This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CHAPTER 2 on...uuoouounonnuoero.IIInnuuuououuoon u    u a n n n nnouoo u u u u . UNIVERSAL BUILDING BLOCKS From now on, we can forget about wires and switches and
work with the abstraction of logic blocks operating on 1’s
and 0’s, a simple step that allows us to pass from the realm
of engineering into the realm of mathematics. This is the
most abstract chapter in the book; it will show you how the
methods used to construct a tictactoe machine can be used
to construct almost any function. In it, we‘ll deﬁne a power
ful set of building blocks: logical functions and ﬁnitestate
machines. With these elements, it’s easy to build a com
puter. LOGICAL FUNCTIONS In constructing the tictactoe machine, we began by writing
the game tree, whch gave us a set of rules for generating the
outputs from the inputs. This turns out to be a generally use—
ful method of attack. Once we write down the rules that
specify what outputs we want for each combination of inputs, we can build a device that implements these rules
using And, 01', and Invert functions. The logic blocks And. 2 22 THE PATTERN ON THE STONE Or, and Invert form a universal construction set, which
can be used to implement any set of rules. [These primi
tive types of logic blocks are sometimes also called logic
gates.) This idea of a universal set of blocks is important: it
means that the set is general enough to build anything. My
favorite toy when I was a child was a set of interlocking plas
tic bricks called Lego blocks, with which I built all kinds of
toys: cars, houses, spaceships, dinosaurs. I loved to play
with these blocks. but they were not quite universal, since
the only objects you could build with them had a certain
squarish, stairsteppy look. Building something with a differ—
ent shape—a cylinder or a sphere, for example—would
require a new type of block. Eventually, I had to switch to
another medium in order to build the things I wanted. But
the And, Or, and Invert blocks of Boolean logic are a univer
sal construction set for converting inputs to outputs. The
best way to see how they form a universal set is to under— stand a general method for using them to implement rules.
To start, we will consider binary rules—rules that specify
inputs and outputs that are either 1 or O. The tictactoe
machine is a good example of a function speciﬁed by binary
rules, because the input switches and the output lights are
either on or off—that is, either 1 or 0. (Later, we will discuss
rules for handling letters, numbers, or even pictures and
sounds as inputs and outputs.) Any set of binary rules can be
completely speciﬁed by showing a table of the outputs for
each possible combination of 1’s and 0’s on the inputs. For
example, the rules for the 01' function are speciﬁed by the following table: Input A Input B Output
0 0 0 1 1
0 1
1 1 UNIVERSAL BUILDING BLOCKS The Invert function is speciﬁed by an even simpler table: Input Output
Invert Function 0 1
1 0 For a binary function with 1: inputs, there are 2n possible
combinations of input signals. Sometimes we won’t bother to
specify all of them, because we don't care about certain com—
binations of inputs. For example, in specifying the function
performed by the tictac—toe machine, we don’t care what
happens if the human player plays in all squares simultane
ously. This move would be disallowed, and we don’t need to
specify the function’s output for this combination of inputs.
Complex logic blocks are constructed by connecting And,
Or, and Invert blocks. In drawings of the connection pattern,
the three blocks are conventionally represented by boxes of
different shape (see Figure 9]; the lines connecting on the left
side represent inputs to the blocks, and the lines connecting
on the right represent the output. Figure 10 shows how a
pair of twoinput 01‘ blocks can be connected to form a three
input 0r function; the output of this function will be 1 if any
one of its three inputs is 1. It’s also possible to string several
And blocks together in a similar manner to make an And block with any number of inputs. mmr A INPWA
WINE? W17“?
INF“1'3 INPUT 8—— INPu r——D)_ 0H.qu INVERT FIGURE 9
And, Or, and Invert Blocks 24 THE PATTERN ON THE STONE i; 2 FIGURE l0
A threeinput Or block made from a pair of twoinput 0r blocks Figure 11 shows how an And block can be constructed by
connecting an Inverter to the inputs and output of an 0r
block. (Here is De Morgan’s theorem again.) The best way to
get a feeling for how this works is to trace through the 1’s and
0’s for every combination of inputs. Notice that this illustra—
tion is essentially the same as Figure 6 in the previous chap—
ter. It points up an interesting fact: we don’t really need And
blocks in our universal building set, because we can always
construct them out of Dr blocks and Inverters. FIGURE I I
Making And out of Dr As in the tictactoe playing machine, And blocks are
used to detect each possible combination of inputs for which
the output is 1, while 0r blocks provide a roster of these
combinations. For example, let’s start with a simple function
of three inputs. Imagine that we want to build a block that
allows the three inputs to vote on the output. In this new UNIVERSAL BUILDING BLOCKS block, majority wins—that is, the output will be i only if two
or more of the inputs are 1. Majority
Inputs Output
ABC
000 0
001 0
010 0
011 1
100 0
101 1
110 1
111 1 Figure 12A shows how this function is irnplemented.An
And block with the appropriate Invert blocks as input is
used to recognize each combination of inputs for which the
output is 1; these blocks are connected by an 0r block,
which produces the output. This strategy can be used to cre
ate any transformation of inputs to outputs: Of course, this particular method of using a separate And
gate to recognize each combination of inputs is not the only
way to implement the function. and it is often not the sim
plest way. Figure 123 shows a simpler way to produce the
majority function. The great thing about the method de
scribed is not that it produces the best implementation but
that it always produces an implementation that works. The
important conclusion to draw is that it is possible to com
bine And, 01', and Invert blocks to implement any binary
function—that is, any function that can be speciﬁed by an
input/output table of 0's and 1’s. Restricting the inputs and output to binary numbers is
not really much of a restriction, because the combinations of
1’s and 0’s can be used to represent other things—letters, 25 mama? MAJOR! TY FIGURE I2 How the voting function is implemented by
And. Or, and Invert Blocks larger numbers, any entity that can be encoded. As an exam
ple of a nonbinary function, suppose we want to build a
machine to act as a judge of the children’s game of Scis
sors/Paper/Rock. This is a game for two players in which
each chooses, in secret, one of three “weapons”—scissors, UNIVERSAL BUILDING BLOCKS paper, or rock. The rules are simple: scissors cuts paper, paper
covers rock, rock crushes scissors. If the two children choose
the same weapon, they tie. Rather than building a machine
that plays the game (which would involve guessing which
weapon the opponent is going to choose], We will build a
machine that judges who wins. Here’s the input/output table
for the function that takes the choices as inputs and declares
the winner as output. The table encodes the rules of the game: Input A Input )3 Output
Scissors Scissors Tie Scissors Paper A wins
Scissors Rock B wins
Paper Scissors B wins
Paper Paper Tie Paper Rock A wins
Rock Scissors A wins
Rock Paper B wins
Rock Rock Tie The ScissorsPaper—Rock judging function is a combinations]
function, but it is not a binary function, since its inputs and
output have more than two possible values. To implement this
function as a combinational logic block, we must convert it to
a function of 1’s and 0's. This requires us to establish some
convention for representing the inputs and outputs. A simple
way to do this would be to use a separate bit for each of the
possibilities. There would be three input signals for each
weapon: a 1 on the ﬁrst input represents Scissors, a 1 on the
second input represents Rock, and a 1 on the third input rep
resents Paper. Similarly, we could use separate output lines to
represent a win for player A, a win for player B, or a tie. So the
box would have six inputs and three outputs. Using three input signals for each weapon is a perfectly
good way to build the function, but if we were doing it
inside a computer we would probably use some kind of 21 "“—— w*. w 7V— " THE FATTERN ON THE STONE UNIVERSAL BUILDING BLOCKS 29 encoding that required a smaller number of inputs and out
puts. For example, we could use two bits for each input and
use the combination 01 to represent Scissors, 10 to represent
Paper, and 11 to represent Rock. We could similarly encode
each of the possible outputs using two bits. This encoding
would result in the simpler threeinput/twooutput table shown below: A Inputs B Inputs Outputs D1 01 DD
01 10 10
Scissors = 01 01 11 01
Paper = 10 10 01 01
Rock = 11 10 10 00
A wins = 10 ‘10 11 10
B wins = 01 11 01 10
Tie = 00 11 10 01
11 11 00 Computers can use combinations of bits to represent any
thing; the number of bits depends on the number of messages
that need to be distinguished. Imagine, for example, a com
puter that works with the letters of the alphabet. Fivebit
input signals can represent thirtytwo different possibilities
[25 = 32]. Functions within the computer that work on letters
sometimes use such a code, although they more often use an
encoding with seven or eight bits, to allow representation of
capitals, punctuation marks, numerals, and so on. Most mod
em computers use the standard representation of alphabet
letters called ASCII (an acronym for American Standard Code
for Information Interchange]. In ASCII, the sequence 1000001
represents the capital letter A, and 1000010 represents the
capital B, and so on. The convention, of course, is arbitrary. Most computers have one or more conventions for repre
senting numbers. One of the most common is the base 2 rep
resentation of numbers, in which the bit sequence 0000000 represents zero, the sequence 0000001 represents the num
ber 1, the sequence 0000010 represents 2,and so on. The
description of computers as “64bit” or “32bi " indicates the
number of bit positions in the representation used by the
computer’s circuits: a 32bit computer uses a combination of
thirtytwo bits to represent a base2 number. The base2
number system is a common convention, but there is nothing
that requires its use. Some computers don’t use it at all, and
most computers that do also represent numbers in other
ways for various purposes. For instance, many computers
use a slightly different convention for representing negative
numbers and also have a convention called a ﬂoating point
to represent numbers that have decimal points. [The position
of the decimal point “ﬂoats” relative to the digits, so that a
ﬁxed number of digits can be used to represent a wide range
of numbers.) The particular representation schemes are often
chosen in such a way as to simplify the logic of the circuits
that perform arithmetical operations, or to make it easy to
convert from one representation to another. Because any logical function can be implemented as a
Boolean logic block, it is possible to build blocks that per
form arithmetical operations like addition or multiplication
by using numbers with any sort of representation. For
instance, imagine that we want to build a functional block
that will add numbers on an eightbit computer. An eightbit
adder block must have sixteen input signals (eight for each
of the numbers to be added], and eight output signals for the
sum. Since each number is represented by eight bits, there
are 256 possible combinations, and each can represent a dif~
ferent number. For example, we could use these combina
tions to represent the numbers between 0 and 255, or
between —100 and +154. Deﬁning the function of the block
would be just a matter of writing down the addition table
and then converting it to 1’5 and 0’s, using the chosen repre sentation. The table of 1’s and 0’s could then be converted to And and Dr blocks by the methods described above. 30 THE PATTERN ON THE STONE By adding two more inputs to the block, we could use
similar techniques to build a block that not only adds but
also subtracts, multiplies, and divides. The two extra control
inputs would specify which of these operations was to take
place. For instance, on every line of the table where the con—
trol inputs were 01, we would specify the output to be the
sum of the input numbers, whereas in every combination
where the control inputs were 10, we would specify the out—
puts to be the product, and so on. Most computers have logi
cal blocks of this type inside them called arithmetic units. Combining Ands and Dre according to this strategy is one
way to build any logical flmction, but it is not always the
most efﬁcient way. Often, by clever design, you can imple
ment a circuit using far fewer building blocks than the pre
ceding strategy requires. It may also be desirable to use other
types of building blocks or to design circuits that min
imize the delay from input to output. Here are some typical
puzzles in logic design: How do you use And blocks and
Inverters to construct Or blocks? [Easy] How do you use a
collection of And and Dr blocks, plus only two Inverters, to
construct the function of three Inverters? (Hard, but possi
ble.) Puzzles like this come up in the course of designing a
computer, which is part of what makes the process fun. FINITESTATE MACHINES IUUIOOI'IIIII'III'IIIIIIIIIDIIIIIII The methods I’ve described can be used to implement any
function that stays constant in time, but a more interesting
class of functions are those that involve sequences in time. To
handle such functions, we use a device called a finitestate
machine. Finite—state machines can be used to implement
timevarying flmctions—functions that depend not just on the
current input but also on the previous history of inputs. Once
you learn to recognize a ﬁnitestate machine, you'll notice i UNIVERSAL BUILDING BLOCKS them everywhere—in combination locks, ballpoint pens, even
legal contracts. The basic idea of a ﬁnitestate machine is to
combine a lookup table, constructed using Boolean logic,
with a memory device. The memory is used to store a sum
mary of the past, which is the state of the ﬁnitestate machine. A combination lock is a simple example of a ﬁnitestate
machine. The state of a combination lock is a summary of
the sequence of numbers dialed into the lock. The lock
doesn't remember all the numbers that have ever been dialed
into it, but it does remember enough about the most recent
numbers to know when they form the sequence that will
open the lock. An even simpler example of a ﬁnitestate
machine is the retractable ballpoint pen. This ﬁnitestate
machine has two possible states—extended and retracted—
and the pen remembers whether its button has been pressed
an odd or an even number of times. All ﬁnitestate machines
have a ﬁxed set of possible states, a set of allowable inputs
that change the state (clicking a pen’s button, or dialing a
number into a combination lock), and a set of possible out
puts (retracting or extending the ballpoint, opening the lock).
The outputs depend only on the state, which in turn
depends only on the history of the sequence of inputs. Another simple example of a ﬁnitestate machine is a
counter, such as the telly counter on a turnstile indicating
the number of people who have passed through. Each time a
new person goes through, the counter’s state is advanced by
one. The counter is a finite state because it can only count up
to a certain number of digits. When it reaches its maximum
count—say, BBQ—the next advance will cause it to return to
zero. Odometers on automobiles work like this. I once drove
an old Checker cab with an odometer that read 70,000, but I
never knew if the cab had traveled 70,000 miles, 170,000
miles, or 270,000 miles, because the odometer had only
100,000 States; all those histories were equivalent as far as
the odometer was concerned. This is why mathematicians
often deﬁne a state as “a set of equivalent histories.” 3 32W W 33 Other familiar examples of ﬁnitestate machines include
trafﬁc lights and elevatorbutton panels. In these machines,
the sequence of states is controlled by some combination of
an internal clock and input buttons such as the "Walk" but
ton at the crosswalk and the elevator call and floorselection
buttons. The next state of the machine depends not only on
the previous state but also on the signals that come from the
input button. The transition from one state to another is
determined by a ﬁxed set of rules. which can be summarized
by a simple state diagram showing the transition between
states. Figure 13 shows a state diagram for a traffic—light FIGURE H ‘ Finitestate machine, with logic block feeding register controller at an intersection where the light turns red in both
directions after the Walk button is pressed. Each drawing of
light represents a state and each arrow represents a transition
between states. The transition depends on whether or not the
“walk” button is pressed. To store the state of the ﬁnitestate machine, we need to
introduce one last building block—a device called a register;
which can be used to store bits. An nbit register has 1: inputs
and n outputs, plus an additional timing input that tells the
register when to change state. Storing new information is
called the state of the register. When the timing
signal tells the register to write a new state, the register
changes its state to match the inputs. The outputs of the reg
ister always indicate its current state. Registers can be imple
mented in many ways. one of which is to use a Boolean logic
block to steer the state information around in a circle. This
type of register is often used in electronic computers, which
is why they lose track of what they’re doing if their power is
interrupted. A ﬁnitestate machine consists of a Boolean logic block
connected to a register, as shown in Figure 14. The ﬁnite
state machine advances its state by writing the output of the i Fl 6 U RE I 3
. D State diagram for a traﬂiclight controller 34 THE PATTERN ON THE STONE Boolean logic block into the register; the logic block then
computes the next state, based on the input and the current
state. This next state is then written into the register on the
next cycle. The process repeats in every cycle. The function of a ﬁnite—state machine can be specified by
a table that shows, for every state and every input, the state
that follows. For example, we can summarize the operation
of the trafficlight controller by the following table: Inputs: Outputs: Walk Current Main Cross Next
Button State Road Road State
Not Pressed A Red Green B
Not Pressed B Red Yellow D
Not Pressed C Yellow Red A
Not Pressed D Green Red C
Not Pressed Walk Walk Walk D
Pressed A Red Green B
Pressed E Red Yellow Walk
Pressed C Yellow Red Walk
Pressed D Green Red C
Pressed Walk Walk Walk Walk The first step in implementing a ﬁnitestate machine is to
generate such a table. The second step is to assign a different
pattern of bits to each state. The five states of the trafﬁclight
controller will require three hits. [Since each bit doubles the
number of possible patterns, it is possible to store up to Zn
states using 11 bits.) By consistently replacing each word in the
preceding table with a binary pattern, we can convert the table
to a function that can be implemented with Boolean logic. In the trafﬁclight system, a timer controls the writing of
the register, which causes the state to change at regular inter
vals. Another example of a ﬁnitestate mac...
View
Full Document
 Spring '10
 ANDREDEHON
 Universal Building Blocks

Click to edit the document details