD]ν Adder.ν Shifter.ν Arithmetic logic unit.ν …ν TOY machine.