講座4B:通用操作符 (Lecture 4B: Generic Operators)

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[MUSIC-- "JESU, JOY OF MAN'S DESIRING" BY
JOHANN SEBASTIAN BACH]
PROFESSOR: So far in this course we've been talking a
lot about data abstraction.
And remember the idea is that we build systems that have
these horizontal barriers in them, these abstraction
barriers that separate use, the way you might use some
data object, from the way you might represent it.
Or another way to think of that is up here you have the
boss who's going to be using some sort of data object.
And down here is George who's implemented it.
Now this notion of separating use from representation so you
can think about these two problems separately is a very,
very powerful programming methodology, data abstraction.
On the other hand, it's not really sufficient for really
complex systems. And the problem with this is George.
Or actually, the problem is that there
are a lot of Georges.
Let's be concrete.
Let's suppose there is George, and there's also Martha.
OK, now George and Martha are both working on this system,
both designing representations, and
absolutely are incompatible.
They wouldn't cooperate on a representation under any
circumstances.
And the problem is you would like to have some system where
both George and Martha are designing representations, and
yet, if you're above this abstraction barrier you don't
want to have to worry about that, whether something is
done by George or by Martha.
And you don't want George and Martha to
interfere with each other.
Somehow in designing a system, you not only want these
horizontal barriers, but you also want some kind of
vertical barrier to keep George and Martha separate.
Let me be a little bit more concrete.
Imagine that you're thinking about personnel records for a
large company with a lot of loosely linked divisions that
don't cooperate very well either.
And imagine even that this company is formed by merging a
whole bunch of companies that already have their personnel
record system set up.
And imagine that once these divisions are all linked in
some kind of very sophisticated satellite
network, and all these databases are put together.
And what you'd like to do is, from any place in the company,
to be able to say things like, oh, what's the name in a
personnel record?
Or, what's the job description in a personnel record?
And not have to worry about the fact that each division
obviously is going to have completely separate
conventions for how you might implement these records.
From this point you don't want to know about that.
Well how could you possibly do that?
One way, of course, is to send down an edict from somewhere
that everybody has to change their format to some fixed
compatible thing.
That's what people often try, and of course it never works.
Another thing that you might want to do is somehow arrange
it so you can have these vertical barriers.
So that when you ask for the name of a personnel record,
somehow, whatever format it happens to be, name will
figure out how to do the right thing.
We want name to be, so-called, a generic operator.
Generic operator means what it sort of precisely does depends
on the kind of data that it's looking at.
More than that, you'd like to design the system so that the
next time a new division comes into the company they don't
have to make any big changes in what they're already doing
to link into this system, and the rest of the company
doesn't have to make any big changes to admit their stuff
to the system.
So that's the problem you should be thinking about.
Like it's sort of just your work.
You want to be able to include new things by
making minimal changes.
OK, well that's the problem that we'll be
talking about today.
And you should have this sort of distributed personnel
record system in your mind.
But actually the one I'll be talking about is a problem
that's a little bit more self-contained than that.
that'll bring up the issues, I think, more clearly.
That's the problem of doing a system that does arithmetic on
complex numbers.
So let's take a look here.
Just as a little review, there are things
called complex numbers.
Complex number you can think of as a point in
the plane, or z.
And you can represent a point either by its real-part and
its imaginary-part.
So if this is z and its real-part is this much, and
its imaginary-part is that much, and you write z
equals x plus iy.
Or another way to represent a complex number is by saying,
what's the distance from the origin, and what's the angle?
So that represents a complex number as its
radius times an angle.
This one's called-- the original one's called
rectangular form, rectangular representation, real- and
imaginary-part, or polar representation.
Magnitude and angle--
and if you know the real- and imaginary-part, you can figure
out the magnitude and angle.
If you know x and y, you can get r by this formula.
Square root of sum of the squares, and you can get the
angle as an arctangent.
Or conversely, if you knew r and A you could
figure out x and y.
x is r times the cosine of A, and y is r times the sine of
A. All right, so there's these two.
They're complex numbers.
You can think of them either in polar form
or rectangular form.
What we would like to do is make a system that does
arithmetic on complex numbers.
In other words, what we'd like--
just like the rational number example--
is to have some operations plus c, which is going to take
two complex numbers and add them, subtract them, and
multiply them, and divide them.
OK, well there's little bit of mathematics behind it.
What are the actual formulas for manipulating such things?
And it's sort of not important where they come from, but just
as an implementer let's see--
if you want to add two complex numbers it's pretty easy to
get its real-part and its imaginary-part.
The real-part of the sum of two complex numbers, the
real-part of the z1 plus z2 is the real-part of z1 plus the
real-part of z2.
And the imaginary-part of z1 plus z2 is the imaginary part
of z1 plus the imaginary part of z2.
So it's pretty easy to add complex numbers.
You just add the corresponding parts and make a new complex
number with those parts.
If you want to multiply them, it's kind of nice to do it in
polar form.
Because if you have two complex numbers, the magnitude
of their product is here, the product of the magnitudes.
And the angle of the product is the sum of the angles.
So that's sort of mathematics that allows you to do
arithmetic on complex numbers.
Let's actually think about the implementation.
Well we do it just like rational numbers.
We come down, we assume we have some
constructors and selectors.
What would we like?
Well let's assume that we make a data object cloud, which is
a complex number that has some stuff in it, and that we can
get out from a complex number the real-part, or the
imaginary-part, or the magnitude, or the angle.
We want some ways of making complex numbers-- not only
selectors, but constructors.
So we'll assume we have a thing called make-rectangular.
What make-rectangular is going to do is take a real-part and
an imaginary-part and construct a complex number
with those parts.
Similarly, we can have make-polar which will take a
magnitude and an angle, and construct a complex number
which has that magnitude and angle.
So here's a system.
We'll have two constructors and four selectors.
And now, just like before, in terms of that abstract data
we'll go ahead and implement our complex number operations.
And here you can see translated into Lisp code just
the arithmetic formulas I put down before.
If I want to add two complex numbers I will make a complex
number out of its real- and imaginary-parts.
The real part of the complex number I'm going to make is
the sum of the real-parts.
The imaginary part of the complex number I'm going to
make is the sum of the imaginary-parts.
I put those together, make a complex number.
That's how I implement complex number addition.
Subtraction is essentially the same.
All I do is subtract the parts rather than add them.
To multiply two complex numbers, I
use the other formula.
I'll make a complex number out of a magnitude and angle.
The magnitude is going to be the product of the magnitudes
of the two complex numbers I'm multiplying.
And the angle is going to be the sum of the angles of the
two complex numbers I'm multiplying.
So there's multiplication.
And then division, division is almost the same.
Here I divide the magnitudes and subtract the angles.
Now I've implemented the operations.
And what do we do?
We call on George.
We've done the use, let's worry about the
representation.
We'll call on George and say to George, go ahead and build
us a complex number representation.
Well that's fine.
George can say, we'll implement a complex number
simply as a pair that has the real-part and the
imaginary-part.
So if I want to make a complex number with a certain
real-part and an imaginary-part, I'll just use
cons to form a pair, and that will-- that's George's
representation of a complex number.
So if I want to get out the real-part of something, I just
extract the car, the first part.
If I want to get the imaginary-part, I extract the
cdr. How do I deal with the magnitude and angle?
Well if I want to extract the magnitude of one of these
things, I get the square root of the sum of the square of
the car plus the square of the cdr. If I want to get the
angle, I compute the arctangent of
the cdr in the car.
This is a list procedure for computing arctangent.
And if somebody hands me a magnitude and an angle and
says, make me a complex number, well I compute the
real-part and the imaginary-part, or our cosine
of a and our sine of a, and stick them
together into a pair.
OK so we're done.
In fact, what I just did, conceptually, is absolutely no
different from the rational number representation that we
looked at last time.
It's the same sort of idea.
You implement the operators, you pick a representation.
Nothing different.
Now let's worry about Martha.
See, Martha has a different idea.
She doesn't want to represent a complex number as a pair of
a real-part and an imaginary-part.
What she would like to do is represent a complex number as
a pair of a magnitude and an angle.
So if instead of calling up George we ask Martha to design
our representation, we get something like this.
We get make-polar.
Sure, if I give you a magnitude and an angle we're
just going to form a pair that has magnitude and angle.
If you want to extract the magnitude, that's easy.
You just pull out the car or the pair.
If you want to extract the angle, sure, that's easy.
You just pull out the cdr. If you want to look for
real-parts and imaginary-parts, well then you
have to do some work.
If you want the real-part, you have to get r cosine a.
In other words, r, the car of the pair, times the cosine of
the cdr of the pair.
So this is r times the cosine of a,
and that's the real-part.
If you want to get the imaginary-part, it's r times
the sine of a.
And if I hand you a real-part and an imaginary-part and say,
make me a complex number with that real-part and
imaginary-part, well I figure out what the magnitude and
angle should be.
The magnitude's the square root of the sum of the squares
and the angle's the arctangent.
I put those together to make a pair.
So there's Martha's idea.
Well which is better?
Well if you're doing a lot of additions, probably George's
is better, because you're doing a lot of real-parts and
imaginary-parts.
If mostly you're going to be doing multiplications and
divisions, then maybe Martha's idea is better.
Or maybe, and this is the real point, you can't decide.
Or maybe you just have to let them both hang around, for
personality reasons.
Maybe you just really can't ever decide
what you would like.
And again, what we would really like is a system that
looks like this.
That somehow there's George over here, who has built
rectangular complex numbers.
And Martha, who has polar complex numbers.
And somehow we have operations that can add, and subtract,
and multiply, and divide, and it shouldn't matter that there
are two incompatible representations of complex
numbers floating around this system.
In other words, not only like an abstraction barrier here
that has things in it like a real-part, and an
imaginary-part, and magnitude, and angle.
So not only is there an abstraction barrier that hides
the actual representation from us, but also there's some kind
of vertical barrier here that allows both of these
representations to exist without
interfering with each other.
The idea is that the things in here--
real-part, imaginary-part, magnitude, and angle--
will be generic operators.
If you ask for the real-part, it will worry about what
representation it's looking at.
OK, well how can we do that?
There's actually a really obvious idea, if you're used
to thinking about complex numbers.
If you're used to thinking about compound data.
See, suppose you could just tell by looking at a complex
number whether it was constructed
by George or Martha.
In other words, so it's not that what's floating around
here are ordinary, just complex numbers, right?
They're fancy, designer complex numbers.
So you look at a complex numbers as it's not just a
complex number, it's got a label on it that says, this
one is by Martha.
Or this is a complex number by George.
Right?
They're signed.
See, and then whenever we looked at a complex number we
could just read the label, and then we'd know how you expect
to operate on that.
In other words, what we want is not just
ordinary data objects.
We want to introduce the notion of what's
called typed data.
Typed data means, again, there's some sort of cloud.
And what it's got in it is an ordinary data object like
we've been thinking about.
Pulled out the contents, sort of the actual data.
But also a thing called a type, but it's signed by
either George or Martha.
So we're going to go from regular data to type data.
How do we build that?
Well that's easy.
We know how to build clouds.
We build them out of pairs.
So here's a little representation that supports
typed data.
There's a thing called take a type and attach it to a piece
of contents, and we just use cons.
And if we have a piece of typed data, we can look at the
type, which is the car.
We can look at the contents, which is the cdr. Now along
with that, the way we use our type data will test, when
we're given a piece of data, what type it is.
So we have some type predicates with us.
For example, to see whether a complex number is one of
George's, whether it's rectangular, we just check to
see if the type of that is the symbol rectangular, right?
The symbol rectangular.
And to check whether a complex number is one of Martha's, we
check to see whether the type is the symbol polar.
So that's a way to test what kind of number
we're looking at.
Now let's think about how we can use that
to build the system.
So let's suppose that George and Martha were off working
separately, and each of them had designed their complex
number representation packages.
What do they have to do to become part of the system, to
exist compatibly?
Well it's really pretty easy.
Remember, George had this package.
Here's George's original package, or half of it.
And underlined in red are the changes he has to make.
So before, when George made a complex number out of an x and
y, he just put them together to make a pair.
And the only difference is that now he signs them.
He attaches the type, which is the symbol
rectangular to that pair.
Everything else George does is the same, except that--
see, George and Martha both have procedures named
real-part and imaginary-part.
So to allow them both to exist in the same Lisp environment,
George had changed the names of his procedures.
So we'll say, this is George's real-part procedure.
It's the real-part rectangular procedure, the imaginary-part
rectangular procedure.
And then here's the rest of George's package.
He'd had magnitude and angle, just renames them magnitude
rectangular and angle rectangular.
And Martha has to do basically the same thing.
Martha previously, when she made a complex number out of a
magnitude and angle, she just cons them.
Now she attaches the type polar, and she changes the
name so her real-part procedure won't conflict in
name with George's.
It's a real-part-polar, imaginary-part-polar,
magnitude polar, and angle polar.
Now we have the system.
Right there's George and Martha.
And now we've got to get some kind of manager to look at
these types.
How are these things actually going to work now that George
and Martha have supplied us with typed data?
Well what we have are a bunch of generic selectors.
Generic selectors for complex numbers real-part,
imaginary-part, magnitude, and angle.
Let's look at them more closely.
What does a real-part do?
If I ask for the real part of a complex number,
well I look at it.
I look at its type.
I say, is it rectangular?
If so, I apply George's real part procedure to the contents
of that complex number.
This is a number that has a type on it.
I strip off the type using contents and
apply George's procedure.
Or is this a polar complex number?
If I want the real part, I apply Martha's real part
procedure to the contents of that number.
So that's how real part works.
And then similarly there's imaginary-part, which is
almost the same.
It looks at the number and if it's rectangular, uses
George's imaginary-part procedure.
If it's polar, uses Martha's.
And then there's a magnitude and an angle.
So there's a system.
Has three parts.
There's sort of George, and Martha, and the manager.
And that's how you get generic operators implemented.
Let's look at just a simple example, just to pin it down.
But exactly how this is going to work, suppose you're going
to be looking at the complex number who's real-part is one,
and who's imaginary-part is two.
So that would be one plus 2i.
What would happen is up here, up here above where the
operations have to happen, that number would be
represented as a pair of 1 and 2 together with typed data.
That would be the contents.
And the whole data would be that thing with the symbol
rectangular added onto that.
And that's the way that complex number would exist in
the system.
When you went to take the real-part, the manager would
look at this and say, oh it's one of George's.
He'll strip off the type and hand down to
George the pair 1, 2.
And that's the kind of data that George developed his
system to use.
So it gets stripped down.
Later on, if you ask George to construct a complex number,
George would construct some complex number as a pair, and
before he passes it back up through the manager would
attach the type rectangular.
So you see what happens.
There's no confusion in this system.
It doesn't matter in the least that the pair 1, 2 means
something completely different in Martha's world.
In Martha's world this pair means the complex number whose
magnitude is 1 and whose angle is 2.
And there's no confusion, because by the time any pair
like this gets handed back through the manager to the
main system it's going to have the type polar attached.
Whereas this one would have the type rectangular attached.
OK, let's take a break.
[MUSIC-- "JESU, JOY OF MAN'S DESIRING" BY
JOHANN SEBASTIAN BACH]
We just looked at a strategy for
implementing generic operators.
That strategy has a name: it's called dispatch type.
And the idea is that you break your system
into a bunch of pieces.
There's George and Martha, who are making representations,
and then there's the manager.
Looks at the types on the data and then dispatches them to
the right person.
Well what criticisms can we make of that as a system
organization?
Well first of all there was this little, annoying problem
that George and Martha had to change the names of their
procedures.
George originally had a real-part procedure, and he
had to go name it real-part rectangular so it wouldn't
interfere with Martha's real-part procedure, which is
now named real-part-polar, so it wouldn't interfere with the
manager's real-part procedure, who's now named real-part.
That's kind of an annoying problem.
But I'm not going to talk about that one now.
We'll see later on when we think about the structure of
Lisp names and environments that there really are ways to
package all those so-called name spaces separately so they
don't interfere with each other.
Not going to think about that problem now.
The problem that I actually want to focus on is what
happens when you bring somebody new into the system.
What has to happen?
Well George and Martha don't care.
George is sitting there in his rectangular world, has his
procedures and his types.
Martha sits in her polar world.
She doesn't care.
But let's look at the manager.
What's the manager have to do?
The manager comes through and had these operations.
There was a test for rectangular
and a test for polar.
If Harry comes in with some new kind of complex number,
and Harry has a new type, Harry type complex number, the
manager has to go in and change all those procedures.
So the inflexibility in the system, the place where work
has to happen to accommodate change, is in the manager.
That's pretty annoying.
It's even more annoying when you realize the manager's not
doing anything.
The manager is just being a paper pusher.
Let's look again at these programs. What are they doing?
What does real-part do?
Real-part says, oh, is it the kind of complex number that
George can handle?
If so, send it off to George.
Is it the kind of complex number that Martha can handle?
If so, send it off to Martha.
So it's really annoying that the bottleneck in this system,
the thing that's preventing flexibility and change, is
completely in the bureaucracy.
It's not in anybody who's doing any of the work.
Not an uncommon situation, unfortunately.
See, what's really going on--
abstractly in the system, there's a table.
So what's really happening is somewhere there's a table.
There're types.
There's polar and rectangular.
And Harry's may be over here.
And there are operators.
There's an operator like real-part.
Or imaginary-part.
Or a magnitude and angle.
And sitting in this table are the right procedures.
So sitting here for the type polar and real-part is
Martha's procedure real-part-polar.
And over here in the table is George's procedure
real-part-rectangular.
And over here would be, say, Martha's procedure
magnitude-polar, and George's procedure
magnitude-rectangular, right, and so on.
The rest of this table's filled in.
And that's really what's going on.
So in some sense, all the manager is doing is acting as
this table.
Well how do we fix our system?
How do you fix bureaucracies a lot of the time?
What you do is you get rid of the manager.
We just take the manager and replace him by a computer.
We're going to automate him out of existence.
Namely, instead of having the manager who basically consults
this table, we'll have our system use the table directly.
What do I mean by that?
Let's assume, again using data abstraction, that we have some
kind of data structure that's a table.
And we have ways of sticking things in and ways of getting
things out.
And to be explicit, let me assume that there's an
operation called "put." And put is going to take, in this
case two things I'll call "keys." Key1 and key2.
And a value.
And that stores the value in the table under key1 and key2.
And then we'll assume there's a thing called "get," such
that if later on I say, get me what's in the table stored
under key1 and key2, it'll retrieve whatever value was
stored there.
And let's not worry about how tables are implemented.
That's yet another data abstraction, George's problem.
And maybe we'll see later--
talk about how you might actually build tables in Lisp.
Well given this organization, what did George and Martha
have to do?
Well when they build their system, they each have the
responsibility to set up their appropriate
column in the table.
So what George does, for example, when he defines his
procedures, all he has to do is go off and put into the
table under the type-rectangular.
And the name of the operation is real-part, his procedure
real-part-rectangular.
So notice what's going into this table.
The two keys here are symbols, rectangular and real-part.
That's the quote.
And what's going into the table is the actual procedure
that he wrote, real-part rectangular.
And then puts an imaginary part into the table, filed
under the keys rectangular- and imaginary-part, and
magnitude under the keys rectangular magnitude, angle
under rectangular-angle.
So that's what George has to do to be part of this system.
Martha similarly sets up the column and
the table under polar.
Polar and real-part.
Is the procedure real-part-polar?
And imaginary-part, and magnitude, and angle.
So this is what Martha has to do to be part of the system.
Everyone who makes a representation has the
responsibility for setting up a column in the table.
And what does Harry do when Harry comes in with his
brilliant idea for implementing complex numbers?
Well he makes whatever procedure he wants and builds
a new column in this table.
OK, well what happened to the manager?
The manager has been automated out of existence and is
replaced by a procedure called operate.
And this is the key procedure in the whole system.
Let's say define operate.
Operate is going to take an operation that you want to do,
the name of an operation, and an object that you would like
to apply that operation to.
So for example, the real-part of some particular complex
number, what does it do?
Well the first thing it does, it looks in the table.
Goes into the table and tries to find a procedure that's
stored in the table.
So it gets from the table, using as keys the type of the
object and the operator, but looks on the table and sees
what's stored under the type of the object and the
operator, sees if anything's stored.
Let's assume that get is implemented.
So if nothing is stored there, it'll return the empty list.
So it says, if there's actually something stored
there, if the procedure here is not no, then it'll take the
procedure that it found in the table and apply it to the
contents of the object.
And otherwise if there was nothing stored there, it'll--
well we can decide.
In this case let's have it put out an error message saying,
undefined operator.
No operator for this type.
Or some appropriate error message.
OK?
And that replaces the manager.
How do we really use it?
Well what we say is we'll go off and define our generic
selectors using operate.
We'll say that the real-part of an object is found by
operating on the object with the name of the operation
being real-part.
And then similarly, imaginary-part is operate
using the name imaginary-part and magnitude and angle.
That's our implementation.
That plus the tape plus the operate procedure.
And the table effectively replaces what the
manager used to do.
Let's just go through that slowly to show you
what's going on.
Suppose I have one of Martha's complex numbers.
It's got magnitude 1 and angle 2.
And it's one of Martha's.
So it's labeled here, polar.
Let's call that z.
Suppose that's z.
And suppose with this implementation someone comes
up and asks for the real-part of z.
Well real-part now is defined in terms of operate.
So that's equivalent to saying operate with the name of the
operator being real-part, the symbol real-part on z.
And now operate comes.
It's going to look in the table, and it's going to try
and find something stored under--
the operation is going to apply by looking in the table
under the type of the object.
And the type of z is polar.
So it's going to look and say, can I get using polar?
And the operation name, which was real-part.
It's going to look in there and apply that to
the contents of z.
And that?
If everything was set up correctly, this thing is the
procedure that Martha put there.
This is real-part-polar.
And this is z without its type.
The thing that Martha originally designed those
procedures to work on, which is 1, 2.
And so operate sort of does uniformly what the manager
used to do sort of all over the system.
It finds the right thing, looks in the table, strips off
the type, and passes it down into the
person who handles it.
This is another, and, you can see, more flexible for most
purposes, way of implementing generic operators.
And it's called data-directed programming.
And the idea of that is in some sense the data objects
themselves, those little complex numbers that are
floating around the system, are carrying with them the
information about how you should operate on them.
Let's break for questions.
Yes.
AUDIENCE: What do you have stored in that data object?
You have the data itself, you have its type, and you have
the operations for that type?
Or where are the operations that you found?
PROFESSOR: OK, let me--
yeah, that's a good question.
Because it raises other possibilities of how
you might do it.
And of course there are a lot of possibilities.
In this particular implementation, what's sitting
in this data object, for example, is the data itself--
which in this case is a pair of 1 and 2--
and also a symbol.
This is the symbol, the word P-O-L-A-R, and that's what's
sitting in this data object.
Where are the operations themselves?
The operations are sitting in the table.
So in this table, the rows and columns of the table are
labeled by symbols.
So when I store something in this table, the key might be
the symbol polar and the symbol magnitude.
And I think by writing it this way I've been very confusing.
Because what's really sitting here isn't--
when I wrote magnitude polar, what I mean is the procedure
magnitude polar.
And probably what I really should have written--
except it's too small for me to write
in this little space--
is something like lambda of z, the thing that
Martha wrote to implement.
And then you can see from that, there's another way that
I alluded to of solving this name conflict problem, which
is that George and Martha never have to name their
procedures at all.
They can just stick the anonymous things generated by
lambda directly into the table.
There's also another thing that your question raises, is
the possibility that maybe what I would like somehow is
to store in this data object not the symbol P-O-L-A-R but
maybe actually all the operations themselves.
And that's another way to organize the system, called
message passing.
So there are a lot of ways you can do it.
AUDIENCE: Therefore if Martha and George had used the same
procedure names, it would be OK because it wouldn't look
[UNINTELLIGIBLE].
PROFESSOR: That's right.
That's right.
See, they wouldn't even have to name their
procedures at all.
What George could have written instead of saying put in the
table under rectangular- and real-part, the procedure
real-part rectangular, George could have written put under
rectangular real-part, lambda of z, such and such,
and such and such.
And the system would work completely the same.
AUDIENCE: My question is, Martha could have put key1
key2 real-part, and George could have put key1 key2
real-part, and as long as they defined them differently they
wouldn't have had any conflicts, right?
PROFESSOR: Yes, that would all be OK except for the fact that
if you imagine George and Martha typing at the same
console with the same meanings for all their names, and it
would get confused by real-part, but there are ways
to arrange that, too.
And in principle you're absolutely right.
If their names didn't conflict--
it's the objects that go in the table, not the names.
OK, let's take a break.
[MUSIC-- "JESU, JOY OF MAN'S DESIRING" BY
JOHANN SEBASTIAN BACH]
All right, well we just looked at data-directed programming
as a way of implementing a system that does arithmetic on
complex numbers.
So I had these operations in it called plus C and minus C,
and multiply, and divide, and maybe some others.
And that sat on top of-- and this is the key point-- sat on
top of two different representations.
A rectangular package here, and a polar package.
And maybe some more.
And we saw that the whole idea is that maybe some more are
now very easy to add.
But that doesn't really show the power of this methodology.
Shows you what's going on.
The power of the methodology only becomes apparent when you
start embedding this in some more complex system.
What I'm going to do now is embed this in some more
complex system.
Let's assume that what we really have is a general kind
of arithmetic system.
So called generic arithmetic system.
And at the top level here, somebody can say add two
things, or subtract two things, or multiply two
things, or divide two things.
And underneath that there's an abstraction barrier.
And underneath this barrier, is, say, a
complex arithmetic package.
And you can say, add two complex numbers.
Or you might also have-- remember we did a rational
number package-- you might have that sitting there.
And there might be a rational thing.
And the rational number package, well, has the things
we implemented.
Plus rat, and times rat, and so on.
Or you might have ordinary Lisp numbers.
You might say add three and four.
So we might have ordinary numbers, in which case we have
the Lisp supplied plus, and minus, and times, and slash.
OK, so we might imagine this complex number system sitting
in a more complicated generic operator structure at
the next level up.
Well how can we make that?
We already have the idea, we're just
going to do it again.
We've implemented a rational number package.
Let's look at how it has to be changed.
In fact, at this level it doesn't have to
be changed at all.
This is exactly the code that we wrote last time.
To add two rational numbers, remember
there was this formula.
You make a rational number whose numerator--
the numerator of the first times the denominator of the
second, plus the denominator of the first times the
numerator of the second.
And who's denominator is the product of the denominators.
And minus rat, and star rat, and slash rat.
And this is exactly the rational number package that
we made before.
We're ignoring the GCD problem, but let's not worry
about that.
As implementers of this rational number package, how
do we install it in the generic arithmetic system?
Well that's easy.
There's only one thing we have to do differently.
Whereas previously we said that to make a rational number
you built a pair of the numerator and denominator,
here we'll not only build the pair, but we'll sign it.
We'll attach the type rational.
That's the only thing we have to do different, make it a
typed data object.
And now we'll stick our operations in the table.
We'll put under the symbol rational and the operation add
our procedure, plus rat.
And, again, note this is a symbol.
Right?
Quote, unquote, but the actual thing we're putting in the
table is the procedure.
And for how to subtract, well you subtract
rationals with minus rat.
And multiply, and divide.
And that is exactly and precisely what we have to do
to fit inside this generic arithmetic system.
Well how does the whole thing work?
See, what we want to do is have some generic operators.
Have add and sub and [UNINTELLIGIBLE]
be generic operators.
So we're going to define add and say, to add x and y, that
will be operate--
we were going to call it operate-2.
This is our operator procedure, but set up for two
arguments using add on x and y.
And so this is the analog to operate.
Let's look at the code for second.
It's almost like operate.
To operate with some operator on an argument 1 and an
argument 2, well the first thing we're going to do is
check and see if the two arguments have the same type.
So we'll say, is the type of the first argument the same as
the type of the second argument?
And if they're not, we'll go off and complain, and say,
that's an error.
We don't know how to do that.
If they do have the same type, we'll do
exactly what we did before.
We'll go look and filed under the type of the argument--
arg 1 and arg 2 have the same type, so it doesn't matter.
So we'll look in the table, find the procedure.
If there is a procedure there, then we'll apply it to the
contents of the argument 1 and the contents of arg 2.
And otherwise we'll say, error.
Undefined operator.
And so there's operate-2.
And that's all we have to do.
We just built the complex number package before.
How do we embed that complex number package in
this generic system?
Almost the same.
We make a procedure called make-complex that takes
whatever George and Martha hand to us and add the
type-complex.
And then we say, to add complex numbers, plus complex,
we use our internal procedure, plus c, and attach a type,
make that a complex number.
So our original package had names plus c and minus c that
we're using to communicate with George and Martha.
And then to communicate with the outside world, we have a
thing called plus-complex and minus-complex.
And so on.
And the only difference is that these return
values that are tight.
So they can be looked at up here.
And these are internal operations.
Let's go look at that slide again.
There's one more thing we do.
After defining plus-complex, we put under the type complex
and the symbol add, that procedure plus complex.
And then similarly for subtracting complex numbers,
and multiplying them, and dividing them.
OK, how do we install ordinary numbers?
Exactly the same way.
Come off and say, well we'll make a thing called
make-number.
Make-number takes a number and attaches a type, which is the
symbol number.
We build a procedure called plus-number, which is simply,
add the two things using the ordinary addition, because in
this case we're talking about ordinary numbers, and attach a
type to it and make that a number.
And then we put into the table under the symbol number and
the operation add, this procedure plus-number, and
then the same thing for subtracting, and multiplying,
and dividing.
Let's look at an example, just to make it clear.
Suppose, for instance, I'm going
to perform the operation.
So I sit up here and I'm going to perform the operation,
which looks like multiplying two complex numbers.
So I would multiply, say, 3 plus 4i and 2 plus 6i.
And that's something that I might want to take
hand that to mul.
I'll write mul as my generic operator here.
How's that going to work?
Well 3 plus 4i, say, sits in the system at this level as
something that looks like this.
Let's say it was one of George's.
So it would have a 3 and a 4.
And attached to that would be George's type, which would say
rectangular, it came from George.
And attached to that--
and this itself would be the data view from the next level
up, which it is--
so that itself would be a type-data object which would
say complex.
So that's what this object would look like up here at the
very highest level, where the really super-generic
operations are looking at it.
Now what happens, mul eventually's going to come
along and say, oh, what's it's type?
It's type is complex.
Go through to operate-2 and say, oh, what I want to do is
apply what's in the table, which is going to be the
procedure star complex, on this thing with the type
stripped off.
So it's going to strip off the type, take that much, and send
that down into the complex world.
The complex world looks at its operations and says, oh, I
have to apply star c.
Star c might say, oh, at some point I want to look at the
magnitude of this object that it's in, that it's got.
And they'll say, oh, it's
rectangular, it's one of George's.
So it'll then strip off the next version of type, and hand
that down to George to take the magnitude of.
So you see what's going on is that there are
these chains of types.
And the length of the chain is sort of the number of levels
that you're going to be going up in this table.
And what a type tells you, every time you have a vertical
barrier in this table, where there's some ambiguity about
where you should go down to the next level, the type is
telling you where to go.
And then everybody at the bottom, as they construct data
and filter it up, they stick their type back on.
So that's the general structure of the system.
OK.
Now that we've got this, let's go and make this thing even
more complex.
Let's talk about adding to the system not only these kinds of
numbers, but it's also meaningful to start talking
about adding polynomials.
Might do arithmetic on polynomials.
Like we could have x to the fifteenth plus 2x to the
seventh plus 5.
That might be some polynomial.
And if we have two such gadgets we can add them or
multiply them.
Let's not worry about dividing them.
Just add them, multiply them, then we'll subtract them.
What do we have to do?
Well let's think about how we might represent a polynomial.
It's going to be some typed data object.
So let's say a polynomial to this system might look like a
thing that starts with the type polynomial.
And then maybe it says the next thing is what
variable its in.
So I might say I'm a polynomial in the variable x.
And then it'll have some information about
what the terms are.
And there're just tons of ways to do this, but one way is to
say we're going to have a thing called a term-list. And
a term-list--
well, in our case we'll use something
that looks like this.
We'll make it a bunch of pairs which have an order in a
coefficient.
So this polynomial would be represented by this term-list.
And what that means is that this polynomial starts off
with a term of order 15 and coefficient 1.
And the next thing in it is a term of order 7 and
coefficient 2, a term of order 0, which is constant in
coefficient 5.
And there are lots and lots of ways, and lots and lots of
trade-offs when you really think about making algebraic
manipulation packages about exactly how you should
represent these things.
But this is a fairly standard one.
It's useful in a lot of contexts.
OK, well how do we implement our polynomial arithmetic?
Let's start out.
What we'll do to make a polynomial--
we'll first have a way to make polynomials.
We're going to make a polynomial out of variable
like x and term-list. And all that does is we'll package
them together someway.
We'll put the variable together with the term list
using cons, and then attached to that the type polynomial.
OK, how do we add two polynomials?
To add a polynomial, p1 and p2, and then just for
simplicity let's say we will only add
things in the same variable.
So if they have the same variable, and same variable
here is going to be some selector we write, whose
details we don't care about.
If the two polynomials have the same variable, then we'll
do something.
If they don't have the same variable, we'll give an error,
polynomials not in the same variable.
And if they do have the same variable, what we'll do is
we'll make a polynomial whose variable is whatever that
variable is, and whose term-list is something we'll
call sum-terms. Plus terms will add the two term lists.
So we'll add the two term lists to the polynomial.
That'll give us a term-list. We'll add on, we'll say it's a
polynomial in the variable with that
term-list. That's plus poly.
And then we're going to put in our table under the type
polynomial, add them using plus poly.
And of course we really haven't done much.
What we've really done is pushed all the work onto this
thing, plus-terms, which is supposed to add term-lists.
Let's look at that.
Here's an overview of how we might add two term-lists.
So L1 and L2 were going to be two term-lists.
And a term-list is a bunch of pairs, coefficient in order.
And it's a big case analysis.
And the first thing we'll check for and see if there are
any terms. We're going to recursively work down these
term-lists, so eventually we'll get to a place where
either L1 or L2 might be empty.
And if either one is empty, our answer will
be the other one.
So if L1 is empty we'll return L2, and if L2 is empty
we'll return L1.
Otherwise there are sort of three interesting cases.
What we're going to do is grab the first term in each of
those lists, called t1 and t2.
And we're going to look at three cases, depending on
whether the order of t1 is greater than the order of t2,
or less than t2, or the same.
Those are the three cases we're going to look at.
Let's look at this case.
If the order of t1 is greater than the order of t2, then
what that means is that our answer is going to start with
this term of the order of t1.
Because it won't combine with any lower order terms. So what
we do is add the lower order terms. We recursively add
together all the terms in the rest of the
term-list in L1 and L2.
That's going to be the lower order terms of the answer.
And then we're going to adjoin to that the
highest order term.
And I'm using here a whole bunch of procedures I haven't
defined, like a adjoin-term, and rest-terms, and selectors
that get order.
But you can imagine what those are.
So if the first term-list has a higher order than the
second, we recursively add all the lower terms and then stick
on that last term.
The other case, the same way.
If the first term has a smaller order, well then we
add the first term-list and the rest of the terms in the
second one, and adjoin on this highest order term.
So so far nothing's much happened, we've just sort of
pushed this thing off into adding lower order terms. The
last case where you actually get to a coefficients that you
have to add, this will be the case where
the orders are equal.
What we do is, well again recursively add the lower
order terms. But now we have to really combine something.
What we do is we make a term whose order is the order of
the term we're looking at.
By now t1 and t2 have the same order.
That's its order.
And its coefficient is gotten by adding the coefficient of
t1 and the coefficient of t2.
This is a big recursive working down of terms, but
really there's only one interesting symbol in this
procedure, only one interesting idea.
The interesting idea is this add.
And the reason that's interesting is because
something completely wonderful just happened.
We reduced adding polynomials, not to sort of plus, but to
the generic add.
In other words, by implementing it that way, not
only do we have our system where we can have rational
numbers, or complex numbers, or ordinary numbers, we've
just added on polynomials.
But the coefficients of the polynomials can be anything
that the system can add.
So these could be polynomials whose coefficients are
rational numbers or complex numbers, which in turn could
be either rectangular, or polar, or ordinary numbers.
So what I mean precisely is our system right now
automatically can handle things like adding together
polynomials that have this one: 2/3 of x squared plus
5/17 x plus 11/4.
Or automatically handle polynomials that look like 3
plus 2i times x to the fifth plus 4 plus 7i, or something.
You can automatically handle those things.
Why is that?
That's merely because, or profoundly because we reduced
adding polynomials to adding their coefficients.
And adding coefficients was done by the generic add
operator, which said, I don't care what your types are as
long as I know how to add you.
So automatically for free we get the
ability to handle that.
What's even better than that, because remember one of the
things we did is we put into the table that the way you add
polynomials is using plus poly.
That means that polynomials themselves are
things that can be added.
So for instance let me write one here.
Here's a polynomial.
So this gadget here I'm writing up, this is a
polynomial in y whose coefficients are
polynomials in x.
So you see, simply by saying, polynomials are themselves
things that can be added, we can go off and say, well not
only can we deal with rationals, or complex, or
ordinary numbers, but we can deal with polynomials whose
coefficients are rationals, or complex, or ordinary numbers,
or polynomials whose coefficients are rationals, or
complex, rectangular, polar, or ordinary numbers, or
polynomials whose coefficients are rationals, complex, or
ordinary numbers.
And so on, and so on, and so on.
So this is sort of an infinite or maybe a recursive tower of
types that we've built up.
And it's all exactly from that one little symbol.
A-D-D. Writing "add" instead of "plus" in
the polynomial thing.
Slightly different way to think about it is that
polynomials are a constructor for types.
Namely you give it a type, like integer, and it returns
for you polynomials in x whose coefficients are integers.
And the important thing about that is that the operations on
polynomials reduce to the operations on the
coefficients.
And there are a lot of things like that.
So for example, let's go back and rational numbers.
We thought about rational numbers as an integer over an
integer, but there's the general notion
of a rational object.
Like we might think about 3x plus 7 over x squared plus 1.
That's general rational object whose numerator and
denominator are polynomials.
And to add two of them we use the same formula, numerator
times denominator plus denominator times numerator
over product of denominators.
How could we install that in our system?
Well here's our original rational
number arithmetic package.
And all we have to do in order to make the entire system
continue working with general rational objects, is replace
these particular pluses and stars by the generic operator.
So if we simply change that procedure to this one, here
we've changed plus and star to add a mul, those are
absolutely the only change, then suddenly our entire
system can start talking about objects that look like this.
So for example, here is a rational object whose
numerator is a polynomial in x whose coefficients are
rational numbers.
Or here is a rational object whose numerator is polynomials
in x whose coefficients are rational objects constructed
out of complex numbers.
And then there are a lot of other things like that.
See, whenever you have a thing where the operations reduce to
operations on the pieces, another example would be two
by two matrices.
I have the idea, there might be a matrix here of general
things that I don't care about.
But if I add two of them, the answer over here is gotten by
adding this one and that one, however they like to add.
So I can implement that the same way.
And if I do that, then again suddenly my system can start
handling things like this.
So here's a matrix whose elements happen to be--
we'll say this element here is a rational object whose
numerator and denominators are polynomials.
And all that comes for free.
What's really going on here?
What's really going on is getting rid of this manager
who's sitting there poking his nose into who everybody's
business is.
We built a system that has decentralized control.
So when you come into and no one's poking around saying,
gee, are you in the official list of
people who can be added?
Rather you say, well go off and add yourself how your
parts like to be added.
And the result of that is you can get this very, very, very
complex hierarchy where a lot of things just get done and
rooted to the right place automatically.
Let's stop for questions.
AUDIENCE: You say you get this for free.
One thing that strikes me is that now you've lost kind of
the cleanness of the break between what's on top and
what's underneath.
In other words, now you're defining some of the
lower-level procedures in terms of things
above their own line.
Isn't that dangerous?
Or, if nothing more, a little less structured?
PROFESSOR: No, I--
the question is whether that's less structured.
Depends on what you mean by structure.
All this is doing is recursion.
See, it's saying that the way you add these
guys is to use that.
And that's not less structured, it's just a
recursive structure.
So I don't think it's particularly any less clean.
AUDIENCE: Now when you want to change the multiplier or the
add operator, suddenly you've got tremendous consequences
underneath that you're not even sure the extent of.
PROFESSOR: That's right, but it depends what you mean.
See, this goes both ways.
What would be a good example?
I ignored greatest common divisor, for instance.
I ignored that problem just to keep the example simple.
But if I suddenly decided that plus rat here should do a GCD
computation and install that, then that immediately becomes
available to all of these, to that guy, and that guy, and
that guy, and all the way down.
So it depends what you mean by the coherence of your system.
It's certainly true that you might want to have a special
different one that didn't filter down through the
coefficients, but the nice thing about this particular
example is that mostly you do.
AUDIENCE: Isn't that the problem, I think, that you're
getting to tied in with the fact that the structuring, the
recursiveness of that structuring there is actually
in execution as opposed to just definition of the actual
types themselves?
PROFESSOR: I think I understand the question.
The point is that these types evolve and get more and more
complex as the thing's actually running.
Is that what--
AUDIENCE: Yes.
As it's running.
PROFESSOR: --what you're saying?
Yes, the point is--
AUDIENCE: As opposed to the basic definitions.
PROFESSOR: Right.
The type structure is sort of recursive.
It's not that you can make this finite list of the actual
things they might look like before the system runs.
It's something that evolves.
So if you want to specify that system, you have to do in some
other way than by this finite list. You have to do it by a
recursive structure.
AUDIENCE: Because the basic structure of the types is
pretty clean and simple.
PROFESSOR: Right.
Yes?
AUDIENCE: I have a question.
I understand once you have your data structure set up,
how it pulls off complex and passes that down, and then
pulls off rect, passes that down.
But if you're just a user and you don't know anything about
rect or polar or whatever, how do you initially set up that
data structure so that everything goes
to the right spot?
If I just have the equation over there on the left and I
just want to add, multiply complex numbers--
PROFESSOR: Well that's the wonderful thing.
If you're just a user you say "mul."
AUDIENCE: And it figures out that I mean complex numbers?
Or how do I tell it that I want--
PROFESSOR: Well you're going to have in your
hands complex numbers.
See what you would have at some level, as a real user, is
a constructor for complex numbers.
AUDIENCE: So then I have to make complex numbers?
PROFESSOR: So you have to make them.
What you would probably have as a user is some little thing
in the reader loop, which would give you some plausible
way to type in a complex number, in
whatever format you like.
Or it might be that you're never typing them in.
Someone's just handing you a complex number.
AUDIENCE: OK, so if I had a complex number that had a
polynomial in it, I'd have to make my polynomial and then
make my complex number.
PROFESSOR: Right if you wanted it constructed from scratch.
At some point you construct them from scratch.
But what you don't have to know of that is when you have
the object you can just say "mul." And it'll multiply.
Yeah?
AUDIENCE: I think the question that was being posed here is,
say if I want to change my presentation of complexes, or
some operation of complex, how much real code I will have to
gets around with, or change to change it in
one specific operation?
PROFESSOR: [UNINTELLIGIBLE] what you have to change.
And the point is that you only have to
change what you're changing.
See if Martha decides that she would rather--
let's see something silly--
like change the order in the pair.
Like angle and magnitude in the other order, she just
makes that change locally.
And the whole thing will propagate through the system
in the right way.
Or if suddenly you said, gee, I have another representation
for rationals.
And I'm going to stick it here, by filing those
operations in the table.
Then suddenly all of these polynomials whose coefficients
are coefficients of coefficients, or whatever,
also can automatically have available that representation.
That's the power of this particular one.
AUDIENCE: I'm not sure if I can even pose an intelligent
sounding question.
But somehow this whole thing went really nicely to this
beautiful finish where all the things seemed
to fall into place.
Sort of seemed a little contrived.
That's all for the sake, I'm sure, of teaching.
I doubt that the guys who first did this--
and I could be wrong--
figured it all out so that when they just all put it all
together, you could all of the sudden, blam, do any kind of
arithmetic on any kind of object.
It seems like maybe they had to play with it for a while
and had to bash it and rework it.
And it seems like that's the kind of problem we're really
faced with we start trying to design a really complex
system, is having lots of different kinds of parts and
not even knowing what kinds of operations we're going to want
to do on those parts.
How to organize the operations in this nice way so that no
matter what you do, when you start putting them together
everything starts falling out for free.
PROFESSOR: OK, well that's certainly a
very intelligent question.
One part is this is a very good methodology that people
have discovered a lot coming from symbolic algebra.
Because there are a lot of complications.
To allow you to implement these things before you decide
what you want all the operations to
be, and all of that.
So in some sense it's an answer that people have
discovered by wading through this stuff.
In another sense, it is a very contrived example.
AUDIENCE: It seems like to be able to do this you do have to
wade through it for a certain amount of time before you can
become good at it.
PROFESSOR: Let me show you how terribly contrived this is.
So you can write all these wonderful things.
But the system that I wrote here, and if we had another
half an hour to give this lecture I would have given
this part of it, which says, notice that it breaks down if
I tell it to do something as foolish as add 3 plus 7/2.
Because what will happen is you'll get to operate-2, and
operate-2 will say, oh this is type number,
and that's type rational.
I don't know how to add them.
So you'd like the system at least to be able to say
something like, gee, before you do that
change that to 3/1.
Turn it into a rational number, hand that to the
rational package.
That's the thing I didn't talk about in this lecture.
It's a little bit in the book, which talks about the problem
of what's called coercion.
Where you wanted--
see, having so carefully set up all of these types as
distinct objects, a lot of times you want to also put in
knowledge about how to view an ordinary number
as a kind of rational.
Or view an ordinary number as a kind of complex.
That's where the complexity in the system really starts
happening, where you talk about, see where
do I put that knowledge?
Is it rational to know that ordinary numbers might be
pieces of [UNINTELLIGIBLE] of them?
Or they're terrible, terrible examples, like if I might want
to add a complex number to a rational number.
Bad example.
5/7.
Then somebody's got to know that I have to convert these
to another type, which is complex numbers whose parts
might be rationals.
And who worries about that?
Does complex worry about that?
Does rational worry about that?
Does plus worry about that?
That's where the real complexity comes in.
And that's where it's pretty well sorted out.
And a lot of, in fact, all of this message passing stuff was
motivated by problems like this.
And when you really push it, people are-- somehow the
algebraic manipulation problem seems to be so complex that
the people who are always at the edge of it are exactly in
the state you said.
They're wading through this thing, mucking around, seeing
what they use, trying to distill stuff.
AUDIENCE: I just want to come back to this issue of
complexity once more.
It certainly seems to be true that you have a great deal of
flexibility in altering the lower level kinds of things.
But it is true that you are, in a sense, freezing higher
level operations.
Or at least if you change them you don't know where all of
the changes are going to show up, or how they are.
PROFESSOR: OK, that's an extremely good question.
What I have to do is, if I decide there's a new general
operation called equality test, then all of these people
have to decide whether or not they would like to have an
equality test by looking in the table.
There're ways to decentralize it even more.
That's what I sort of hinted at last time, where I said you
could not only have this type as a symbol, but you actually
might store in each object the operations
that it knows of that.
So you might have things like greatest common divisor, which
is a thing here which is defined only for integers, and
not in general for rational numbers.
So it might be a very, very fragmented system.
And then depending on where you want your flexibility,
there's a whole spectrum of places that you
can build that in.
But you're pointing at the place where this starts being
weak, that there has to be some agreement on top here
about these general operations.
Or at least people have to think about them.
Or you might decide, you might have a table that's very
sparse, that only has a few things in it.
But there are lot of ways to play that game.
OK, thank you.
[MUSIC: "JESU, JOY OF MAN'S DESIRING" BY
JOHANN SEBASTIAN BACH]