[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.