[MUSIC PLAYING]
PROFESSOR: I'd like to
welcome you to this
course on computer science.
Actually, that's a terrible
way to start.
Computer science is a terrible
name for this business.
First of all, it's
not a science.
It might be engineering or it
might be art, but we'll
actually see that computer
so-called science actually has
a lot in common with magic,
and we'll see
that in this course.
So it's not a science.
It's also not really very
much about computers.
And it's not about computers in
the same sense that physics
is not really about particle
accelerators, and biology is
not really about microscopes
and petri dishes.
And it's not about computers
in the same sense that
geometry is not really about
using surveying instruments.
In fact, there's a lot of
commonality between computer
science and geometry.
Geometry, first of all,
is another subject
with a lousy name.
The name comes from Gaia,
meaning the Earth, and metron,
meaning to measure.
Geometry originally
meant measuring
the Earth or surveying.
And the reason for that was
that, thousands of years ago,
the Egyptian priesthood
developed the rudiments of
geometry in order to figure
out how to restore the
boundaries of fields that were
destroyed in the annual
flooding of the Nile.
And to the Egyptians who did
that, geometry really was the
use of surveying instruments.
Now, the reason that we think
computer science is about
computers is pretty much the
same reason that the Egyptians
thought geometry was about
surveying instruments.
And that is, when some field
is just getting started and
you don't really understand it
very well, it's very easy to
confuse the essence of what
you're doing with the tools
that you use.
And indeed, on some absolute
scale of things, we probably
know less about the essence of
computer science than the
ancient Egyptians really
knew about geometry.
Well, what do I mean by the
essence of computer science?
What do I mean by the
essence of geometry?
See, it's certainly true that
these Egyptians went off and
used surveying instruments, but
when we look back on them
after a couple of thousand
years, we say, gee, what they
were doing, the important stuff
they were doing, was to
begin to formalize notions about
space and time, to start
a way of talking about
mathematical truths formally.
That led to the axiomatic
method.
That led to sort of all of
modern mathematics, figuring
out a way to talk precisely
about so-called declarative
knowledge, what is true.
Well, similarly, I think in the
future people will look
back and say, yes, those
primitives in the 20th century
were fiddling around with
these gadgets called
computers, but really what they
were doing is starting to
learn how to formalize
intuitions about process, how
to do things, starting to
develop a way to talk
precisely about how-to
knowledge, as opposed to
geometry that talks about
what is true.
Let me give you an
example of that.
Let's take a look.
Here is a piece of mathematics
that says what
a square root is.
The square root of X is the
number Y, such that Y squared
is equal to X and Y
is greater than 0.
Now, that's a fine piece of
mathematics, but just telling
you what a square root is
doesn't really say anything
about how you might go
out and find one.
So let's contrast that with a
piece of imperative knowledge,
how you might go out and
find a square root.
This, in fact, also comes
from Egypt, not
ancient, ancient Egypt.
This is an algorithm due to
Heron of Alexandria, called
how to find a square root
by successive averaging.
And what it says is that, in
order to find a square root,
you make a guess, you
improve that guess--
and the way you improve the
guess is to average the guess
and X over the guess, and we'll
talk a little bit later
about why that's a reasonable
thing--
and you keep improving the guess
until it's good enough.
That's a method.
That's how to do something
as opposed to declarative
knowledge that says what
you're looking for.
That's a process.
Well, what's a process
in general?
It's kind of hard to say.
You can think of it as like a
magical spirit that sort of
lives in the computer
and does something.
And the thing that directs a
process is a pattern of rules
called a procedure.
So procedures are the spells,
if you like, that control
these magical spirits that
are the processes.
I guess you know everyone needs
a magical language, and
sorcerers, real sorcerers, use
ancient Arcadian or Sumerian
or Babylonian or whatever.
We're going to conjure our
spirits in a magical language
called Lisp, which is a language
designed for talking
about, for casting the spells
that are procedures to direct
the processes.
Now, it's very easy
to learn Lisp.
In fact, in a few minutes,
I'm going to teach you,
essentially, all of Lisp.
I'm going to teach you,
essentially, all of the rules.
And you shouldn't find that
particularly surprising.
That's sort of like saying it's
very easy to learn the
rules of chess.
And indeed, in a few minutes,
you can tell somebody the
rules of chess.
But of course, that's very
different from saying you
understand the implications of
those rules and how to use
those rules to become a
masterful chess player.
Well, Lisp is the same way.
We're going to state the rules
in a few minutes, and it'll be
very easy to see.
But what's really hard is going
to be the implications
of those rules, how you exploit
those rules to be a
master programmer.
And the implications of those
rules are going to take us
the, well, the whole rest of
the subject and, of course,
way beyond.
OK, so in computer science,
we're in the business of
formalizing this sort of how-to
imperative knowledge,
how to do stuff.
And the real issues of computer
science are, of
course, not telling people
how to do square roots.
Because if that was
all it was, there
wouldn't be no big deal.
The real problems come when we
try to build very, very large
systems, computer programs that
are thousands of pages
long, so long that nobody can
really hold them in their
heads all at once.
And the only reason that that's
possible is because
there are techniques for
controlling the complexity of
these large systems. And these
techniques that are
controlling complexity
are what this
course is really about.
And in some sense, that's
really what
computer science is about.
Now, that may seem like a very
strange thing to say.
Because after all, a lot of
people besides computer
scientists deal with controlling
complexity.
A large airliner is an extremely
complex system, and
the aeronautical engineers who
design that are dealing with
immense complexity.
But there's a difference
between that kind of
complexity and what we deal
with in computer science.
And that is that computer
science, in some
sense, isn't real.
You see, when an engineer is
designing a physical system,
that's made out of real parts.
The engineers who worry about
that have to address problems
of tolerance and approximation
and noise in the system.
So for example, as an electrical
engineer, I can go
off and easily build a one-stage
amplifier or a
two-stage amplifier, and I can
imagine cascading a lot of
them to build a million-stage
amplifier.
But it's ridiculous to build
such a thing, because long
before the millionth stage,
the thermal noise in those
components way at the beginning
is going to get
amplified and make the whole
thing meaningless.
Computer science deals with
idealized components.
We know as much as we want about
these little program and
data pieces that we're fitting
things together.
We don't have to worry
about tolerance.
And that means that, in building
a large program,
there's not all that much
difference between what I can
build and what I can imagine,
because the parts are these
abstract entities that I
know as much as I want.
I know about them as precisely
as I'd like.
So as opposed to other kinds
of engineering, where the
constraints on what you can
build are the constraints of
physical systems, the
constraints of physics and
noise and approximation, the
constraints imposed in
building large software systems
are the limitations of
our own minds.
So in that sense, computer
science is like an abstract
form of engineering.
It's the kind of engineering
where you ignore the
constraints that are
imposed by reality.
Well, what are some of
these techniques?
They're not special to
computer science.
First technique, which is used
in all of engineering, is a
kind of abstraction called
black-box abstraction.
Take something and build
a box about it.
Let's see, for example, if we
looked at that square root
method, I might want to take
that and build a box.
That sort of says, to find the
square root of X. And that
might be a whole complicated
set of rules.
And that might end up being a
kind of thing where I can put
in, say, 36 and say, what's
the square root of 36?
And out comes six.
And the important thing is that
I'd like to design that
so that if George comes along
and would like to compute,
say, the square root of A plus
the square root of B, he can
take this thing and use it as
a module without having to
look inside and build something
that looks like
this, like an A and a B and a
square root box and another
square root box and then
something that adds that would
put out the answer.
And you can see, just from the
fact that I want to do that,
is from George's point of view,
the internals of what's
in here should not
be important.
So for instance, it shouldn't
matter that, when I wrote
this, I said I want to find the
square root of X. I could
have said the square root of Y,
or the square root of A, or
anything at all.
That's the fundamental notion of
putting something in a box
using black-box abstraction
to suppress detail.
And the reason for that is you
want to go off and build
bigger boxes.
Now, there's another reason
for doing black-box
abstraction other than you want
to suppress detail for
building bigger boxes.
Sometimes you want to say that
your way of doing something,
your how-to method, is an
instance of a more general
thing, and you'd like your
language to be able to express
that generality.
Let me show you another example
sticking with square roots.
Let's go back and take another
look at that slide with the
square root algorithm on it.
Remember what that says.
That says, in order to do
something, I make a guess, and
I improve that guess,
and I sort of keep
improving that guess.
So there's the general strategy
of, I'm looking for
something, and the way
I find it is that I
keep improving it.
Now, that's a particular case
of another kind of strategy
for finding a fixed point
of something.
So you have a fixed point
of a function.
A fixed point of a function
is something, is a value.
A fixed point of a function F is
a value Y, such that F of Y
equals Y. And the way I might do
that is start with a guess.
And then if I want something
that doesn't change when I
keep applying F, is I'll keep
applying F over and over until
that result doesn't
change very much.
So there's a general strategy.
And then, for example, to
compute the square root of X,
I can try and find a fixed point
of the function which
takes Y to the average of X/Y.
And the idea that is that if I
really had Y equal to the square
root of X, then Y and
X/Y would be the same value.
They'd both be the square root
of X, because X over the
square root of X is the
square root of X.
And so the average if Y were
equal to the square of X, then
the average wouldn't change.
So the square root of X
is a fixed point of
that particular function.
Now, what I'd like to have,
I'd like to express the
general strategy for finding
fixed points.
So what I might imagine doing,
is to find, is to be able to
use my language to define a box
that says "fixed point,"
just like I could make a box
that says "square root." And
I'd like to be able to express
this in my language.
So I'd like to express not only
the imperative how-to
knowledge of a particular thing
like square root, but
I'd like to be able to express
the imperative knowledge of
how to do a general thing like
how to find fixed point.
And in fact, let's go back and
look at that slide again.
See, not only is this a piece
of imperative knowledge, how
to find a fixed point, but
over here on the bottom,
there's another piece of
imperative knowledge which
says, one way to compute square
root is to apply this
general fixed point method.
So I'd like to also
be able to express
that imperative knowledge.
What would that look like?
That would say, this fixed point
box is such that if I
input to it the function that
takes Y to the average of Y
and X/Y, then what should come
out of that fixed point box is
a method for finding
square roots.
So in these boxes we're
building, we're not only
building boxes that you input
numbers and output numbers,
we're going to be building in
boxes that, in effect, compute
methods like finding
square root.
And my take is their inputs
functions, like Y goes to the
average of Y and X/Y. The reason
we want to do that, the
reason this is a procedure, will
end up being a procedure,
as we'll see, whose value is
another procedure, the reason
we want to do that is because
procedures are going to be our
ways of talking about imperative
knowledge.
And the way to make that very
powerful is to be able to talk
about other kinds
of knowledge.
So here is a procedure that, in
effect, talks about another
procedure, a general strategy
that itself talks about
general strategies.
Well, our first topic in this
course-- there'll be three
major topics-- will be black-box
abstraction.
Let's look at that in a little
bit more detail.
What we're going to do is we
will start out talking about
how Lisp is built up out
of primitive objects.
What does the language
supply with us?
And we'll see that there are
primitive procedures and
primitive data.
Then we're going to see, how do
you take those primitives
and combine them to make more
complicated things, means of
combination?
And what we'll see is that
there are ways of putting
things together, putting
primitive procedures together
to make more complicated
procedures.
And we'll see how to put
primitive data together to
make compound data.
Then we'll say, well, having
made those compounds things,
how do you abstract them?
How do you put those black boxes
around them so you can
use them as components in
more complex things?
And we'll see that's done by
defining procedures and a
technique for dealing with
compound data called data
abstraction.
And then, what's maybe the most
important thing, is going
from just the rules to how
does an expert work?
How do you express common
patterns of doing things, like
saying, well, there's a general
method of fixed point
and square root is a particular
case of that?
And we're going to use--
I've already hinted at it--
something called higher-order
procedures, namely procedures
whose inputs and outputs are
themselves procedures.
And then we'll also see
something very interesting.
We'll see, as we go further and
further on and become more
abstract, there'll be very--
well, the line between what we
consider to be data and what
we consider to be procedures
is going to blur at an
incredible rate.
Well, that's our first
subject, black-box
abstraction.
Let's look at the
second topic.
I can introduce it like this.
See, suppose I want to
express the idea--
remember, we're talking
about ideas--
suppose I want to express the
idea that I can take something
and multiply it by the sum
of two other things.
So for example, I might say,
if I had one and three and
multiply that by two,
I get eight.
But I'm talking about the
general idea of what's called
linear combination, that you
can add two things and
multiply them by
something else.
It's very easy when I think
about it for numbers, but
suppose I also want to use that
same idea to think about,
I could add two vectors, a1 and
a2, and then scale them by
some factor x and get
another vector.
Or I might say, I want to think
about a1 and a2 as being
polynomials, and I might want
to add those two polynomials
and then multiply them by two to
get a more complicated one.
Or a1 and a2 might be electrical
signals, and I
might want to think about
summing those two electrical
signals and then putting the
whole thing through an
amplifier, multiplying
it by some
factor of two or something.
The idea is I want to
think about the
general notion of that.
Now, if our language is going
to be good language for
expressing those kind of general
ideas, if I really,
really can do that, I'd like to
be able to say I'm going to
multiply by x the sum of a1 and
a2, and I'd like that to
express the general idea of all
different kinds of things
that a1 and a2 could be.
Now, if you think about that,
there's a problem, because
after all, the actual primitive
operations that go
on in the machine are obviously
going to be
different if I'm adding two
numbers than if I'm adding two
polynomials, or if I'm adding
the representation of two
electrical signals
or wave forms.
Somewhere, there has to be the
knowledge of the kinds of
various things that you
can add and the
ways of adding them.
Now, to construct such a system,
the question is, where
do I put that knowledge?
How do I think about
the different kinds
of choices I have?
And if tomorrow George comes up
with a new kind of object
that might be added and
multiplied, how do I add
George's new object to the
system without screwing up
everything that was
already there?
Well, that's going to be the
second big topic, the way of
controlling that kind
of complexity.
And the way you do that is by
establishing conventional
interfaces, agreed upon ways of
plugging things together.
Just like in electrical
engineering, people have
standard impedances for
connectors, and then you know
if you build something with
one of those standard
impedances, you can plug it
together with something else.
So that's going to be our
second large topic,
conventional interfaces.
What we're going to see is,
first, we're going to talk
about the problem of generic
operations, which is the one I
alluded to, things like "plus"
that have to work with all
different kinds of data.
So we talk about generic
operations.
Then we're going to talk about
really large-scale structures.
How do you put together very
large programs that model the
kinds of complex systems
in the real world that
you'd like to model?
And what we're going to see
is that there are two very
important metaphors for putting
together such systems.
One is called object-oriented
programming, where you sort of
think of your system as a kind
of society full of little
things that interact by sending
information between them.
And then the second one is
operations on aggregates,
called streams, where you think
of a large system put
together kind of like a signal
processing engineer puts
together a large electrical
system.
That's going to be
our second topic.
Now, the third thing we're going
to come to, the third
basic technique for controlling
complexity, is
making new languages.
Because sometimes, when you're
sort of overwhelmed by the
complexity of a design, the
way that you control that
complexity is to pick a
new design language.
And the purpose of the new
design language will be to
highlight different aspects
of the system.
It will suppress some kinds of
details and emphasize other
kinds of details.
This is going to be the most
magical part of the course.
We're going to start out by
actually looking at the
technology for building new
computer languages.
The first thing we're going to
do is actually build in Lisp.
We're going to express in Lisp
the process of interpreting
Lisp itself.
And that's going to be a very
sort of self-circular thing.
There's a little mystical
symbol that
has to do with that.
The process of interpreting Lisp
is sort of a giant wheel
of two processes, apply and
eval, which sort of constantly
reduce expressions
to each other.
Then we're going to see all
sorts of other magical things.
Here's another magical symbol.
This is sort of the Y operator,
which is, in some
sense, the expression
of infinity inside
our procedural language.
We'll take a look at that.
In any case, this section
of the course is called
Metalinguistic Abstraction,
abstracting by talking about
how you construct
new languages.
As I said, we're going to start
out by looking at the
process of interpretation.
We're going to look
at this apply-eval
loop, and build Lisp.
Then, just to show you that this
is very general, we're
going to use exactly the same
technology to build a very
different kind of language, a
so-called logic programming
language, where you don't really
talk about procedures
at all that have inputs
and outputs.
What you do is talk about
relations between things.
And then finally, we're going
to talk about how you
implement these things very
concretely on the very
simplest kind of machines.
We'll see something like this.
This is a picture of a chip,
which is the Lisp interpreter
that we will be talking about
then in hardware.
Well, there's an outline of the
course, three big topics.
Black-box abstraction,
conventional interfaces,
metalinguistic abstraction.
Now, let's take a break now and
then we'll get started.
[MUSIC PLAYING]
Let's actually start in
learning Lisp now.
Actually, we'll start out by
learning something much more
important, maybe the very most
important thing in this
course, which is not Lisp, in
particular, of course, but
rather a general framework for
thinking about languages that
I already alluded to.
When somebody tells you they're
going to show you a
language, what you should say
is, what I'd like you to tell
me is what are the primitive
elements?
What does the language
come with?
Then, what are the ways you
put those together?
What are the means
of combination?
What are the things that allow
you to take these primitive
elements and build bigger
things out of them?
What are the ways of putting
things together?
And then, what are the
means of abstraction?
How do we take those complicated
things and draw
those boxes around them?
How do we name them so that we
can now use them as if they
were primitive elements
in making still
more complex things?
And so on, and so
on, and so on.
So when someone says to you,
gee, I have a great new
computer language, you don't
say, how many characters does
it take to invert a matrix?
It's irrelevant.
What you say is, if the language
did not come with
matrices built in or with
something else built in, how
could I then build that thing?
What are the means of
combination which would allow
me to do that?
And then, what are the means of
abstraction which allow me
then to use those as elements
in making more complicated
things yet?
Well, we're going to see that
Lisp has some primitive data
and some primitive procedures.
In fact, let's really start.
And here's a piece of
primitive data in
Lisp, number three.
Actually, if I'm being very
pedantic, that's not the
number three.
That's some symbol that
represents Plato's concept of
the number three.
And here's another.
Here's some more primitive
data in Lisp, 17.4.
Or actually, some representation
of 17.4.
And here's another one, five.
Here's another primitive
object that's
built in Lisp, addition.
Actually, to use the same kind
of pedantic-- this is a name
for the primitive method
of adding things.
Just like this is a name for
Plato's number three, this is
a name for Plato's concept
of how you add things.
So those are some primitive
elements.
I can put them together.
I can say, gee, what's the sum
of three and 17.4 and five?
And the way I do that is to
say, let's apply the sum
operator to these
three numbers.
And I should get,
what? eight, 17.
25.4.
So I should be able to ask Lisp
what the value of this
is, and it will return 25.4.
Let's introduce some names.
This thing that I typed is
called a combination.
And a combination consists,
in general,
of applying an operator--
so this is an operator--
to some operands.
These are the operands.
And of course, I can make
more complex things.
The reason I can get complexity
out of this is
because the operands themselves,
in general, can be
combinations.
So for instance, I could say,
what is the sum of three and
the product of five and
six and eight and two?
And I should get-- let's see--
30, 40, 43.
So Lisp should tell
me that that's 43.
Forming combinations is the
basic needs of combination
that we'll be looking at.
And then, well, you see
some syntax here.
Lisp uses what's called prefix
notation, which means that the
operator is written to the
left of the operands.
It's just a convention.
And notice, it's fully
parenthesized.
And the parentheses make it
completely unambiguous.
So by looking at this, I can see
that there's the operator,
and there are one, two,
three, four operands.
And I can see that the second
operand here is itself some
combination that has one
operator and two operands.
Parentheses in Lisp are a little
bit, or are very unlike
parentheses in conventional
mathematics.
In mathematics, we sort of use
them to mean grouping, and it
sort of doesn't hurt if
sometimes you leave out
parentheses if people
understand
that that's a group.
And in general, it doesn't
hurt if you put in extra
parentheses, because that
maybe makes the
grouping more distinct.
Lisp is not like that.
In Lisp, you cannot leave out
parentheses, and you cannot
put in extra parentheses,
because putting in parentheses
always means, exactly and
precisely, this is a
combination which has
meaning, applying
operators to operands.
And if I left this out, if I
left those parentheses out, it
would mean something else.
In fact, the way to think about
this, is really what I'm
doing when I write something
like this is writing a tree.
So this combination is a tree
that has a plus and then a
thee and then a something else
and an eight and a two.
And then this something else
here is itself a little
subtree that has a star
and a five and a six.
And the way to think of that
is, really, what's going on
are we're writing these trees,
and parentheses are just a way
to write this two-dimensional
structure as a linear
character string.
Because at least when Lisp first
started and people had
teletypes or punch cards or
whatever, this was more
convenient.
Maybe if Lisp started today,
the syntax of Lisp
would look like that.
Well, let's look at
what that actually
looks like on the computer.
Here I have a Lisp interaction
set up.
There's a editor.
And on the top, I'm going to
type some values and ask Lisp
what they are.
So for instance, I can say
to Lisp, what's the
value of that symbol?
That's three.
And I ask Lisp to evaluate it.
And there you see Lisp has
returned on the bottom, and
said, oh yeah, that's three.
Or I can say, what's the sum of
three and four and eight?
What's that combination?
And ask Lisp to evaluate it.
That's 15.
Or I can type in something
more complicated.
I can say, what's the sum of the
product of three and the
sum of seven and 19.5?
And you'll notice here that Lisp
has something built in
that helps me keep track of
all these parentheses.
Watch as I type the next closed
parentheses, which is
going to close the combination
starting with the star.
The opening one will flash.
Here, I'll rub those out
and do it again.
Type close, and you see
that closes the plus.
Close again, that
closes the star.
Now I'm back to the sum, and
maybe I'm going to add that
all to four.
That closes the plus.
Now I have a complete
combination, and I can ask
Lisp for the value of that.
That kind of paren balancing is
something that's built into
a lot of Lisp systems to help
you keep track, because it is
kind of hard just by hand doing
all these parentheses.
There's another kind of
convention for keeping track
of parentheses.
Let me write another complicated
combination.
Let's take the sum of the
product of three and five and
add that to something.
And now what I'm going to do is
I'm going to indent so that
the operands are written
vertically.
Which the sum of that and
the product of 47 and--
let's say the product
of 47 with a
difference of 20 and 6.8.
That means subtract
6.8 from 20.
And then you see the
parentheses close.
Close the minus.
Close the star.
And now let's get another
operator.
You see the Lisp editor here
is indenting to the right
position automatically to
help me keep track.
I'll do that again.
I'll close that last
parentheses again.
You see it balances the plus.
Now I can say, what's
the value of that?
So those two things, indenting
to the right level, which is
called pretty printing, and
flashing parentheses, are two
things that a lot of Lisp
systems have built in to help
you keep track.
And you should learn
how to use them.
Well, those are the
primitives.
There's a means of
combination.
Now let's go up to the
means of abstraction.
I'd like to be able to take
the idea that I do some
combination like this, and
abstract it and give it a
simple name, so I can use
that as an element.
And I do that in Lisp with
"define." So I can say, for
example, define A to be the
product of five and five.
And now I could say, for
example, to Lisp, what is the
product of A and A?
And this should be 25, and
this should be 625.
And then, crucial thing,
I can now use A--
here I've used it in
a combination--
but I could use that in other
more complicated things that I
name in turn.
So I could say, define B to be
the sum of, we'll say, A and
the product of five and A.
And then close the plus.
Let's take a look at that
on the computer and
see how that looks.
So I'll just type what
I wrote on the board.
I could say, define A to be the
product of five and five.
And I'll tell that to Lisp.
And notice what Lisp responded
there with
was an A in the bottom.
In general, when you type in
a definition in Lisp, it
responds with the symbol
being defined.
Now I could say to Lisp, what
is the product of A and A?
And it says that's 625.
I can define B to be the sum of
A and the product of five
and A. Close a paren
closes the star.
Close the plus.
Close the "define." Lisp says,
OK, B, there on the bottom.
And now I can say to Lisp,
what's the value of B?
And I can say something more
complicated, like what's the
sum of A and the quotient
of B and five?
That slash is divide, another
primitive operator.
I've divided B by five,
added it to A. Lisp
says, OK, that's 55.
So there's what it looks like.
There's the basic means
of defining something.
It's the simplest kind of
naming, but it's not really
very powerful.
See, what I'd really
like to name--
remember, we're talking about
general methods--
I'd like to name, oh, the
general idea that, for
example, I could multiply five
by five, or six by six, or
1,001 by 1,001, 1,001.7
by 1,001.7.
I'd like to be able to name
the general idea of
multiplying something
by itself.
Well, you know what that is.
That's called squaring.
And the way I can do that in
Lisp is I can say, define to
square something x, multiply
x by itself.
And then having done that,
I could say to Lisp, for
example, what's the
square of 10?
And Lisp will say 100.
So now let's actually look at
that a little more closely.
Right, there's the definition
of square.
To square something, multiply
it by itself.
You see this x here.
That x is kind of a pronoun,
which is the something that
I'm going to square.
And what I do with it
is I multiply x, I
multiply it by itself.
OK.
So there's the notation for
defining a procedure.
Actually, this is a little bit
confusing, because this is
sort of how I might
use square.
And I say square root of x or
square root of 10, but it's
not making it very clear that
I'm actually naming something.
So let me write this definition
in another way that
makes it a little
bit more clear
that I'm naming something.
I'll say, "define" square to
be lambda of x times xx.
Here, I'm naming something
square, just like over here,
I'm naming something A. The
thing that I'm naming square--
here, the thing I named A was
the value of this combination.
Here, the thing that I'm naming
square is this thing
that begins with lambda, and
lambda is Lisp's way of saying
make a procedure.
Let's look at that more
closely on the slide.
The way I read that definition
is to say, I define square to
be make a procedure--
that's what the lambda is--
make a procedure with
an argument named x.
And what it does is return
the results of
multiplying x by itself.
Now, in general, we're going to
be using this top form of
defining, just because it's a
little bit more convenient.
But don't lose sight of the fact
that it's really this.
In fact, as far as the Lisp
interpreter's concerned,
there's no difference between
typing this to it and typing
this to it.
And there's a word for that,
sort of syntactic sugar.
What syntactic sugar means,
it's having somewhat more
convenient surface forms
for typing something.
So this is just really syntactic
sugar for this
underlying Greek thing
with the lambda.
And the reason you should
remember that is don't forget
that, when I write something
like this, I'm
really naming something.
I'm naming something square,
and the something that I'm
naming square is a procedure
that's getting constructed.
Well, let's look at that
on the computer, too.
So I'll come and I'll say,
define square of
x to be times xx.
Now I'll tell Lisp that.
It says "square." See, I've
named something "square." Now,
having done that, I can
ask Lisp for, what's
the square of 1,001?
Or in general, I could say,
what's the square of the sum
of five and seven?
The square of 12's 144.
Or I can use square itself
as an element in some
combination.
I can say, what's the sum of
the square of three and the
square of four?
nine and 16 is 25.
Or I can use square as an
element in some much more
complicated thing.
I can say, what's the square
of, the sqare of,
the square of 1,001?
And there's the square of the
square of the square of 1,001.
Or I can say to Lisp, what
is square itself?
What's the value of that?
And Lisp returns some
conventional way of telling me
that that's a procedure.
It says, "compound procedure
square." Remember, the value
of square is this procedure, and
the thing with the stars
and the brackets are just Lisp's
conventional way of
describing that.
Let's look at two more
examples of defining.
Here are two more procedures.
I can define the average of x
and y to be the sum of x and y
divided by two.
Or having had average and mean
square, having had average and
square, I can use that to talk
about the mean square of
something, which is the average
of the square of x and
the square of y.
So for example, having done
that, I could say, what's the
mean square of two and three?
And I should get the
average of four and
nine, which is 6.5.
The key thing here is that,
having defined square, I can
use it as if it were
primitive.
So if we look here on the
slide, if I look at mean
square, the person defining mean
square doesn't have to
know, at this point, whether
square was something built
into the language or
whether it was a
procedure that was defined.
And that's a key thing in Lisp,
that you do not make
arbitrary distinctions between
things that happen to be
primitive in the language
and things that
happen to be built in.
A person using that shouldn't
even have to know.
So the things you construct get
used with all the power
and flexibility as if they
were primitives.
In fact, you can drive that
home by looking on the
computer one more time.
We talked about plus.
And in fact, if I come here on
the computer screen and say,
what is the value of plus?
Notice what Lisp types out.
On the bottom there, it typed
out, "compound procedure
plus." Because, in this system,
it turns out that the
addition operator is itself
a compound procedure.
And if I didn't just type that
in, you'd never know that, and
it wouldn't make any
difference anyway.
We don't care.
It's below the level of
the abstraction that
we're dealing with.
So the key thing is you cannot
tell, should not be able to
tell, in general, the difference
between things that
are built in and things
that are compound.
Why is that?
Because the things that are
compound have an abstraction
wrapper wrapped around them.
We've seen almost all the
elements of Lisp now.
There's only one more we have to
look at, and that is how to
make a case analysis.
Let me show you what I mean.
We might want to think about the
mathematical definition of
the absolute value functions.
I might say the absolute value
of x is the function which has
the property that it's
negative of x.
For x less than zero, it's
zero for x equal to zero.
And it's x for x greater
than zero.
And Lisp has a way of making
case analyses.
Let me define for you
absolute value.
Say define the absolute value
of x is conditional.
This means case analysis,
COND.
If x is less than zero, the
answer is negate x.
What I've written here
is a clause.
This whole thing is a
conditional clause,
and it has two parts.
This part here is a predicate
or a condition.
That's a condition.
And the condition is expressed
by something called a
predicate, and a predicate in
Lisp is some sort of thing
that returns either
true or false.
And you see Lisp has a
primitive procedure,
less-than, that tests whether
something is true or false.
And the other part of a clause
is an action or a thing to do,
in the case where that's true.
And here, what I'm doing
is negating x.
The negation operator, the
minus sign in Lisp is
a little bit funny.
If there's two or more
arguments, if there's two
arguments it subtracts the
second one from the first, and
we saw that.
And if there's one argument,
it negates it.
So this corresponds to that.
And then there's another
COND clause.
It says, in the case where
x is equal to zero,
the answer is zero.
And in the case where x
is greater than zero,
the answer is x.
Close that clause.
Close the COND.
Close the definition.
And there's the definition
of absolute value.
And you see it's the case
analysis that looks very much
like the case analysis you
use in mathematics.
There's a somewhat different
way of writing a restricted
case analysis.
Often, you have a case analysis
where you only have
one case, where you test
something, and then depending
on whether it's true or false,
you do something.
And here's another definition of
absolute value which looks
almost the same, which says,
if x is less than zero, the
result is negate x.
Otherwise, the answer is x.
And we'll be using "if" a lot.
But again, the thing to remember
is that this form of
absolute value that you're
looking at here, and then this
one over here that I wrote
on the board, are
essentially the same.
And "if" and COND are--
well, whichever way
you like it.
You can think of COND as
syntactic sugar for "if," or
you can think of "if" as
syntactic sugar for COND, and
it doesn't make any
difference.
The person implementing a Lisp
system will pick one and
implement the other
in terms of that.
And it doesn't matter
which one you pick.
Why don't we break now, and
then take some questions.
How come sometimes when I write
define, I put an open
paren here and say, define open
paren something or other,
and sometimes when
I write this, I
don't put an open paren?
The answer is, this particular
form of "define," where you
say define some expression, is
this very special thing for
defining procedures.
But again, what it really means
is I'm defining this
symbol, square, to be that.
So the way you should think
about it is what "define" does
is you write "define," and the
second thing you write is the
symbol here-- no open paren--
the symbol you're defining and
what you're defining it to be.
That's like here
and like here.
That's sort of the basic way
you use "define." And then,
there's this special syntactic
trick which allows you to
define procedures that
look like this.
So the difference is, it's
whether or not you're defining
a procedure.
[MUSIC PLAYING]
Well, believe it or not, you
actually now know enough Lisp
to write essentially any
numerical procedure that you'd
write in a language like FORTRAN
or Basic or whatever,
or, essentially, any
other language.
And you're probably saying,
that's not believable, because
you know that these languages
have things like "for
statements," and "do until
while" or something.
But we don't really
need any of that.
In fact, we're not going
to use any of
that in this course.
Let me show you.
Again, looking back at square
root, let's go back to this
square root algorithm of
Heron of Alexandria.
Remember what that said.
It said, to find an
approximation to the square
root of X, you make a guess,
you improve that guess by
averaging the guess and
X over the guess.
You keep improving that until
the guess is good enough.
I already alluded to the idea.
The idea is that, if the initial
guess that you took
was actually equal to the square
root of X, then G here
would be equal to X/G.
So if you hit the square
root, averaging them
wouldn't change it.
If the G that you picked was
larger than the square root of
X, then X/G will be smaller than
the square root of X, so
that when you average
G and X/G, you get
something in between.
So if you pick a G that's
too small, your
answer will be too large.
If you pick a G that's too
large, if your G is larger
than the square root of X and
X/G will be smaller than the
square root of X.
So averaging always gives you
something in between.
And then, it's not quite
trivial, but it's possible to
show that, in fact, if G misses
the square root of X by
a little bit, the average of G
and X/G will actually keep
getting closer to the square
root of X. So if you keep
doing this enough, you'll
eventually get as
close as you want.
And then there's another fact,
that you can always start out
this process by using 1
as an initial guess.
And it'll always converge to
the square root of X. So
that's this method of successive
averaging due to
Heron of Alexandria.
Let's write it in Lisp.
Well, the central idea is, what
does it mean to try a
guess for the square
root of X?
Let's write that.
So we'll say, define to try a
guess for the square root of
X, what do we do?
We'll say, if the guess is good
enough to be a guess for
the square root of X,
then, as an answer,
we'll take the guess.
Otherwise, we will try
the improved guess.
We'll improve that guess for
the square root of X, and
we'll try that as a guess for
the square root of X. Close
the "try." Close the "if." Close
the "define." So that's
how we try a guess.
And then, the next part of the
process said, in order to
compute square roots, we'll
say, define to compute the
square root of X, we will try
one as a guess for the square
root of X. Well, we have to
define a couple more things.
We have to say, how is
a guess good enough?
And how do we improve a guess?
So let's look at that.
The algorithm to improve a guess
for the square root of
X, we average--
that was the algorithm--
we average the guess with
the quotient of
dividing X by the guess.
That's how we improve a guess.
And to tell whether a guess is
good enough, well, we have to
decide something.
This is supposed to be a guess
for the square root of X, so
one possible thing you can do
is say, when you take that
guess and square it, do you get
something very close to X?
So one way to say that is to
say, I square the guess,
subtract X from that, and see if
the absolute value of that
whole thing is less than some
small number, which depends on
my purposes.
So there's a complete procedure
for how to compute
the square root of X. Let's look
at the structure of that
a little bit.
I have the whole thing.
I have the notion of how to
compute a square root.
That's some kind of module.
That's some kind of black box.
It's defined in terms of how to
try a guess for the square
root of X.
"Try" is defined in terms of,
well, telling whether
something is good enough
and telling
how to improve something.
So good enough.
"Try" is defined in terms of
"good enough" and "improve."
And let's see what
else I fill in.
Well, I'll go down this tree.
"Good enough" was defined
in terms of
absolute value, and square.
And improve was defined in
terms of something called
averaging and then some other
primitive operator.
Square root's defined in terms
of "try." "Try" is defined in
terms of "good enough"
and "improve,"
but also "try" itself.
So "try" is also defined in
terms of how to try itself.
Well, that may give you some
problems. Your high school
geometry teacher probably told
you that it's naughty to try
and define things in terms of
themselves, because it doesn't
make sense.
But that's false.
Sometimes it makes perfect
sense to define things in
terms of themselves.
And this is the case.
And we can look at that.
We could write down what this
means, and say, suppose I
asked Lisp what the square
root of two is.
What's the square root
of two mean?
Well, that means I try one
as a guess for the
square root of two.
Now I look.
I say, gee, is one a good enough
guess for the square
root of two?
And that depends on the test
that "good enough" does.
And in this case, "good enough"
will say, no, one is
not a good enough guess for
the square root of two.
So that will reduce to saying,
I have to try an improved--
improve one as a guess for the
square root of two, and try
that as a guess for the
square root of two.
Improving one as a guess for the
square root of two means I
average one and two
divided by one.
So this is going
to be average.
This piece here will be the
average of one and the
quotient of two by one.
That's this piece here.
And this is 1.5.
So this square root of two
reduces to trying one for the
square root of two, which
reduces to trying 1.5 as a
guess for the square
root of two.
So that makes sense.
Let's look at the rest
of the process.
If I try 1.5, that reduces.
1.5 turns out to be not good
enough as a guess for the
square root of two.
So that reduces to trying the
average of 1.5 and two divided
by 1.5 as a guess for the
square root of two.
That average turns
out to be 1.333.
So this whole thing reduces to
trying 1.333 as a guess for
the square root of two.
And then so on.
That reduces to another called
a "good enough," 1.4
something or other.
And then it keeps going until
the process finally stops with
something that "good enough"
thinks is good enough, which,
in this case, is 1.4142
something or other.
So the process makes
perfect sense.
This, by the way, is called
a recursive definition.
And the ability to make
recursive definitions is a
source of incredible power.
And as you can already see I've
hinted at, it's the thing
that effectively allows you to
do these infinite computations
that go on until something is
true, without having any other
constricts other than the
ability to call a procedure.
Well, let's see, there's
one more thing.
Let me show you a variant of
this definition of square root
here on the slide.
Here's sort of the same thing.
What I've done here is packaged
the definitions of
"improve" and "good enough"
and "try" inside "square
root." So, in effect, what
I've done is I've built a
square root box.
So I've built a box that's the
square root procedure that
someone can use.
They might put in 36
and get out six.
And then, packaged inside this
box are the definitions of
"try" and "good enough"
and "improve."
So they're hidden
inside this box.
And the reason for doing that
is that, if someone's using
this square root, if George is
using this square root, George
probably doesn't care very much
that, when I implemented
square root, I had things inside
there called "try" and
"good enough" and "improve." And
in fact, Harry might have
a cube root procedure that has
"try" and "good enough" and
"improve." And in order to not
get the whole system confused,
it'd be good for Harry to
package his internal
procedures inside his
cube root procedure.
Well, this is called block
structure, this particular way
of packaging internals inside
of a definition.
And let's go back and look
at the slide again.
The way to read this kind of
procedure is to say, to define
"square root," well, inside that
definition, I'll have the
definition of an "improve" and
the definition of "good
enough" and the definition of
"try." And then, subject to
those definitions, the way I do
square root is to try one.
And notice here, I don't have to
say one as a guess for the
square root of X, because since
it's all inside the
square root, it sort of
has this X known.
Let me summarize.
We started out with the idea
that what we're going to be
doing is expressing imperative
knowledge.
And in fact, here's a slide
that summarizes the way we
looked at Lisp.
We started out by looking at
some primitive elements in
addition and multiplication,
some predicates for testing
whether something is less-than
or something's equal.
And in fact, we saw really
sneakily in the system we're
actually using, these aren't
actually primitives, but it
doesn't matter.
What matters is we're going
to use them as if they're
primitives.
We're not going to
look inside.
We also have some primitive
data and some numbers.
We saw some means of
composition, means of
combination, the basic one being
composing functions and
building combinations with
operators and operands.
And there were some other
things, like COND and "if" and
"define." But the main thing
about "define," in particular,
was that it was the means
of abstraction.
It was the way that
we name things.
You can also see from this slide
not only where we've
been, but holes we
have to fill in.
At some point, we'll have to
talk about how you combine
primitive data to get compound
data, and how you abstract
data so you can use large
globs of data as
if they were primitive.
So that's where we're going.
But before we do that, for the
next couple of lectures we're
going to be talking about, first
of all, how it is that
you make a link between these
procedures we write and the
processes that happen
in the machine.
And then, how it is that you
start using the power of Lisp
to talk not only about these
individual little
computations, but about general
conventional methods
of doing things.
OK, are there any questions?
AUDIENCE: Yes.
If we defined A using
parentheses instead of as we
did, what would be
the difference?
PROFESSOR: If I wrote this, if
I wrote that, what I would be
doing is defining a procedure
named A. In this case, a
procedure of no arguments,
which, when I ran it, would
give me back five times five.
AUDIENCE: Right.
I mean, you come up with the
same thing, except for you
really got a different--
PROFESSOR: Right.
And the difference would
be, in the old one--
Let me be a little
bit clearer here.
Let's call this A, like here.
And pretend here, just for
contrast, I wrote, define D to
be the product of
five and five.
And the difference between
those, let's think about
interactions with the
Lisp interpreter.
I could type in A and Lisp
would return 25.
I could type in D, if I just
typed in D, Lisp would return
compound procedure D, because
that's what it is.
It's a procedure.
I could run D. I could say,
what's the value of running D?
Here is a combination
with no operands.
I see there are no operands.
I didn't put any after D. And
it would say, oh, that's 25.
Or I could say, just for
completeness, if I typed in,
what's the value of running A?
I get an error.
The error would be the same
one as over there.
It'd be the error would say,
sorry, 25, which is the value
of A, is not an operator that
I can apply to something.
