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CSE 599 Lecture 3: Digital Computing
 In the previous lectures, we examined:
 Theory of Computation
 Turing Machines and Automata
 Computability and Decidability
 Time and Space Complexity
 Today: Theory and Implementation of Digital Computers
 Guest Lecture by Prof. Chris Diorio on silicon integrated-circuit
technology
 Digital logic
 Digital computer organization and design
 Moore’s law and technology scaling
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History of Digital Computing
 ~1850: George Boole invents Boolean algebra
 Maps logical propositions to symbols
 Allows us to manipulate logic statements using mathematics
 1936: Alan Turing develops the formalism of Turing Machines
 1945: John von Neumann proposes the stored computer program concept
 1946: ENIAC: 18,000 tubes, several hundred multiplications per minute
 1947: Shockley, Brattain, and Bardeen invent the transistor
 1956: Harris introduces the first logic gate
 1972: Intel introduces the 4004 microprocessor
 Present: <0.2 m feature sizes; processors with >20-million transistors
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The mathematics: Boolean algebra
 A Boolean algebra consists of
 A set of elements B
 Binary operations {+ , •}
 A unary operation { ' }
 And the following axioms:
1. The set B contains at least two elements, a, b, such that a  b
2. Closure:
a + b is in B
a • b is in B
3. Commutative:
a+b=b+a
a•b=b•a
4. Associative:
a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c
5. Identity:
a+0=a
a•1=a
6. Distributive:
a + (b•c)=(a + b)•(a + c) a•(b + c)=(a•b) + (a•c)
7. Complementarity: a + a' = 1
a • a' = 0
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Binary logic is a Boolean algebra
 Substitute
 {0, 1} for B
 OR for +, AND for •
 NOT for '
 All the axioms hold for binary logic
 Definitions
 Boolean function: Maps inputs from the set {0,1} to the set {0,1}
 Boolean expression: An algebraic statement of Boolean variables and
operators
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What is digital hardware?
 Physical quantities (voltages) represent logical values
 If (0V < voltage < 0.8V), then symbol is a “0”
 If (2.0V < voltage < 5V), then symbol is a “1”
 Physical devices compute logical functions of their inputs
 E.g. AND, OR, NOT
 Set of n wires allow binary integers from 0 to 2n - 1
 How do we compute using digital hardware?
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Lowest Level: Transistors
 Transistors implement switches e.g. NOT, NAND, etc.
Output
Transfer Function
Output
Input
Input
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Switches allow digital logic
 Map problems (e.g. addition) to logical expressions
 Map logical expressions to switching devices
AND
A
B
Z = A and B
A
OR
Z = A or B
B
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Digital logic allows computation
 A NOR gate:
X
Y
Z
X
0
0
1
1
Y
0
1
0
1
Z
1
0
0
0
 NOR or NAND each form a complete operator
 Can form any Boolean expression using either of them
 Using only NOR gates and wire, you can build a general
purpose digital computer
 E.g. A one-bit memory (flip-flop)
R
Q
S
Q'
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Why do digital computers work like this?
 There is no compelling theoretical reason.
 Nothing from physics or chemistry, information theory, or CS
 The reason is mere expediency
 We build computers this way because we can.
 All the technology “fits”
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The Digital Computing Hierarchy
 A hierarchical approach allows general-purpose digital
computing:
 Transistors  switches  gates  combinational and
sequential logic  finite-state behavior  register-transfer
behavior  …
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Logic in digital computer design
 Digital logic: Circuit elements coding binary symbols
 Transistor switches have 2 simple states (on/off)
 Encode binary symbols implicitly
 Combinational logic: Circuits without memory
 Logic devices act as Boolean primitives
 Example: a NOR gate
 Allow arithmetic operators such as ADD to be constructed
 Sequential logic: Circuits with memory
 Feedback stores logic values
 Example: a flip-flop (also known as a latch)
 Allows registers and memory to be implemented
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Combinational versus sequential systems
 Combinational systems are memoryless
 The outputs depend only on the present inputs
Inputs
System
Outputs
 Sequential systems have memory
 The outputs depend on the present inputs and on the previous inputs
Inputs
System
Outputs
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Combinational logic gates
X•Y
X
Y
 OR
X+Y
X
 Buffer
X
X
 NOT
X
X
 AND
Y
Z
X
0
0
1
1
Y
0
1
0
1
Z
0
0
0
1
Z
X
0
0
1
1
Y
0
1
0
1
Z
0
1
1
1
Y
X
0
1
Y
0
1
Y
X
0
1
Y
1
0
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Combinational logic gates (cont.)
 NAND
 NOR
XY
XY
X
Y
XY
Y
0
1
0
1
Z
1
1
1
0
Z
X
0
0
1
1
Y
0
1
0
1
Z
1
0
0
0
Z
X
0
0
1
1
Y
0
1
0
1
Z
0
1
1
0
X
Y
 XOR
Z
X
0
0
1
1
X
Y
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Complete operators
 Can implement any logic function using only NOR or only NAND
 E.g. Logical inversion (NOT)
 NOR with both inputs tied together gives NOT
X
0
1
Y
0
1
X nor Y
1
0
 Noninverting functions
 Example: (X or Y) = not (X nor Y)
 In the above, use “not” constructed from a “nor” gate
 Can implement NAND and NOR from each other
 Example: X nand Y = not ((not X) nor (not Y))
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Mapping Boolean expressions to logic gates
c b
gh
 Example: Z  A  B  C  D  A  B  C  D
A
A
Z
B
C
D
Z
Z
B
C
D
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A binary decoder circuit
 Input: 2-digit binary number; Output: turn on 1 of 4 wires
 Truth Table:
A
B
Wire #
0
0
0
0
1
1
1
0
2
1
1
3
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A binary decoder circuit
 Input: 2-digit binary number AB; Output: 1 of 4 wires
 Circuit:
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A multiplexer circuit
 Goal: Select one of 4 input lines and pass the information on
that line to the single output line
 Circuit: Uses binary decoder plus an OR gate
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A multiplexer circuit
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Exercise: An Adder Circuit
 Design a circuit for adding two binary numbers
 First, write the truth table (input bits A and B, output bits SUM and
CARRY)
 Construct circuit using logic gates
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An Adder Circuit
 Truth table:
A
B
CARRY
SUM
0
0
0
0
0
1
0
1
1
0
0
1
1
1
1
0
 Circuit:
 Pick gates that match
the two outputs
SUM = A xor B
CARRY = A • B (i.e. A and B)
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A Full Adder
 Suppose you want to add 2 n-bit numbers
 Can you do this by using the previous 1-bit adder with two
inputs and two outputs?
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A Full Adder
 No, you need a 1-bit adder with three inputs: A, B and the
CARRY bit from the previous digit
 Then, to add 2 n-bit numbers, you can chain n 1-bit adders
together, with the CARRY output of one adder feeding into
the next adder
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A Full Adder
 Truth Table:
 SUM = ?
 CARRY = ?
A
B
C
CARRY
SUM
0
0
0
0
0
0
0
1
0
1
0
1
0
0
1
0
1
1
1
0
1
0
0
0
1
1
0
1
1
0
1
1
0
1
0
1
1
1
1
1
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A Full Adder
 Truth Table:
 SUM = (A xor B) xor C
 CARRY =
(A • B) + (A • C) + (B • C)
A
B
C
CARRY
SUM
0
0
0
0
0
0
0
1
0
1
0
1
0
0
1
0
1
1
1
0
1
0
0
0
1
1
0
1
1
0
1
1
0
1
0
1
1
1
1
1
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An Aside: Reversible logic gates
 Most Boolean gates are not reversible: Cannot construct input from
output (exceptions: NOT and buffer)
 Destroying information consumes energy – we will address this later when
discussing thermodynamics and quantum computers
 Two reversible gates: controlled not (CN) and controlled controlled not
(CCN).
A CCN gate
A CN gate
A
A'
A
A'
B
B'
B
B'
C
C'
A
0
0
1
1
B
0
1
0
1
A’
0
0
1
1
B’
0
1
1
0
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
A’
0
0
0
0
1
1
1
1
B’ C’
0 0
0 1
1 0
1 1
0 0
0 1
1 1
1 0
CCN is complete: we can form any Boolean
function using only CCN gates: e.g. AND if C = 0
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Sequential logic
 The devices
 Flip-flops
 Shift registers
 Finite state machines
 The concepts
 Sequential systems have memory
 The memory of a system is its state
 Sequential systems employ feedback
 Present inputs affect future outputs
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RS Flip-Flops
 Inputs: Set and Reset, Output: 2 stored bits that are
complementary. Example: Using NOR gates
Present Q S
R
Next Q
0
0
0
0
1
0
0
1
0
0
1
0
1
0
1
0
0
1
0
1
1
1
0
1
S
Not(Q)
Q
R
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The D flip-flop
 At the clock edge, the D flip-flop takes D to Q
 Internal feedback holds Q until next clock edge
Clock is a
periodic signal
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Shift registers
 Chain of D flip-flops: Stores sequences of bits
 Assume ABC stores some binary number xyz initially
 Stores 1 bit per clock cycle: ABC = xyz, 0yz, 10z, 010
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Finite state machines (FSMs)
 Consists of combinational logic and storage elements
 Localized feedback loops
 Sequential logic allows control of sequential algorithms
Inputs
Combinational
Logic
State Inputs
Outputs
State Outputs
Storage Elements
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Generalized FSM model
 State variables (state vector) describes circuit state
 We store state variables in memory (registers)
 Combinational logic computes next state and outputs
 Next state is a function of current state and inputs
Inputs
output
logic
next state
logic
Outputs
Next State
Current State
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Synchronous design using a clock
 Digital designs are almost always synchronous
 All voltages change at particular instants in time
 At a rising clock edge
 The computation is paced by the clock
 The clock hides transient behavior
 The clock forces the circuit to a known state at regular intervals
 Error-free sequencing of our algorithms
The circuit transitions to one among a finite number of states
at every clock edge
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Computer organization and design
 Computer design is an application of digital logic design
 Combinational and sequential logic
 Computer = Central processing unit + memory subsystem
 Central processing unit (CPU) = datapath + control
 Datapath = functional units + registers
 Functional units = ALU, adders, multipliers, dividers, etc.
 Registers = program counter, shifters, storage registers
 Control = finite state machine
 Instructions (fetch, decode, execute) tell the FSM what to do
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Computer structure
address
Processor
read/write
central processing
unit (CPU)
Control
data
control signals
Memory
System
Data Path
data conditions
instruction unit:
instruction fetch and
interpretation FSM
execution unit:
functional units
registers
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The processing unit
 First topic: The datapath
 Functional units operate on data
 ALU, adders, multipliers, ROM lookup tables, etc.
 Registers store and shift data and addresses
 Program counter, shifters, registers
 Second topic: The controller (control FSM)
 Finite state machine coordinates the processor’s operations
 Instructions (fetch, decode, execute) tell the FSM what to do
 Inputs = machine instruction, datapath conditions
 Outputs = register-transfer control signals, ALU op codes
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Datapath: Registers
 A collection of synchronous D flip-flops
 Load selectively using LD
 Read using OE (output enable)
LD
OE
D7
D6
D5
D4
D3
D2
D1
D0
Q7
Q6
Q5
Q4
Q3
Q2
Q1
Q0
CLK
8 bit register
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Datapath: Register files
 Collections of registers
 Two-dimensional array of flip-flops
 An address indexes a particular word
 Can have separate read and write addresses
 Can read and write simultaneously
 Example: 8 by 8 register file
 Uses 64 D flip-flops or eight 8-bit registers
(as in previous slide)
 Can store 8 words of 8 bits each
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Datapath: ALU
 General-purpose arithmetic logic unit
 Input: data and operation (derived from an op-code)
 Output: result and status
 Built from combinational logic like our ADDER circuit
Data
A
B
16
16
Operation
16
N
S
Z
Result and status
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Controlling the datapath: The control FSM
Reset
 Top level state diagram
 Reset
 Fetch instruction
 Decode
 Execute
 3 classes of instructions
 Branch
 Load/store
 Register-to-register operation
 Different sequence of states
for each instruction type
(PC = program counter)
Init
Initialize
Machine
Fetch
Instr.
Branch
Load/
Store
Branch
Taken
RegReg
Registerto-Register
Branch
Not Taken Incr.
PC
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Inside the control FSM
 Standard state-machine elements
 State registers
 Next-state combinational logic
 Output combinational logic (datapath/control signaling)
 “Control" registers
 Instruction register (IR)
 Program counter (PC)
 Inputs/outputs
 Outputs control datapath
 Inputs from datapath may alter program flow (e.g. branch if (x-y) =
0)
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Instructions versus Data: Harvard architecture
 Instructions and data stored
16
in two separate memories
 OP from control FSM
specifies ALU operation
load
path
REG
AC
16
16
store
path
OP
N
rd wr
data
Data Memory
(16-bit words)
addr
16
Z
16
IR
PC
16
16
Control
FSM
OP
data
Inst Memory
(8-bit words)
addr
16
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Communication: Buses
 Real processors have multiple buses
 Trade communication bandwidth versus hardware complexity
 Control FSM coordinates data transfer between registers
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The Key Points
 Digital computers are built from simple logic devices
 NOR, NAND, or other logic gates built from switches, which are
built from transistors, which are built on silicon wafers
 Hierarchy allows us to build complex computers
 Datapath comprises combinational circuits and registers
 Controller comprises finite state machines
 With NORs and wire, you can build the entire internet, with
every computer on it!
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So, where is digital computing headed?
 Technology has scaled exponentially the past few decades in
accordance with Moore’s law
 Chip complexity (transistor density) has doubled every 1.5
years, as “feature” sizes on a chip keep decreasing
Graph: Transistor density
versus minimum feature size
(feature size = width of wire on a chip)
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Clock speed has scaled exponentially
 Clock frequencies have doubled every ~3 years
Graph: Clock speed versus minimum feature size
From Sasaki, Multimedia:
Future and impact for
semiconductor scaling,
IEDM, 1997
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Drivers of semiconductor scaling
 Shrinking feature dimensions reduces energy consumption,
physical size, and interconnect length
 Energy consumption and physical size
 Power dissipation dominates chip design
 Power dissipation and size drive computer infrastructure
 Fans, heat sinks, etc. to get rid of heat
 More chips  bigger boards
 Interconnect (wire)
 Wire parasitics can dominate chip speed
 Resistance, capacitance, inductance, delay
 Increased noise (switching, coupling) and delay
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But, there are problems…
 Approaching physical, practical, and economic limits.
 Photolithography: etching circuits on silicon wafers
 Component sizes (~ 0.1 m) getting close to the wavelength of light
used for etching (mercury, pulsed excimer laser, x-rays (?)…)
 Tunneling effects: tiny distances between wires cause
electrons to leak across wire, corrupting the circuit…
 Clock speed so fast, signals can only travel a fraction of a
mm in one cycle – can’t reach all components…
 Component sizes at atomic scale – quantum laws take effect
 Economics: Fab lines too expensive, transistors too cheap…
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The end of scaling?
 Reasonable projections: We will be able to engineer devices
down to 0.03µm feature sizes
 ~10 more years of scaling
 Projected transistor density at a 0.03µm: 5 million / mm2
 A 15mm×15mm die can have ~ 1 billion transistors
 Issue 1: Power loss increases
 Issue 2: Building the interconnect becomes hard
 Projected clock rate at 0.03µm: 40GHz
 Signals can travel only 4mm in one clock period: can’t reach other
components?
 More details in the handouts…
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Conclusions
 Silicon technology takes a simple basis and allows us to
build complex computers
 Technology scaling will continue for ~10 years
 What do we do at the end?
 Bigger chips?
 Only if we increase the energy efficiency of our computations
 Deal with the interconnect
 Keep computations local and parallel
 Allow for transistor failure
 Reliable systems from unreliable components
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Conclusions (cont.)
 Change the information representation?
 Use transistors as more than switches
 Find alternative ways of computing in silicon
 Use alternative paradigms?
 The subject of this class
 Build chips based on neurons and biology?
 Use quantum effects of microscopic particles?
 Use molecules (DNA) for computation?
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Next Class: DNA Computing
 NOTE: Room Change for next class
 The next class will be held in SIEG 134
 Guest Lecture by Anne Condon, University of British
Columbia
 Followed by Discussion and Review
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Things to do this week…
Finish Homework Assignment # 2 (due next class 1/25)
Read handouts for information on Moore’s law and scaling
Browse the DNA computing links on course web page
Have a great weekend!
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