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PROFESSOR: Well, last time Gerry
really let the cat out

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of the bag.

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He introduced the idea
of assignment.

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Assignment and state.

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And as we started to see, the
implications of introducing

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assignment and state into the
language are absolutely

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frightening.

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First of all, the substitution
model of

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evaluation breaks down.

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And we have to use this much
more complicated environment

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model and this very mechanistic
thing with

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diagrams, even to say what
statements in the programming

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language mean.

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And that's not a mere
technical point.

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See, it's not that we had this
particular substitution model

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and, well, it doesn't quite
work, so we have to do

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something else.

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It's that nothing like the
substitution model can work.

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Because suddenly, a variable
is not just something that

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stands for a value.

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A variable now has to somehow
specify a place

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that holds a value.

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And the value that's in
that place can change.

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Or for instance, an expression
like f of x might have a side

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effect in it.

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So if we say f of x and it has
some value, and then later we

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say f of x again, we might
get a different value

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depending on the order.

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So suddenly, we have to think
not only about values but

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about time.

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And then things like pairs are
no longer just their CARs and

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their CDRs.

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A pair now is not quite its CAR
and its CDR. It's rather

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its identity.

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So a pair has identity.

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It's an object.

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And two pairs that have the same
CAR and CDR might be the

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same or different, because
suddenly we have to worry

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about sharing.

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So all of these things enter
as soon as we introduce

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assignment.

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See, this is a really far cry
from where we started with

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substitution.

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It's a technically harder way
of looking at things because

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we have to think more
mechanistically about our

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programming language.

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We can't just think about
it as mathematics.

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It's philosophically harder,
because suddenly there are all

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these funny issues about what
does it mean that something

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changes or that two things
are the same.

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And also, it's programming
harder, because as Gerry

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showed last time, there are all
these bugs having to do

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with bad sequencing and aliasing
that just don't exist

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in a language where we don't
worry about objects.

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Well, how'd we get
into this mess?

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Remember what we did, the reason
we got into this is

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because we were looking to
build modular systems. We

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wanted to build systems that
fall apart into chunks that

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seem natural.

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So for instance, we want to take
a random number generator

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and package up the state of that
random number generator

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inside of it so that we can
separate the idea of picking

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random numbers from the general
Monte Carlo strategy

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of estimating something and
separate that from the

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particular way that you work
with random numbers in that

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formula developed by
Cesaro for pi.

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And similarly, when we go off
and construct some models of

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things, if we go off and model
a system that we see in the

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real world, we'd like our
program to break into natural

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pieces, pieces that mirror the
parts of the system that we

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see in the real world.

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So for example, if we look at
a digital circuit, we say,

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gee, there's a circuit and
it has a piece and

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it has another piece.

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And these different pieces
sort of have identity.

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They have state.

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And the state sits
on these wires.

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And we think of this piece as
an object that's different

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from that as an object.

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And when we watch the system
change, we think about a

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signal coming in here and
changing a state that might be

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here and going here and
interacting with a state that

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might be stored there,
and so on and so on.

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So what we'd like is we'd like
to build in the computer

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systems that fall into pieces
that mirror our view of

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reality, of the way that the
actual systems we're modeling

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seem to fall into pieces.

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Well, maybe the reason that
building systems like this

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seems to introduce such
technical complications has

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nothing to do with computers.

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See, maybe the real reason that
we pay such a price to

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write programs that mirror our
view of reality is that we

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have the wrong view
of reality.

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See, maybe time is just
an illusion, and

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nothing ever changes.

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See, for example, if I take this
chalk, and we say, gee,

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this is an object and
it has a state.

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At each moment it has a position
and a velocity.

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And if we do something,
that state can change.

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But if you studied any
relativity, for instance, you

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know that you don't think of
the path of that chalk as

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something that goes on
instant by instant.

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It's more insightful to think
of that whole chalk's

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existence as a path
in space-time.

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that's all splayed out.

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There aren't individual
positions and velocities.

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There's just its unchanging
existence in space-time.

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Similarly, if we look at this
electrical system, if we

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imagine this electrical system
is implementing some sort of

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signal processing system, the
signal processing engineer who

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put that thing together doesn't
think of it as, well,

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at each instance there's
a voltage coming in.

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And that translates
into something.

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And that affects the state over
here, which changes the

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state over here.

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Nobody putting together a
signal processing system

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thinks about it like that.

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Instead, you say there's
this signal that's

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splayed out over time.

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And if this is acting as a
filter, this whole thing

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transforms this whole thing for
some sort of other output.

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You don't think of it as what's
happening instant by

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instant as the state
of these things.

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And somehow you think of this
box as a whole thing, not as

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little pieces sending messages
of state to each other at

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particular instants.

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Well, today we're going to
look at another way to

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decompose systems that's more
like the signal processing

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engineer's view of the world
than it is like thinking about

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objects that communicate
sending messages.

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That's called stream
processing.

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And we're going to start by
showing how we can make our

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programs more uniform and see
a lot more commonality if we

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throw out of these programs
what you might say is an

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inordinate concern with
worrying about time.

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Let me start by comparing
two procedures.

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The first one does this.

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We imagine that there's
a tree.

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Say there's a tree
of integers.

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It's a binary tree.

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So it looks like this.

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And there's integers in
each of the nodes.

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And what we would like to
compute is for each odd number

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sitting here, we'd like to find
the square and then sum

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up all those squares.

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Well, that should be a familiar
kind of thing.

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There's a recursive strategy
for doing it.

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We look at each leaf, and
either it's going to

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contribute the square of
the number if it's odd

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or 0 if it's even.

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And then recursively, we can say
at each tree, the sum of

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all of them is the sum coming
from the right branch and the

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left branch, and recursively
down through the nodes.

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And that's a familiar way of
thinking about programming.

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Let's actually look at
that on the slide.

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We say to sum the odd squares
in a tree, well, there's a

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test. Either it's a leaf node,
and we're going to check to

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see if it's an integer, and then
either it's odd, in which

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we take the square,
or else it's 0.

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And then the sum of the whole
thing is the sum coming from

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the left branch and
the right branch.

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OK, well, let me contrast that
with a second problem.

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Suppose I give you an integer
n, and then some function to

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compute of the first of each
integer in 1 through n.

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And then I want to collect
together in a list all those

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function values that satisfy
some property.

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That's a general
kind of thing.

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Let's say to be specific, let's
imagine that for each

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integer, k, we're
going to compute

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the k Fibonacci number.

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And then we'll see which of
those are odd and assemble

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those into a list.

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So here's a procedure
that does that.

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189
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Find the odd Fibonacci numbers
among the first n.

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And here is a standard loop the
way we've been writing it.

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This is a recursion.

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It's a loop on k, and says if
k is bigger than n, it's the

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empty list. Otherwise we compute
the k-th Fibonacci

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number, call that f.

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If it's odd, we CONS it on
to the list starting

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with the next one.

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And otherwise, we just
take the next one.

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And this is the standard
way we've been

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writing iterative loops.

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And we start off calling
that loop with 1.

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OK, so there are
two procedures.

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Those procedures look
very different.

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00:11:02,900 --> 00:11:04,390
They have very different
structures.

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Yet from a certain point of
view, those procedures are

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00:11:07,740 --> 00:11:11,330
really doing very much
the same thing.

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00:11:11,330 --> 00:11:14,930
So if I was talking like a
signal processing engineer,

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00:11:14,930 --> 00:11:25,730
what I might say is that the
first procedure enumerates the

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00:11:25,730 --> 00:11:26,980
leaves of a tree.

209
00:11:26,980 --> 00:11:31,160


210
00:11:31,160 --> 00:11:33,510
And then we can think of a
signal coming out of that,

211
00:11:33,510 --> 00:11:35,330
which is all the leaves.

212
00:11:35,330 --> 00:11:43,970
We'll filter them to see which
ones are odd, put them through

213
00:11:43,970 --> 00:11:45,190
some kind of filter.

214
00:11:45,190 --> 00:11:49,000
We'll then put them through
a kind of transducer.

215
00:11:49,000 --> 00:11:51,420
And for each one of those
things, we'll take the square.

216
00:11:51,420 --> 00:11:54,200


217
00:11:54,200 --> 00:11:58,290
And then we'll accumulate
all of those.

218
00:11:58,290 --> 00:12:00,570
We'll accumulate them by
sticking them together with

219
00:12:00,570 --> 00:12:03,340
addition starting from 0.

220
00:12:03,340 --> 00:12:07,140


221
00:12:07,140 --> 00:12:08,210
That's the first program.

222
00:12:08,210 --> 00:12:10,620
The second program, I can
describe in a very, very

223
00:12:10,620 --> 00:12:11,780
similar way.

224
00:12:11,780 --> 00:12:17,450
I'll say, we'll enumerate the
numbers on this interval, for

225
00:12:17,450 --> 00:12:19,080
the interval 1 through n.

226
00:12:19,080 --> 00:12:22,500


227
00:12:22,500 --> 00:12:28,080
We'll, for each one, compute the
Fibonacci number, put them

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00:12:28,080 --> 00:12:29,270
through a transducer.

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00:12:29,270 --> 00:12:31,780
We'll then take the result
of that, and we'll

230
00:12:31,780 --> 00:12:35,976
filter it for oddness.

231
00:12:35,976 --> 00:12:39,350
And then we'll take those and
put them into an accumulator.

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00:12:39,350 --> 00:12:41,730
This time we'll build up a list,
so we'll accumulate with

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00:12:41,730 --> 00:12:47,110
CONS starting from
the empty list.

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00:12:47,110 --> 00:12:50,940
So this way of looking at the
program makes the two seem

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00:12:50,940 --> 00:12:51,900
very, very similar.

236
00:12:51,900 --> 00:12:55,880
The problem is that that
commonality is completely

237
00:12:55,880 --> 00:12:58,050
obscured when we look at the
procedures we wrote.

238
00:12:58,050 --> 00:13:02,670
Let's go back and look at some
odd squares again, and say

239
00:13:02,670 --> 00:13:06,300
things like, where's
the enumerator?

240
00:13:06,300 --> 00:13:08,140
Where's the enumerator
in this program?

241
00:13:08,140 --> 00:13:11,230
Well, it's not in one place.

242
00:13:11,230 --> 00:13:15,990
It's a little bit in this
leaf-node test,

243
00:13:15,990 --> 00:13:17,160
which is going to stop.

244
00:13:17,160 --> 00:13:19,380
It's a little bit in the
recursive structure of the

245
00:13:19,380 --> 00:13:20,630
thing itself.

246
00:13:20,630 --> 00:13:23,150


247
00:13:23,150 --> 00:13:24,120
Where's the accumulator?

248
00:13:24,120 --> 00:13:25,680
The accumulator isn't
in one place either.

249
00:13:25,680 --> 00:13:32,180
It's partly in this 0 and
partly in this plus.

250
00:13:32,180 --> 00:13:34,510
It's not there as a thing
that we can look at.

251
00:13:34,510 --> 00:13:40,550
Similarly, if we look at odd
Fibs, that's also, in some

252
00:13:40,550 --> 00:13:42,940
sense, an enumerator and
an accumulator, but

253
00:13:42,940 --> 00:13:44,470
it looks very different.

254
00:13:44,470 --> 00:13:49,260
Because partly, the enumerator
is here in this greater than

255
00:13:49,260 --> 00:13:52,100
sign in the test. And partly
it's in this whole recursive

256
00:13:52,100 --> 00:13:55,680
structure in the loop, and
the way that we call it.

257
00:13:55,680 --> 00:13:58,100
And then similarly, that's also
mixed up in there with

258
00:13:58,100 --> 00:14:01,010
the accumulator, which is partly
over there and partly

259
00:14:01,010 --> 00:14:03,600
over there.

260
00:14:03,600 --> 00:14:09,790
So these very, very natural
pieces, these very natural

261
00:14:09,790 --> 00:14:13,770
boxes here don't appear in our
programs. Because they're kind

262
00:14:13,770 --> 00:14:14,360
of mixed up.

263
00:14:14,360 --> 00:14:16,290
The programs don't chop things
up in the right way.

264
00:14:16,290 --> 00:14:19,450


265
00:14:19,450 --> 00:14:22,240
Going back to this fundamental
principle of computer science

266
00:14:22,240 --> 00:14:24,620
that in order to control
something, you need the name

267
00:14:24,620 --> 00:14:27,820
of it, we don't really have
control over thinking about

268
00:14:27,820 --> 00:14:30,500
things this way because we don't
have our hands in them

269
00:14:30,500 --> 00:14:31,060
explicitly.

270
00:14:31,060 --> 00:14:35,510
We don't have a good language
for talking about them.

271
00:14:35,510 --> 00:14:42,850
Well, let's invent an
appropriate language in which

272
00:14:42,850 --> 00:14:44,515
we can build these pieces.

273
00:14:44,515 --> 00:14:48,650
The key to the language is these
guys, is what is these

274
00:14:48,650 --> 00:14:50,480
things I called signals?

275
00:14:50,480 --> 00:14:52,070
What are these things that
are flying on the

276
00:14:52,070 --> 00:14:53,320
arrows between the boxes?

277
00:14:53,320 --> 00:14:56,880


278
00:14:56,880 --> 00:15:02,840
Well, those things are going to
be data structures called

279
00:15:02,840 --> 00:15:04,770
streams. That's going
to be the key to

280
00:15:04,770 --> 00:15:07,980
inventing this language.

281
00:15:07,980 --> 00:15:08,600
What's a stream?

282
00:15:08,600 --> 00:15:10,820
Well, a stream is, like
anything else, a data

283
00:15:10,820 --> 00:15:12,220
abstraction.

284
00:15:12,220 --> 00:15:15,000
So I should tell you what
its selectors and

285
00:15:15,000 --> 00:15:16,870
constructors are.

286
00:15:16,870 --> 00:15:20,185
For a stream, we're going to
have one constructor that's

287
00:15:20,185 --> 00:15:21,435
called CONS-stream.

288
00:15:21,435 --> 00:15:25,690


289
00:15:25,690 --> 00:15:29,060
CONS-stream is going to put two
things together to form a

290
00:15:29,060 --> 00:15:32,040
thing called a stream.

291
00:15:32,040 --> 00:15:34,250
And then to extract things from
the stream, we're going

292
00:15:34,250 --> 00:15:38,010
to have a selector called
the head of the stream.

293
00:15:38,010 --> 00:15:41,340
So if I have a stream, I
can take its head or I

294
00:15:41,340 --> 00:15:44,720
can take its tail.

295
00:15:44,720 --> 00:15:48,290
And remember, I have to tell you
George's contract here to

296
00:15:48,290 --> 00:15:53,160
tell you what the axioms
are that relate these.

297
00:15:53,160 --> 00:16:04,080
And it's going to be for any
x and y, if I form the

298
00:16:04,080 --> 00:16:11,420
CONS-stream and take the head,
the head of CONS-stream of x

299
00:16:11,420 --> 00:16:26,590
and y is going to be x and the
tail of CONS-stream of x and y

300
00:16:26,590 --> 00:16:28,440
is going to be y.

301
00:16:28,440 --> 00:16:31,180
So those are the constructor,
two selectors for

302
00:16:31,180 --> 00:16:34,750
streams, and an axiom.

303
00:16:34,750 --> 00:16:36,980
There's something fishy here.

304
00:16:36,980 --> 00:16:41,060
So you might notice that these
are exactly the axioms for

305
00:16:41,060 --> 00:16:46,100
CONS, CAR, and CDR. If instead
of writing CONS-stream I wrote

306
00:16:46,100 --> 00:16:50,810
CONS and I said head was the
CAR and tail was the CDR,

307
00:16:50,810 --> 00:16:52,810
those are exactly the
axioms for pairs.

308
00:16:52,810 --> 00:16:55,130
And in fact, there's
another thing here.

309
00:16:55,130 --> 00:17:02,930
We're going to have a thing
called the-empty-stream, which

310
00:17:02,930 --> 00:17:08,319
is like the-empty-list.

311
00:17:08,319 --> 00:17:10,030
So why am I introducing
this terminology?

312
00:17:10,030 --> 00:17:12,780
Why don't I just keep talking
about pairs and lists?

313
00:17:12,780 --> 00:17:15,510
Well, we'll see.

314
00:17:15,510 --> 00:17:18,440
For now, if you like, why don't
you just pretend that

315
00:17:18,440 --> 00:17:21,560
streams really are just a
terminology for lists.

316
00:17:21,560 --> 00:17:24,890
And we'll see in a little while
why we want to keep this

317
00:17:24,890 --> 00:17:28,150
extra abstraction layer and
not just call them lists.

318
00:17:28,150 --> 00:17:32,300


319
00:17:32,300 --> 00:17:34,860
OK, now that we have streams, we
can start constructing the

320
00:17:34,860 --> 00:17:38,990
pieces of the language to
operate on streams. And there

321
00:17:38,990 --> 00:17:41,330
are a whole bunch of very useful
things that we could

322
00:17:41,330 --> 00:17:42,120
start making.

323
00:17:42,120 --> 00:17:54,850
For instance, we'll make our map
box to take a stream, s,

324
00:17:54,850 --> 00:18:00,400
and a procedure, and to generate
a new stream which

325
00:18:00,400 --> 00:18:03,640
has as its elements the
procedure applied to all the

326
00:18:03,640 --> 00:18:05,666
successive elements of s.

327
00:18:05,666 --> 00:18:07,400
In fact, we've seen
this before.

328
00:18:07,400 --> 00:18:10,950
This is the procedure map
that we did with lists.

329
00:18:10,950 --> 00:18:14,000
And you see it's exactly map,
except we're testing for

330
00:18:14,000 --> 00:18:14,650
empty-stream.

331
00:18:14,650 --> 00:18:15,560
Oh, I forgot to mention that.

332
00:18:15,560 --> 00:18:19,420
Empty-stream is like the null
test. So if it's empty, we

333
00:18:19,420 --> 00:18:20,510
generate the empty stream.

334
00:18:20,510 --> 00:18:24,700
Otherwise, we form a new stream
whose first element is

335
00:18:24,700 --> 00:18:28,950
the procedure applied to the
head of the stream, and whose

336
00:18:28,950 --> 00:18:31,570
rest is gotten by mapping along
with the procedure down

337
00:18:31,570 --> 00:18:33,140
the tail of the stream.

338
00:18:33,140 --> 00:18:34,920
So that looks exactly like
the map procedure

339
00:18:34,920 --> 00:18:37,030
we looked at before.

340
00:18:37,030 --> 00:18:38,350
Here's another useful thing.

341
00:18:38,350 --> 00:18:40,460
Filter, this is our
filter box.

342
00:18:40,460 --> 00:18:43,890
We're going to have a predicate
and a stream.

343
00:18:43,890 --> 00:18:46,720
We're going to make a new stream
that consists of all

344
00:18:46,720 --> 00:18:48,310
the elements of the
original one

345
00:18:48,310 --> 00:18:50,160
that satisfy the predicate.

346
00:18:50,160 --> 00:18:51,270
That's case analysis.

347
00:18:51,270 --> 00:18:53,140
When there's nothing
in the stream, we

348
00:18:53,140 --> 00:18:56,280
return the empty stream.

349
00:18:56,280 --> 00:19:00,060
We test the predicate on
the head of the stream.

350
00:19:00,060 --> 00:19:03,520
And if it's true, we add the
head of the stream onto the

351
00:19:03,520 --> 00:19:08,220
result of filtering the
tail of the stream.

352
00:19:08,220 --> 00:19:10,870
And otherwise, if that predicate
was false, we just

353
00:19:10,870 --> 00:19:13,500
filter the tail of the stream.

354
00:19:13,500 --> 00:19:16,595
Right, so there's filter.

355
00:19:16,595 --> 00:19:18,560
Let me run through a couple
more rather quickly.

356
00:19:18,560 --> 00:19:20,880
They're all in the book and
you can look at them.

357
00:19:20,880 --> 00:19:22,110
Let me just flash through.

358
00:19:22,110 --> 00:19:23,260
Here's accumulate.

359
00:19:23,260 --> 00:19:27,690
Accumulate takes a way of
combining things and an

360
00:19:27,690 --> 00:19:31,560
initial value in a stream and
sticks them all together.

361
00:19:31,560 --> 00:19:33,970
If the stream's empty, it's
just the initial value.

362
00:19:33,970 --> 00:19:36,930
Otherwise, we combine the head
of the stream with the result

363
00:19:36,930 --> 00:19:39,550
of accumulating the tail of the
stream starting from the

364
00:19:39,550 --> 00:19:40,900
initial value.

365
00:19:40,900 --> 00:19:42,830
So that's what I'd use to add
up everything in the stream.

366
00:19:42,830 --> 00:19:45,830
I'd accumulate with plus.

367
00:19:45,830 --> 00:19:48,060
How would I enumerate the
leaves of a tree?

368
00:19:48,060 --> 00:19:54,530
Well, if the tree is just a leaf
itself, I make something

369
00:19:54,530 --> 00:19:56,640
which only has that
node in it.

370
00:19:56,640 --> 00:20:01,100
Otherwise, I append together the
stuff of enumerating the

371
00:20:01,100 --> 00:20:04,340
left branch and the
right branch.

372
00:20:04,340 --> 00:20:08,130
And then append here is like the
ordinary append on lists.

373
00:20:08,130 --> 00:20:13,190


374
00:20:13,190 --> 00:20:13,850
You can look at that.

375
00:20:13,850 --> 00:20:16,410
That's analogous to the
ordinary procedure for

376
00:20:16,410 --> 00:20:19,150
appending two lists.

377
00:20:19,150 --> 00:20:21,810
How would I enumerate
an interval?

378
00:20:21,810 --> 00:20:24,500
This will take two integers, low
and high, and generate a

379
00:20:24,500 --> 00:20:28,106
stream of the integers going
from low to high.

380
00:20:28,106 --> 00:20:31,890
And we can make a whole
bunch of pieces.

381
00:20:31,890 --> 00:20:34,860
So that's a little language of
talking about streams. Once we

382
00:20:34,860 --> 00:20:37,670
have streams, we can build
things for manipulating them.

383
00:20:37,670 --> 00:20:40,200
Again, we're making
a language.

384
00:20:40,200 --> 00:20:41,270
And now we can start expressing

385
00:20:41,270 --> 00:20:43,060
things in this language.

386
00:20:43,060 --> 00:20:46,590
Here's our original procedure
for summing the odd

387
00:20:46,590 --> 00:20:47,310
squares in a tree.

388
00:20:47,310 --> 00:20:52,210
And you'll notice it looks
exactly now like the block

389
00:20:52,210 --> 00:20:54,590
diagram, like the signal
processing block diagram.

390
00:20:54,590 --> 00:21:00,230
So to sum the odd squares in a
tree, we enumerate the leaves

391
00:21:00,230 --> 00:21:01,320
of the tree.

392
00:21:01,320 --> 00:21:04,830
We filter that for oddness.

393
00:21:04,830 --> 00:21:06,220
We map that for squareness.

394
00:21:06,220 --> 00:21:09,320


395
00:21:09,320 --> 00:21:12,460
And we accumulate the result
of that using addition,

396
00:21:12,460 --> 00:21:14,760
starting from 0.

397
00:21:14,760 --> 00:21:17,290
So we can see the pieces
that we wanted.

398
00:21:17,290 --> 00:21:22,050
Similarly, the Fibonacci one,
how do we get the odd Fibs?

399
00:21:22,050 --> 00:21:27,900
Well, we enumerate the interval
from 1 to n, we map

400
00:21:27,900 --> 00:21:30,920
along that, computing the
Fibonacci of each one.

401
00:21:30,920 --> 00:21:34,810
We filter the result of
those for oddness.

402
00:21:34,810 --> 00:21:38,460
And we accumulate all of that
stuff using CONS starting from

403
00:21:38,460 --> 00:21:43,650
the empty-list.

404
00:21:43,650 --> 00:21:47,680
OK, what's the advantage
of this?

405
00:21:47,680 --> 00:21:50,260
Well, for one thing, we now have
pieces that we can start

406
00:21:50,260 --> 00:21:51,880
mixing and matching.

407
00:21:51,880 --> 00:21:58,230
So for instance, if I wanted to
change this, if I wanted to

408
00:21:58,230 --> 00:22:00,400
compute the squares of the
integers and then filter them,

409
00:22:00,400 --> 00:22:03,810
all I need to do is pick up a
standard piece like this in

410
00:22:03,810 --> 00:22:06,210
that square and put it in.

411
00:22:06,210 --> 00:22:10,150
Or if we wanted to do this whole
Fibonacci computation on

412
00:22:10,150 --> 00:22:12,980
the leaves of a tree rather than
a sequence, all I need to

413
00:22:12,980 --> 00:22:18,030
do is replace this enumerator
with that one.

414
00:22:18,030 --> 00:22:20,650
See, the advantage of this
stream processing is that

415
00:22:20,650 --> 00:22:21,995
we're establishing--

416
00:22:21,995 --> 00:22:25,330
this is one of the big themes
of the course--

417
00:22:25,330 --> 00:22:35,570
we're establishing conventional
interfaces that

418
00:22:35,570 --> 00:22:38,130
allow us to glue things
together.

419
00:22:38,130 --> 00:22:41,730
Things like map and filter are
a standard set of components

420
00:22:41,730 --> 00:22:43,900
that we can start using for
pasting together programs in

421
00:22:43,900 --> 00:22:45,750
all sorts of ways.

422
00:22:45,750 --> 00:22:50,090
It allows us to see the
commonality of programs.

423
00:22:50,090 --> 00:22:52,390
I just ought to mention, I've
only showed you two

424
00:22:52,390 --> 00:22:53,860
procedures.

425
00:22:53,860 --> 00:22:57,800
But let me emphasize that this
way of putting things together

426
00:22:57,800 --> 00:22:59,780
with maps, filters,
and accumulators

427
00:22:59,780 --> 00:23:01,410
is very, very general.

428
00:23:01,410 --> 00:23:08,010
It's the generate and test
paradigm for programs. And as

429
00:23:08,010 --> 00:23:11,970
an example of that, Richard
Waters, who was at MIT when he

430
00:23:11,970 --> 00:23:14,060
was a graduate student, as part
of his thesis research

431
00:23:14,060 --> 00:23:17,700
went and analyzed a large chunk
of the IBM scientific

432
00:23:17,700 --> 00:23:22,340
subroutine library, and
discovered that about 60% of

433
00:23:22,340 --> 00:23:26,830
the programs in it could be
expressed exactly in terms

434
00:23:26,830 --> 00:23:28,940
using no more than what
we've put here--

435
00:23:28,940 --> 00:23:30,710
map, filter, and accumulate.

436
00:23:30,710 --> 00:23:31,960
All right, let's take a break.

437
00:23:31,960 --> 00:23:36,620


438
00:23:36,620 --> 00:23:37,870
Questions?

439
00:23:37,870 --> 00:23:40,470


440
00:23:40,470 --> 00:23:43,033
AUDIENCE: It seems like the
essence of this whole thing is

441
00:23:43,033 --> 00:23:45,980
just that you have a very
uniform, simple data structure

442
00:23:45,980 --> 00:23:48,380
to work with, the stream.

443
00:23:48,380 --> 00:23:48,920
PROFESSOR: Right.

444
00:23:48,920 --> 00:23:51,670
The essence is that you, again,
it's this sense of

445
00:23:51,670 --> 00:23:53,710
conventional interfaces.

446
00:23:53,710 --> 00:23:55,610
So you can start putting a
lot of things together.

447
00:23:55,610 --> 00:23:59,830
And the stream is as you say,
the uniform data structure

448
00:23:59,830 --> 00:24:00,890
that supports that.

449
00:24:00,890 --> 00:24:03,600
This is very much like
APL, by the way.

450
00:24:03,600 --> 00:24:06,330
APL is very much the same idea,
except in APL, instead

451
00:24:06,330 --> 00:24:09,560
of this stream, you have
arrays and vectors.

452
00:24:09,560 --> 00:24:13,565
And a lot of the power of APL is
exactly the same reason of

453
00:24:13,565 --> 00:24:14,815
the power of this.

454
00:24:14,815 --> 00:24:19,910


455
00:24:19,910 --> 00:24:20,910
OK, thank you.

456
00:24:20,910 --> 00:24:22,160
Let's take a break.

457
00:24:22,160 --> 00:24:57,470


458
00:24:57,470 --> 00:24:57,610
All right.

459
00:24:57,610 --> 00:25:02,830
We've been looking at ways of
organizing computations using

460
00:25:02,830 --> 00:25:07,560
streams. What I want to do now
is just show you two somewhat

461
00:25:07,560 --> 00:25:10,810
more complicated examples
of that.

462
00:25:10,810 --> 00:25:15,000
Let's start by thinking about
the following kind of utility

463
00:25:15,000 --> 00:25:16,810
procedure that will
come in useful.

464
00:25:16,810 --> 00:25:19,960
Suppose I've got a stream.

465
00:25:19,960 --> 00:25:23,730
And the elements of this stream
are themselves streams.

466
00:25:23,730 --> 00:25:26,530
So the first thing
might be 1, 2, 3.

467
00:25:26,530 --> 00:25:32,600


468
00:25:32,600 --> 00:25:33,880
So I've got a stream.

469
00:25:33,880 --> 00:25:40,100
And each element of the stream
is itself a stream.

470
00:25:40,100 --> 00:25:45,580
And what I'd like to do is build
a stream that collects

471
00:25:45,580 --> 00:25:47,870
together all of the elements,
pulls all of the elements out

472
00:25:47,870 --> 00:25:50,840
of these sub-streams and
strings them all

473
00:25:50,840 --> 00:25:52,080
together in one thing.

474
00:25:52,080 --> 00:25:56,220
So just to show you the use of
this language, how easy it is,

475
00:25:56,220 --> 00:25:56,960
call that flatten.

476
00:25:56,960 --> 00:26:13,020
And I can define to flatten this
stream of streams. Well,

477
00:26:13,020 --> 00:26:13,960
what is that?

478
00:26:13,960 --> 00:26:16,240
That's just an accumulation.

479
00:26:16,240 --> 00:26:25,240
I want to accumulate
using append, by

480
00:26:25,240 --> 00:26:26,450
successively appending.

481
00:26:26,450 --> 00:26:36,590
So I accumulate using append
streams, starting with

482
00:26:36,590 --> 00:26:54,370
the-empty-stream down that
stream of streams.

483
00:26:54,370 --> 00:26:58,290
OK, so there's an example of how
you can start using these

484
00:26:58,290 --> 00:27:00,830
higher order things to do some
interesting operations.

485
00:27:00,830 --> 00:27:04,230
In fact, there's another
useful thing

486
00:27:04,230 --> 00:27:05,100
that I want to do.

487
00:27:05,100 --> 00:27:18,700
I want to define a procedure
called flat-map, flat map of

488
00:27:18,700 --> 00:27:21,840
some function and a stream.

489
00:27:21,840 --> 00:27:23,920
And what this is going
to do is f will

490
00:27:23,920 --> 00:27:25,720
be a stream of elements.

491
00:27:25,720 --> 00:27:28,930
f is going to be a function that
for each element in the

492
00:27:28,930 --> 00:27:31,950
stream produces another
stream.

493
00:27:31,950 --> 00:27:33,950
And what I want to do is take
all of the elements and all of

494
00:27:33,950 --> 00:27:36,000
those streams and combine
them together.

495
00:27:36,000 --> 00:27:51,350
So that's just going to be the
flatten of map f down s.

496
00:27:51,350 --> 00:27:54,290
Each time I apply f to an
element of s, I get a stream.

497
00:27:54,290 --> 00:27:56,690
If I map it all the way down, I
get a stream of streams, and

498
00:27:56,690 --> 00:27:58,385
I'll flatten that.

499
00:27:58,385 --> 00:28:04,670
Well, I want to use that to
show you a new way to do a

500
00:28:04,670 --> 00:28:06,360
familiar kind of problem.

501
00:28:06,360 --> 00:28:12,310
The problem's going to be like a
lot of problems you've seen,

502
00:28:12,310 --> 00:28:14,190
although maybe not this
particular one.

503
00:28:14,190 --> 00:28:15,490
I'm going to give you
an integer, n.

504
00:28:15,490 --> 00:28:18,480


505
00:28:18,480 --> 00:28:31,020
And the problem is going to be
find all pairs and integers i

506
00:28:31,020 --> 00:28:42,740
and j, between 0 and i, with j
less than i, up to n, such

507
00:28:42,740 --> 00:28:51,910
that i plus j is prime.

508
00:28:51,910 --> 00:28:55,740


509
00:28:55,740 --> 00:29:00,520
So for example, if n equals 6,
let's make a little table

510
00:29:00,520 --> 00:29:06,640
here, i and j and i plus j.

511
00:29:06,640 --> 00:29:09,700


512
00:29:09,700 --> 00:29:15,520
So for, say, i equals 2 and
j equals 1, I'd get 3.

513
00:29:15,520 --> 00:29:18,940
And for i equals 3, I could
have j equals 2, and that

514
00:29:18,940 --> 00:29:21,210
would be 5.

515
00:29:21,210 --> 00:29:28,400
And 4 and 1 would be 5 and so
on, up until i goes to 6.

516
00:29:28,400 --> 00:29:33,640
And what I'd like to return is
to produce a stream of all the

517
00:29:33,640 --> 00:29:37,350
triples like this, let's
say i, j, and i plus j.

518
00:29:37,350 --> 00:29:41,530
So for each n, I want to
generate this stream.

519
00:29:41,530 --> 00:29:43,680
OK, well, that's easy.

520
00:29:43,680 --> 00:29:47,230
Let's build it up.

521
00:29:47,230 --> 00:29:50,150
We start like this.

522
00:29:50,150 --> 00:29:55,510
We're going to say for
each i, we're going

523
00:29:55,510 --> 00:29:56,440
to generate a stream.

524
00:29:56,440 --> 00:29:58,830
For each i in the interval 1
through n, we're going to

525
00:29:58,830 --> 00:30:00,660
generate a stream.

526
00:30:00,660 --> 00:30:02,230
What's that stream
going to be?

527
00:30:02,230 --> 00:30:04,180
We're going to start by
generating all the pairs.

528
00:30:04,180 --> 00:30:11,840
So for each i, we're going to
generate, for each j in the

529
00:30:11,840 --> 00:30:19,450
interval 1 to i minus 1, we'll
generate the pair, or the list

530
00:30:19,450 --> 00:30:20,710
with two elements i and j.

531
00:30:20,710 --> 00:30:23,780


532
00:30:23,780 --> 00:30:30,712
So we map along the interval,
generating the pairs.

533
00:30:30,712 --> 00:30:33,170
And for each i, that generates
a stream of pairs.

534
00:30:33,170 --> 00:30:34,590
And we flatmap it.

535
00:30:34,590 --> 00:30:37,390
Now we have all the pairs
i and j, such that i

536
00:30:37,390 --> 00:30:38,730
is less than j.

537
00:30:38,730 --> 00:30:39,850
So that builds that.

538
00:30:39,850 --> 00:30:42,990
Now we're got to test them.

539
00:30:42,990 --> 00:30:47,160
Well, we take that thing we just
built, the flatmap, and

540
00:30:47,160 --> 00:30:50,090
we filter it to see
whether the i--

541
00:30:50,090 --> 00:30:51,660
see, we had an i and a j.

542
00:30:51,660 --> 00:30:55,180
i was the first thing in the
list, j was the second thing

543
00:30:55,180 --> 00:30:59,030
in the list. So we have a
predicate which says in that

544
00:30:59,030 --> 00:31:00,870
list of two elements
is the sum of the

545
00:31:00,870 --> 00:31:02,070
CAR and the CDR prime.

546
00:31:02,070 --> 00:31:06,540
And we filter that collection
of pairs we just built.

547
00:31:06,540 --> 00:31:09,420
So those are the
pairs we want.

548
00:31:09,420 --> 00:31:13,340
Now we go ahead and we take the
result of that filter and

549
00:31:13,340 --> 00:31:19,610
we map along it, generating the
list i and j and i plus j.

550
00:31:19,610 --> 00:31:22,910
And that's our procedure
prime-sum-pairs.

551
00:31:22,910 --> 00:31:24,480
And then just to flash it up,
here's the whole procedure.

552
00:31:24,480 --> 00:31:27,945


553
00:31:27,945 --> 00:31:30,750
A map, a filter, a flatmap.

554
00:31:30,750 --> 00:31:34,850


555
00:31:34,850 --> 00:31:36,350
There's the whole thing,
even though this isn't

556
00:31:36,350 --> 00:31:37,120
particularly readable.

557
00:31:37,120 --> 00:31:40,000
It's just expanding
that flatmap.

558
00:31:40,000 --> 00:31:45,090
So there's an example which
illustrates the general point

559
00:31:45,090 --> 00:31:49,350
that nested loops in this
procedure start looking like

560
00:31:49,350 --> 00:31:52,370
compositions of flatmaps of
flatmaps of flatmaps of maps

561
00:31:52,370 --> 00:31:54,200
and things.

562
00:31:54,200 --> 00:31:57,900
So not only can we enumerate
individual things, but by

563
00:31:57,900 --> 00:32:00,890
using flatmaps, we can do what
would correspond to nested

564
00:32:00,890 --> 00:32:03,230
loops in most other languages.

565
00:32:03,230 --> 00:32:06,870
Of course, it's pretty awful to
keep writing these flatmaps

566
00:32:06,870 --> 00:32:08,410
of flatmaps of flatmaps.

567
00:32:08,410 --> 00:32:13,830
Prime-sum-pairs you saw looked
fairly complicated, even

568
00:32:13,830 --> 00:32:15,480
though the individual
pieces were easy.

569
00:32:15,480 --> 00:32:17,800
So what you can do, if you
like, is introduced some

570
00:32:17,800 --> 00:32:21,040
syntactic sugar that's
called collect.

571
00:32:21,040 --> 00:32:23,570
And collect is just an
abbreviation for that nest of

572
00:32:23,570 --> 00:32:26,160
flatmaps and filters arranged
in that particular way.

573
00:32:26,160 --> 00:32:29,620
Here's prime-sum-pairs again,
written using collect.

574
00:32:29,620 --> 00:32:32,670
It says to find all those pairs,
I'm going to collect

575
00:32:32,670 --> 00:32:40,910
together a result, which is the
list i, j, and i plus j,

576
00:32:40,910 --> 00:32:44,510
that's going to be generated as
i runs through the interval

577
00:32:44,510 --> 00:32:51,440
from 1 to n and as j runs
through the interval from 1 to

578
00:32:51,440 --> 00:32:58,040
i minus 1, such that
i plus j is prime.

579
00:32:58,040 --> 00:33:00,690
So I'm not going to say what
collect does in general.

580
00:33:00,690 --> 00:33:03,420
You can look at that by looking
at it in the book.

581
00:33:03,420 --> 00:33:06,010
But pretty much, you can see
that the pieces of this are

582
00:33:06,010 --> 00:33:08,820
the pieces of that original
procedure I wrote.

583
00:33:08,820 --> 00:33:11,550
And this collect is just some
syntactic sugar for

584
00:33:11,550 --> 00:33:16,310
automatically generating that
nest of flatmaps and flatmaps.

585
00:33:16,310 --> 00:33:21,120
OK, well, let me do one more
example that shows you the

586
00:33:21,120 --> 00:33:22,120
same kind of thing.

587
00:33:22,120 --> 00:33:25,740
Here's a very famous problem
that's used to illustrate a

588
00:33:25,740 --> 00:33:28,980
lot of so-called backtracking
computer algorithms. This is

589
00:33:28,980 --> 00:33:30,200
the eight queens problem.

590
00:33:30,200 --> 00:33:32,370
This is a chess board.

591
00:33:32,370 --> 00:33:34,570
And the eight queens problem
says, find a way to put down

592
00:33:34,570 --> 00:33:37,660
eight queens on a chess board
so that no two are attacking

593
00:33:37,660 --> 00:33:38,000
each other.

594
00:33:38,000 --> 00:33:39,685
And here's a particular
solution to the

595
00:33:39,685 --> 00:33:41,430
eight queens problem.

596
00:33:41,430 --> 00:33:44,450
So I have to make sure to put
down queens so that no two are

597
00:33:44,450 --> 00:33:48,570
in the same row or the
same column or sit

598
00:33:48,570 --> 00:33:51,410
along the same diagonal.

599
00:33:51,410 --> 00:33:56,400
Now, there's sort of a standard
way of doing that.

600
00:33:56,400 --> 00:33:59,740


601
00:33:59,740 --> 00:34:03,200
Well, first we need
to do is below the

602
00:34:03,200 --> 00:34:04,940
surface, at George's level.

603
00:34:04,940 --> 00:34:07,340
We have to find some way to
represent a board, and

604
00:34:07,340 --> 00:34:08,095
represent positions.

605
00:34:08,095 --> 00:34:09,800
And we'll not worry
about that.

606
00:34:09,800 --> 00:34:12,540
But let's assume that there's
a predicate called safe.

607
00:34:12,540 --> 00:34:16,040


608
00:34:16,040 --> 00:34:19,090
And what safe is going to do is
going to say given that I

609
00:34:19,090 --> 00:34:22,520
have a bunch of queens down on
the chess board, is it OK to

610
00:34:22,520 --> 00:34:25,400
put a queen in this
particular spot?

611
00:34:25,400 --> 00:34:32,889
So safe is going to take
a row and a column.

612
00:34:32,889 --> 00:34:34,510
That's going to be a place where
I'm going to try and put

613
00:34:34,510 --> 00:34:42,370
down the next queen, and
the rest of positions.

614
00:34:42,370 --> 00:34:45,420


615
00:34:45,420 --> 00:34:48,679
And what safe will say is given
that I already have

616
00:34:48,679 --> 00:34:53,920
queens down in these positions,
is it safe to put

617
00:34:53,920 --> 00:34:58,300
another queen down in that
row and that column?

618
00:34:58,300 --> 00:34:59,360
And let's not worry
about that.

619
00:34:59,360 --> 00:35:01,380
That's George's problem. and
it's not hard to write.

620
00:35:01,380 --> 00:35:06,350
You just have to check whether
this thing contains any things

621
00:35:06,350 --> 00:35:10,530
on that row or that column
or in that diagonal.

622
00:35:10,530 --> 00:35:13,590
Now, how would you organize
the program given that?

623
00:35:13,590 --> 00:35:18,010
And there's sort of a
traditional way to organize it

624
00:35:18,010 --> 00:35:20,116
called backtracking.

625
00:35:20,116 --> 00:35:27,570
And it says, well, let's think
about all the ways of putting

626
00:35:27,570 --> 00:35:31,290
the first queen down in
the first column.

627
00:35:31,290 --> 00:35:32,580
There are eight ways.

628
00:35:32,580 --> 00:35:35,880
Well, let's say try
the first column.

629
00:35:35,880 --> 00:35:37,300
Try column 1, row 1.

630
00:35:37,300 --> 00:35:41,300
These branches are going to
represent the possibilities at

631
00:35:41,300 --> 00:35:43,360
each level.

632
00:35:43,360 --> 00:35:45,875
So I'll try and put a queen
down in the first column.

633
00:35:45,875 --> 00:35:48,360
And now given that it's in the
first column, I'll try and put

634
00:35:48,360 --> 00:35:49,980
the next queen down in
the first column.

635
00:35:49,980 --> 00:35:53,035


636
00:35:53,035 --> 00:35:55,470
I'll try and put the first
queen, the one in the first

637
00:35:55,470 --> 00:35:56,920
column, down in the first row.

638
00:35:56,920 --> 00:35:59,050
I'm sorry.

639
00:35:59,050 --> 00:36:00,780
And then given that, we'll
put the next queen down

640
00:36:00,780 --> 00:36:01,390
in the first row.

641
00:36:01,390 --> 00:36:02,090
And that's no good.

642
00:36:02,090 --> 00:36:04,200
So I'll back up to here.

643
00:36:04,200 --> 00:36:06,280
And I'll say, oh, can I put the
first queen down in the

644
00:36:06,280 --> 00:36:07,510
second row?

645
00:36:07,510 --> 00:36:08,550
Well, that's no good.

646
00:36:08,550 --> 00:36:09,760
Oh, can I put it down
in the third row?

647
00:36:09,760 --> 00:36:12,790
Well, that's good.

648
00:36:12,790 --> 00:36:14,290
Well, now can I put the
next queen down

649
00:36:14,290 --> 00:36:15,380
in the first column?

650
00:36:15,380 --> 00:36:18,030
Well, I can't visualize this
chess board anymore, but I

651
00:36:18,030 --> 00:36:19,195
think that's right.

652
00:36:19,195 --> 00:36:20,450
And I try the next one.

653
00:36:20,450 --> 00:36:24,170
And at each place, I go as far
down this tree as I can.

654
00:36:24,170 --> 00:36:25,640
And I back up.

655
00:36:25,640 --> 00:36:28,970
If I get down to here and find
no possibilities below there,

656
00:36:28,970 --> 00:36:31,740
I back all the way up to here,
and now start again generating

657
00:36:31,740 --> 00:36:33,260
this sub-tree.

658
00:36:33,260 --> 00:36:35,050
And I sort of walk around.

659
00:36:35,050 --> 00:36:37,870
And finally, if I ever manage to
get all the way down, I've

660
00:36:37,870 --> 00:36:40,090
found a solution.

661
00:36:40,090 --> 00:36:45,020
So that's a typical sort of
paradigm that's used a lot in

662
00:36:45,020 --> 00:36:45,930
AI programming.

663
00:36:45,930 --> 00:36:47,300
It's called backtracking
search.

664
00:36:47,300 --> 00:36:57,470


665
00:36:57,470 --> 00:37:03,860
And it's really unnecessary.

666
00:37:03,860 --> 00:37:06,550
You saw me get confused when I
was visualizing this thing.

667
00:37:06,550 --> 00:37:08,550
And you see the complication.

668
00:37:08,550 --> 00:37:10,760
This is a complicated
thing to say.

669
00:37:10,760 --> 00:37:12,390
Why is it complicated?

670
00:37:12,390 --> 00:37:16,190
Its because somehow this program
is too inordinately

671
00:37:16,190 --> 00:37:18,580
concerned with time.

672
00:37:18,580 --> 00:37:19,200
It's too much--

673
00:37:19,200 --> 00:37:21,670
I try this one, and I try this
one, and I go back to the last

674
00:37:21,670 --> 00:37:22,320
possibility.

675
00:37:22,320 --> 00:37:24,340
And that's a complicated
thing.

676
00:37:24,340 --> 00:37:28,590
If I stop worrying about time
so much, then there's a much

677
00:37:28,590 --> 00:37:31,200
simpler way to describe this.

678
00:37:31,200 --> 00:37:40,320
It says, let's imagine that I
have in my hands the tree down

679
00:37:40,320 --> 00:37:43,400
to k minus 1 levels.

680
00:37:43,400 --> 00:37:50,670
See, suppose I had in my hands
all possible ways to put down

681
00:37:50,670 --> 00:37:53,560
queens in the first k columns.

682
00:37:53,560 --> 00:37:54,610
Suppose I just had that.

683
00:37:54,610 --> 00:37:57,070
Let's not worry about
how we get it.

684
00:37:57,070 --> 00:37:59,200
Well, then, how do
I extend that?

685
00:37:59,200 --> 00:38:01,420
How do I find all possible ways
to put down queens in the

686
00:38:01,420 --> 00:38:02,480
next column?

687
00:38:02,480 --> 00:38:03,620
It's really easy.

688
00:38:03,620 --> 00:38:12,210
For each of these positions I
have, I think about putting

689
00:38:12,210 --> 00:38:16,160
down a queen in each row
to make the next thing.

690
00:38:16,160 --> 00:38:18,930
And then for each one I put
down, I filter those by the

691
00:38:18,930 --> 00:38:22,080
ones that are safe.

692
00:38:22,080 --> 00:38:24,190
So instead of thinking about
this tree as generated step by

693
00:38:24,190 --> 00:38:26,860
step, suppose I had
it all there.

694
00:38:26,860 --> 00:38:29,680


695
00:38:29,680 --> 00:38:32,990
And to extend it from level k
minus 1 to level k, I just

696
00:38:32,990 --> 00:38:36,840
need to extend each thing in
all possible ways and only

697
00:38:36,840 --> 00:38:37,800
keep the ones that are safe.

698
00:38:37,800 --> 00:38:39,300
And that will give me
the tree to level k.

699
00:38:39,300 --> 00:38:41,675
And that's a recursive strategy
for solving the eight

700
00:38:41,675 --> 00:38:44,530
queens problem.

701
00:38:44,530 --> 00:38:45,780
All right, well, let's
look at it.

702
00:38:45,780 --> 00:38:50,280


703
00:38:50,280 --> 00:38:54,360
To solve the eight queens
problem on a board of some

704
00:38:54,360 --> 00:39:00,390
specified size, we write
a sub-procedure called

705
00:39:00,390 --> 00:39:01,030
fill-columns.

706
00:39:01,030 --> 00:39:04,050
Fill-columns is going to
put down queens up

707
00:39:04,050 --> 00:39:06,086
through column k.

708
00:39:06,086 --> 00:39:07,700
And here's the pattern
of the recursion.

709
00:39:07,700 --> 00:39:12,990
I'm going to call fill-columns
with the size eventually.

710
00:39:12,990 --> 00:39:15,630
So fill-columns says how to put
down queens safely in the

711
00:39:15,630 --> 00:39:19,255
first k columns of this chess
board with a size number of

712
00:39:19,255 --> 00:39:20,360
rows in it.

713
00:39:20,360 --> 00:39:22,946
If k is equal to 0, well,
then I don't have to

714
00:39:22,946 --> 00:39:23,940
put anything down.

715
00:39:23,940 --> 00:39:26,710
So my solution is just
an empty chess board.

716
00:39:26,710 --> 00:39:28,070
Otherwise, I'm going
to do some stuff.

717
00:39:28,070 --> 00:39:30,522
And I'm going to use collect.

718
00:39:30,522 --> 00:39:31,772
And here's the collect.

719
00:39:31,772 --> 00:39:34,530


720
00:39:34,530 --> 00:39:40,590
I find all ways to put
down queens in the

721
00:39:40,590 --> 00:39:41,910
first k minus 1 columns.

722
00:39:41,910 --> 00:39:43,320
And this was just
what I set for.

723
00:39:43,320 --> 00:39:48,880
Imagine I have this tree down
to k minus 1 levels.

724
00:39:48,880 --> 00:39:53,230
And then I find all ways of
trying a row, that's just each

725
00:39:53,230 --> 00:39:54,130
of the possible rows.

726
00:39:54,130 --> 00:39:58,040
They're size rows, so that's
enumerate interval.

727
00:39:58,040 --> 00:40:03,950
And now what I do is I collect
together the new row I'm going

728
00:40:03,950 --> 00:40:08,950
to try and column k with
the rest of the queens.

729
00:40:08,950 --> 00:40:10,200
I adjoin a position.

730
00:40:10,200 --> 00:40:11,290
This is George's problem.

731
00:40:11,290 --> 00:40:13,640
An adjoined position
is like safe.

732
00:40:13,640 --> 00:40:16,530
It's a thing that takes a row
and a column and the rest of

733
00:40:16,530 --> 00:40:19,660
the positions and makes a
new position collection.

734
00:40:19,660 --> 00:40:26,230
So I adjoin a position of a new
row and a new column to

735
00:40:26,230 --> 00:40:30,310
the rest of the queens, where
the rest of the queens runs

736
00:40:30,310 --> 00:40:32,870
through all possible ways
of solving the problem

737
00:40:32,870 --> 00:40:34,620
in k minus 1 columns.

738
00:40:34,620 --> 00:40:39,730
And the new row runs through all
possible rows such that it

739
00:40:39,730 --> 00:40:43,240
was safe to put one there.

740
00:40:43,240 --> 00:40:46,500
And that's the whole program.

741
00:40:46,500 --> 00:40:49,840
There's the whole procedure.

742
00:40:49,840 --> 00:40:51,990
Not only that, that doesn't just
solve the eight queens

743
00:40:51,990 --> 00:40:56,010
problem, it gives you
all solutions to the

744
00:40:56,010 --> 00:40:56,680
eight queens problem.

745
00:40:56,680 --> 00:40:58,480
When you're done, you
have a stream.

746
00:40:58,480 --> 00:41:00,650
And the elements of that stream
are all possible ways

747
00:41:00,650 --> 00:41:01,900
of solving that problem.

748
00:41:01,900 --> 00:41:05,310


749
00:41:05,310 --> 00:41:06,260
Why is that simpler?

750
00:41:06,260 --> 00:41:10,170
Well, we threw away the whole
idea that this is some process

751
00:41:10,170 --> 00:41:12,720
that happens in time
with state.

752
00:41:12,720 --> 00:41:14,420
And we just said it's a whole
collection of stuff.

753
00:41:14,420 --> 00:41:18,260
And that's why it's simpler.

754
00:41:18,260 --> 00:41:20,110
We've changed our view.

755
00:41:20,110 --> 00:41:22,820
Remember, that's where
we started today.

756
00:41:22,820 --> 00:41:26,230
We've changed our view of what
it is we're trying to model.

757
00:41:26,230 --> 00:41:30,570
we stop modeling things that
evolve in time and have steps

758
00:41:30,570 --> 00:41:31,750
and have state.

759
00:41:31,750 --> 00:41:33,990
And instead, we're trying to
model this global thing like

760
00:41:33,990 --> 00:41:37,950
the whole flight of the
chalk, rather than its

761
00:41:37,950 --> 00:41:40,750
state at each instant.

762
00:41:40,750 --> 00:41:42,000
Any questions?

763
00:41:42,000 --> 00:41:43,810


764
00:41:43,810 --> 00:41:46,190
AUDIENCE: It looks to me like
backtracking would be

765
00:41:46,190 --> 00:41:49,970
searching for the first solution
it can find, whereas

766
00:41:49,970 --> 00:41:54,030
this recursive search would be
looking for all solutions.

767
00:41:54,030 --> 00:41:58,090
And it seems that if you have a
large enough area to search,

768
00:41:58,090 --> 00:42:01,360
that the second is going
to become impossible.

769
00:42:01,360 --> 00:42:07,610
PROFESSOR: OK, the answer to
that question is the whole

770
00:42:07,610 --> 00:42:08,570
rest of this lecture.

771
00:42:08,570 --> 00:42:10,540
It's exactly the
right question.

772
00:42:10,540 --> 00:42:13,522


773
00:42:13,522 --> 00:42:15,540
And without trying to anticipate
the lecture too

774
00:42:15,540 --> 00:42:19,910
much, you should start being
suspicious at this point, and

775
00:42:19,910 --> 00:42:22,220
exactly those kinds
of suspicions.

776
00:42:22,220 --> 00:42:24,830
It's wonderful, but isn't it
so terribly inefficient?

777
00:42:24,830 --> 00:42:28,100
That's where we're going.

778
00:42:28,100 --> 00:42:30,020
So I won't answer now, but
I'll answer later.

779
00:42:30,020 --> 00:42:33,350


780
00:42:33,350 --> 00:42:34,600
OK, let's take a break.

781
00:42:34,600 --> 00:43:29,650


782
00:43:29,650 --> 00:43:35,600
Well, by now you should be
starting to get suspicious.

783
00:43:35,600 --> 00:43:41,450
See, I've showed your this
simple, elegant way of putting

784
00:43:41,450 --> 00:43:46,440
programs together, very unlike
these other traditional

785
00:43:46,440 --> 00:43:50,490
programs that sum the odd
squares or compute the odd

786
00:43:50,490 --> 00:43:53,740
Fibonacci numbers.

787
00:43:53,740 --> 00:43:57,080
Very unlike these programs that
mix up the enumerator and

788
00:43:57,080 --> 00:44:00,440
the filter and the
accumulator.

789
00:44:00,440 --> 00:44:04,770
And by mixing it up, we don't
have all of these wonderful

790
00:44:04,770 --> 00:44:07,990
conceptual advantages of these
streams pieces, these

791
00:44:07,990 --> 00:44:09,840
wonderful mix and match
components for putting

792
00:44:09,840 --> 00:44:13,800
together lots and lots
of programs.

793
00:44:13,800 --> 00:44:15,810
On the other hand, most of the
programs you've seen look like

794
00:44:15,810 --> 00:44:18,340
these ugly ones.

795
00:44:18,340 --> 00:44:19,460
Why's that?

796
00:44:19,460 --> 00:44:23,705
Can it possibly be that computer
scientists are so

797
00:44:23,705 --> 00:44:28,370
obtuse that they don't notice
that if you'd merely did this

798
00:44:28,370 --> 00:44:33,620
thing, then you can get this
great programming elegance?

799
00:44:33,620 --> 00:44:36,760
There's got to be a catch.

800
00:44:36,760 --> 00:44:39,510
And it's actually pretty easy
to see what the catch is.

801
00:44:39,510 --> 00:44:42,030
Let's think about the
following problem.

802
00:44:42,030 --> 00:44:47,510
Suppose I tell you to find the
second prime between 10,000

803
00:44:47,510 --> 00:44:51,020
and 1 million, or if your
computer's larger, say between

804
00:44:51,020 --> 00:44:54,105
10,000 and 100 billion,
or something.

805
00:44:54,105 --> 00:44:55,550
And you say, oh, that's easy.

806
00:44:55,550 --> 00:44:57,080
I can do that with a stream.

807
00:44:57,080 --> 00:45:01,530
All I do is I enumerate
the interval

808
00:45:01,530 --> 00:45:04,160
from 10,000 to 1 million.

809
00:45:04,160 --> 00:45:06,800
So I get all those integers
from 10,000 to 1 million.

810
00:45:06,800 --> 00:45:10,520
I filter them for prime-ness, so
test all of them and see if

811
00:45:10,520 --> 00:45:11,762
they're prime.

812
00:45:11,762 --> 00:45:13,170
And I take the second element.

813
00:45:13,170 --> 00:45:16,130
That's the head of the tail.

814
00:45:16,130 --> 00:45:17,380
Well, that's clearly
pretty ridiculous.

815
00:45:17,380 --> 00:45:21,660


816
00:45:21,660 --> 00:45:24,620
We'd not even have room in the
machine to store the integers

817
00:45:24,620 --> 00:45:27,040
in the first place, much
less to test them.

818
00:45:27,040 --> 00:45:29,810
And then I only want
the second one.

819
00:45:29,810 --> 00:45:36,500
See, the power of this
traditional programming style

820
00:45:36,500 --> 00:45:39,860
is exactly its weakness, that
we're mixing up the

821
00:45:39,860 --> 00:45:45,090
enumerating and the testing
and the accumulating.

822
00:45:45,090 --> 00:45:46,670
So we don't do it all.

823
00:45:46,670 --> 00:45:52,580
So the very thing that makes
it conceptually ugly is the

824
00:45:52,580 --> 00:45:55,210
very thing that makes
it efficient.

825
00:45:55,210 --> 00:45:57,800
It's this mixing up.

826
00:45:57,800 --> 00:45:59,840
So it seems that all I've done
this morning so far is just

827
00:45:59,840 --> 00:46:00,420
confuse you.

828
00:46:00,420 --> 00:46:02,930
I showed you this wonderful
way that programming might

829
00:46:02,930 --> 00:46:05,840
work, except that it doesn't.

830
00:46:05,840 --> 00:46:09,040
Well, here's where the wonderful
thing happens.

831
00:46:09,040 --> 00:46:13,210
It turns out in this game that
we really can have our cake

832
00:46:13,210 --> 00:46:14,870
and eat it too.

833
00:46:14,870 --> 00:46:20,280
And what I mean by that is
that we really can write

834
00:46:20,280 --> 00:46:24,210
stream programs exactly like the
ones I wrote and arrange

835
00:46:24,210 --> 00:46:28,830
things so that when the machine
actually runs, it's as

836
00:46:28,830 --> 00:46:31,690
efficient as running this
traditional programming style

837
00:46:31,690 --> 00:46:36,310
that mixes up the generation
and the test.

838
00:46:36,310 --> 00:46:40,770
Well, that sounds
pretty magic.

839
00:46:40,770 --> 00:46:43,690
The key to this is that
streams are not lists.

840
00:46:43,690 --> 00:46:48,090


841
00:46:48,090 --> 00:46:50,070
We'll see this carefully in a
second, but for now, let's

842
00:46:50,070 --> 00:46:52,115
take a look at that
slide again.

843
00:46:52,115 --> 00:46:55,060
The image you should have here
of this signal processing

844
00:46:55,060 --> 00:47:00,940
system is that what's going to
happen is there's this box

845
00:47:00,940 --> 00:47:05,360
that has the integers
sitting in it.

846
00:47:05,360 --> 00:47:08,680
And there's this filter that's
connected to it and it's

847
00:47:08,680 --> 00:47:10,940
tugging on them.

848
00:47:10,940 --> 00:47:13,680
And then there's someone who's
tugging on this stuff saying

849
00:47:13,680 --> 00:47:16,790
what comes out of the filter.

850
00:47:16,790 --> 00:47:19,630
And the image you should have
is that someone says, well,

851
00:47:19,630 --> 00:47:24,590
what's the first prime, and
tugs on this filter.

852
00:47:24,590 --> 00:47:28,020
And the filter tugs
on the integers.

853
00:47:28,020 --> 00:47:29,830
And you look only at that much,
and then say, oh, I

854
00:47:29,830 --> 00:47:30,930
really wanted the second one.

855
00:47:30,930 --> 00:47:33,710
What's the second prime?

856
00:47:33,710 --> 00:47:37,730
And that no computation gets
done except when you tug on

857
00:47:37,730 --> 00:47:40,500
these things.

858
00:47:40,500 --> 00:47:41,410
Let me try that again.

859
00:47:41,410 --> 00:47:43,815
This is a little device.

860
00:47:43,815 --> 00:47:46,400
This is a little stream machine
invented by Eric

861
00:47:46,400 --> 00:47:49,830
Grimson who's been teaching
this course at MIT.

862
00:47:49,830 --> 00:47:52,940
And the image is here's a stream
of stuff, like a whole

863
00:47:52,940 --> 00:47:54,780
bunch of the integers.

864
00:47:54,780 --> 00:47:58,700
And here's some processing
elements.

865
00:47:58,700 --> 00:48:02,600
And if, say, it's filter of
filter of map, or something.

866
00:48:02,600 --> 00:48:05,570


867
00:48:05,570 --> 00:48:08,760
And if I really tried to
implement that with streams as

868
00:48:08,760 --> 00:48:11,520
lists, what I'd say is, well,
I've got this list of things,

869
00:48:11,520 --> 00:48:12,670
and now I do the first filter.

870
00:48:12,670 --> 00:48:14,070
So do all this processing.

871
00:48:14,070 --> 00:48:18,570
And I take this and I process
and I process and I process

872
00:48:18,570 --> 00:48:19,610
and I process.

873
00:48:19,610 --> 00:48:21,910
And now I'm got this
new stream.

874
00:48:21,910 --> 00:48:24,070
Now I take that result
in my hand someplace.

875
00:48:24,070 --> 00:48:25,260
And I put that through
the second one.

876
00:48:25,260 --> 00:48:28,110
And I process the whole thing.

877
00:48:28,110 --> 00:48:29,510
And there's this new stream.

878
00:48:29,510 --> 00:48:32,130


879
00:48:32,130 --> 00:48:35,230
And then I take the result and
I put it all the way through

880
00:48:35,230 --> 00:48:36,360
this one the same way.

881
00:48:36,360 --> 00:48:41,760
That's what would happen to
these stream programs if

882
00:48:41,760 --> 00:48:43,860
streams were just lists.

883
00:48:43,860 --> 00:48:46,065
But in fact, streams aren't
lists, they're streams. And

884
00:48:46,065 --> 00:48:47,240
the image you should have
is something a little

885
00:48:47,240 --> 00:48:50,230
bit more like this.

886
00:48:50,230 --> 00:48:55,880
I've got these gadgets connected
up by this data

887
00:48:55,880 --> 00:48:57,130
that's flowing out of them.

888
00:48:57,130 --> 00:48:59,960


889
00:48:59,960 --> 00:49:04,190
And here's my original source
of the streams. It might be

890
00:49:04,190 --> 00:49:05,980
starting to generate
the integers.

891
00:49:05,980 --> 00:49:07,580
And now, what happens
if I want a result?

892
00:49:07,580 --> 00:49:10,200
I tug on the end here.

893
00:49:10,200 --> 00:49:13,090
And this element says, gee,
I need some more data.

894
00:49:13,090 --> 00:49:15,830
So this one comes here
and tugs on that one.

895
00:49:15,830 --> 00:49:17,890
And it says, gee, I need
some more data.

896
00:49:17,890 --> 00:49:19,960
And this one tugs on this
thing, which might be a

897
00:49:19,960 --> 00:49:21,640
filter, and says, gee, I
need some more data.

898
00:49:21,640 --> 00:49:24,755
And only as much of this thing
at the end here gets generated

899
00:49:24,755 --> 00:49:25,780
as I tugged.

900
00:49:25,780 --> 00:49:28,030
And only as much of this stuff
goes through the processing

901
00:49:28,030 --> 00:49:30,760
units as I'm pulling
on the end I need.

902
00:49:30,760 --> 00:49:33,720
That's the image you should have
of the difference between

903
00:49:33,720 --> 00:49:36,580
implementing what we're actually
going to do and if

904
00:49:36,580 --> 00:49:37,830
streams were lists.

905
00:49:37,830 --> 00:49:40,600


906
00:49:40,600 --> 00:49:42,430
Well, how do we make
this thing?

907
00:49:42,430 --> 00:49:43,400
I hope you have the image.

908
00:49:43,400 --> 00:49:44,947
The trick is how to make it.

909
00:49:44,947 --> 00:49:47,930


910
00:49:47,930 --> 00:49:52,080
We want to arrange for a stream
to be a data structure

911
00:49:52,080 --> 00:49:55,670
that computes itself
incrementally, an on-demand

912
00:49:55,670 --> 00:49:56,920
data structure.

913
00:49:56,920 --> 00:49:59,220


914
00:49:59,220 --> 00:50:02,700
And the basic idea is, again,
one of the very basic ideas

915
00:50:02,700 --> 00:50:04,490
that we're seeing throughout
the whole course.

916
00:50:04,490 --> 00:50:07,440
And that is that there's not
a firm distinction between

917
00:50:07,440 --> 00:50:09,240
programs and data.

918
00:50:09,240 --> 00:50:12,260
So what a stream is going to be
is simultaneously this data

919
00:50:12,260 --> 00:50:15,270
structure that you think of,
like the stream of the leaves

920
00:50:15,270 --> 00:50:16,810
of this tree.

921
00:50:16,810 --> 00:50:18,880
But at the same time, it's
going to be a very clever

922
00:50:18,880 --> 00:50:23,550
procedure that has the method
of computing in it.

923
00:50:23,550 --> 00:50:25,930
Well, let me try this.

924
00:50:25,930 --> 00:50:28,460
It's going to turn out that we
don't need any more mechanism.

925
00:50:28,460 --> 00:50:31,150
We already have everything we
need simply from the fact that

926
00:50:31,150 --> 00:50:32,770
we know how to handle
procedures

927
00:50:32,770 --> 00:50:35,460
as first-class objects.

928
00:50:35,460 --> 00:50:36,880
Well, let's go back
to the key.

929
00:50:36,880 --> 00:50:39,030
The key is, remember, we
had these operations.

930
00:50:39,030 --> 00:50:48,080
CONS-stream and head and tail.

931
00:50:48,080 --> 00:50:51,580
When I started, I said you can
think about this as CONS and

932
00:50:51,580 --> 00:50:53,340
think about this as CAR and
think about that as

933
00:50:53,340 --> 00:50:55,080
CDR, but it's not.

934
00:50:55,080 --> 00:50:57,550
Now, let's look at what
they really are.

935
00:50:57,550 --> 00:51:09,360
Well, CONS-stream of x and y is
going to be an abbreviation

936
00:51:09,360 --> 00:51:19,540
for the following thing.

937
00:51:19,540 --> 00:51:24,470
CONS form a pair, ordinary CONS,
of x to a thing called

938
00:51:24,470 --> 00:51:28,000
delay of y.

939
00:51:28,000 --> 00:51:31,188


940
00:51:31,188 --> 00:51:34,670
And before I explain that, let
me go and write the rest. The

941
00:51:34,670 --> 00:51:39,790
head of a stream is going
to be just the CAR.

942
00:51:39,790 --> 00:51:42,380


943
00:51:42,380 --> 00:51:47,610
And the tail of a stream is
going to be a thing called

944
00:51:47,610 --> 00:51:56,120
force the CDR of the stream.

945
00:51:56,120 --> 00:51:58,060
Now let me explain this.

946
00:51:58,060 --> 00:52:01,420
Delay is going to be a
special magic thing.

947
00:52:01,420 --> 00:52:06,240
What delay does is take an
expression and produce a

948
00:52:06,240 --> 00:52:08,380
promise to compute
that expression

949
00:52:08,380 --> 00:52:10,600
when you ask for it.

950
00:52:10,600 --> 00:52:11,980
It doesn't do any computation
here.

951
00:52:11,980 --> 00:52:14,820
It just gives you
a rain check.

952
00:52:14,820 --> 00:52:17,110
It produces a promise.

953
00:52:17,110 --> 00:52:23,280
And CONS-stream says I'm going
to put together in a pair x

954
00:52:23,280 --> 00:52:25,360
and a promise to compute y.

955
00:52:25,360 --> 00:52:28,230


956
00:52:28,230 --> 00:52:30,200
Now, if I want the head, that's
just the CAR that I put

957
00:52:30,200 --> 00:52:31,840
in the pair.

958
00:52:31,840 --> 00:52:34,350
And the key is that the
tail is going to be--

959
00:52:34,350 --> 00:52:39,110
force calls in that promise.

960
00:52:39,110 --> 00:52:43,690
Tail says, well, take
that promise and now

961
00:52:43,690 --> 00:52:44,610
call in that promise.

962
00:52:44,610 --> 00:52:47,430
And then we compute
that thing.

963
00:52:47,430 --> 00:52:48,740
That's how this is
going to work.

964
00:52:48,740 --> 00:52:51,550
That's what CONS-stream, head,
and tail really are.

965
00:52:51,550 --> 00:52:54,196


966
00:52:54,196 --> 00:52:55,570
Now, let's see how this works.

967
00:52:55,570 --> 00:52:58,410
And we'll go through this
fairly carefully.

968
00:52:58,410 --> 00:53:01,990
We're going to see how this
works in this example of

969
00:53:01,990 --> 00:53:08,650
computing the second prime
between 10,000 and a million.

970
00:53:08,650 --> 00:53:11,610
OK, so we start off and we
have this expression.

971
00:53:11,610 --> 00:53:15,820


972
00:53:15,820 --> 00:53:20,380
The second prime-- the head of
the tail of the result of

973
00:53:20,380 --> 00:53:24,060
filtering for primality
the integers between

974
00:53:24,060 --> 00:53:26,710
10,000 and 1 million.

975
00:53:26,710 --> 00:53:28,400
Now, what is that?

976
00:53:28,400 --> 00:53:35,790
What that is, that interval
between 10,000 and 1 million,

977
00:53:35,790 --> 00:53:37,480
well, if you trace through
enumerate interval, there

978
00:53:37,480 --> 00:53:40,250
builds a CONS-stream.

979
00:53:40,250 --> 00:53:45,880
And the CONS-stream is the CONS
of 10,000 to a promise to

980
00:53:45,880 --> 00:53:54,480
compute the integers between
10,001 and 1 million.

981
00:53:54,480 --> 00:53:55,750
So that's what this
expression is.

982
00:53:55,750 --> 00:53:57,640
Here I'm using the substitution
model.

983
00:53:57,640 --> 00:53:59,690
And we can use the substitution
model because we

984
00:53:59,690 --> 00:54:01,010
don't have side effects
and state.

985
00:54:01,010 --> 00:54:04,270


986
00:54:04,270 --> 00:54:07,860
So I have CONS of 10,000 to a
promise to compute the rest of

987
00:54:07,860 --> 00:54:08,380
the integers.

988
00:54:08,380 --> 00:54:09,850
So only one integer, so
far, got enumerated.

989
00:54:09,850 --> 00:54:14,380


990
00:54:14,380 --> 00:54:16,580
Well, I'm going to filter that
thing for primality.

991
00:54:16,580 --> 00:54:19,900


992
00:54:19,900 --> 00:54:22,360
Again, you go back and look
at the filter code.

993
00:54:22,360 --> 00:54:25,460
What the filter will first
do is test the head.

994
00:54:25,460 --> 00:54:31,580
So in this case, the filter will
test 10,000 and say, oh,

995
00:54:31,580 --> 00:54:33,500
10,000's not prime.

996
00:54:33,500 --> 00:54:36,260
Therefore, what I have
to do recursively

997
00:54:36,260 --> 00:54:39,220
is filter the tail.

998
00:54:39,220 --> 00:54:42,550
And what's the tail of it, well,
that's the tail of this

999
00:54:42,550 --> 00:54:46,340
pair with a promise in it.

1000
00:54:46,340 --> 00:54:49,680
Tail now comes in and says,
well, I'm going to force that.

1001
00:54:49,680 --> 00:54:53,790
I'm going to force that promise,
which means now I'm

1002
00:54:53,790 --> 00:55:00,880
going to compute the integers
between 10,001 and 1 million.

1003
00:55:00,880 --> 00:55:02,970
OK, so this filter now
is looking at that.

1004
00:55:02,970 --> 00:55:07,810


1005
00:55:07,810 --> 00:55:10,100
That enumerate itself, well, now
we're back in the original

1006
00:55:10,100 --> 00:55:11,960
enumerate situation.

1007
00:55:11,960 --> 00:55:16,920
The enumerate is the CONS of the
first thing, 10,001, onto

1008
00:55:16,920 --> 00:55:19,740
a promise to compute the rest.

1009
00:55:19,740 --> 00:55:23,060
So now the primality filter is
going to go look at 10,001.

1010
00:55:23,060 --> 00:55:25,120
It's going to decide if
it likes that or not.

1011
00:55:25,120 --> 00:55:27,550
It turns out 10,001
isn't prime.

1012
00:55:27,550 --> 00:55:29,610
So it'll force it again
and again and again.

1013
00:55:29,610 --> 00:55:32,920


1014
00:55:32,920 --> 00:55:37,100
And finally, I think the first
prime it hits is 10,009.

1015
00:55:37,100 --> 00:55:40,465
And at that point, it'll stop.

1016
00:55:40,465 --> 00:55:42,500
And that will be the first
prime, and then eventually,

1017
00:55:42,500 --> 00:55:45,240
it'll need the second prime.

1018
00:55:45,240 --> 00:55:47,030
So at that point, it
will go again.

1019
00:55:47,030 --> 00:55:51,880
So you see what happens is that
no more gets generated

1020
00:55:51,880 --> 00:55:53,130
than you actually need.

1021
00:55:53,130 --> 00:55:56,690


1022
00:55:56,690 --> 00:56:00,060
That enumerator is not going to
generate any more integers

1023
00:56:00,060 --> 00:56:02,410
than the filter asks it for as
it's pulling in things to

1024
00:56:02,410 --> 00:56:04,930
check for primality.

1025
00:56:04,930 --> 00:56:07,290
And the filter is not going to
generate any more stuff than

1026
00:56:07,290 --> 00:56:11,255
you ask it for, which is
the head of the tail.

1027
00:56:11,255 --> 00:56:17,180
You see, what's happened is
we've put that mixing of

1028
00:56:17,180 --> 00:56:20,130
generation and test into what
actually happens in the

1029
00:56:20,130 --> 00:56:24,250
computer, even though that's
not apparently what's

1030
00:56:24,250 --> 00:56:28,160
happening from looking
at our programs.

1031
00:56:28,160 --> 00:56:30,230
OK, well, that seemed easy.

1032
00:56:30,230 --> 00:56:33,326
All of this mechanism got put
into this magic delay.

1033
00:56:33,326 --> 00:56:36,900
So you're saying, gee, that must
be where the magic is.

1034
00:56:36,900 --> 00:56:39,070
But see there's no magic
there either.

1035
00:56:39,070 --> 00:56:40,610
You know what delay is.

1036
00:56:40,610 --> 00:56:50,040
Delay on some expression is
just an abbreviation for--

1037
00:56:50,040 --> 00:56:53,400


1038
00:56:53,400 --> 00:56:56,490
well, what's a promise to
compute an expression?

1039
00:56:56,490 --> 00:57:00,700
Lambda of nil, procedure of no
arguments, which is that

1040
00:57:00,700 --> 00:57:03,000
expression.

1041
00:57:03,000 --> 00:57:03,930
That's what a procedure is.

1042
00:57:03,930 --> 00:57:06,050
It says I'm going to compute
an expression.

1043
00:57:06,050 --> 00:57:07,460
What's force?

1044
00:57:07,460 --> 00:57:10,800
How do I take up a promise?

1045
00:57:10,800 --> 00:57:15,890
Well, force of some procedure,
a promise, is just run it.

1046
00:57:15,890 --> 00:57:18,710


1047
00:57:18,710 --> 00:57:20,120
Done.

1048
00:57:20,120 --> 00:57:23,580
So there's no magic
there at all.

1049
00:57:23,580 --> 00:57:26,440
Well, what have we done?

1050
00:57:26,440 --> 00:57:29,510
We said the old style,
traditional style of

1051
00:57:29,510 --> 00:57:30,960
programming is more efficient.

1052
00:57:30,960 --> 00:57:35,260
And the stream thing is
more perspicuous.

1053
00:57:35,260 --> 00:57:40,070
And we managed to make the
stream procedures run like the

1054
00:57:40,070 --> 00:57:43,350
other procedures
by using delay.

1055
00:57:43,350 --> 00:57:46,880
And the thing that delay did
for us was to de-couple the

1056
00:57:46,880 --> 00:57:52,150
apparent order of events in our
programs from the actual

1057
00:57:52,150 --> 00:57:54,440
order of events that happened
in the machine.

1058
00:57:54,440 --> 00:57:56,540
That's really what
delay is doing.

1059
00:57:56,540 --> 00:57:58,290
That's exactly the
whole point.

1060
00:57:58,290 --> 00:58:04,720
We've given up the idea that our
procedures, as they run,

1061
00:58:04,720 --> 00:58:09,182
or as we look at them, mirror
some clear notion of time.

1062
00:58:09,182 --> 00:58:12,960
And by giving that up, we give
delay the freedom to arrange

1063
00:58:12,960 --> 00:58:16,690
the order of events in the
computation the way it likes.

1064
00:58:16,690 --> 00:58:17,610
That's the whole idea.

1065
00:58:17,610 --> 00:58:20,640
We de-couple the apparent
order of events in our

1066
00:58:20,640 --> 00:58:24,200
programs from the actual order
of events in the computer.

1067
00:58:24,200 --> 00:58:25,770
OK, well there's one
more detail.

1068
00:58:25,770 --> 00:58:27,750
It's just a technical detail,
but it's actually

1069
00:58:27,750 --> 00:58:29,730
an important one.

1070
00:58:29,730 --> 00:58:32,190
As you run through these
recursive programs unwinding,

1071
00:58:32,190 --> 00:58:35,360
you'll see a lot of things that
look like tail of the

1072
00:58:35,360 --> 00:58:39,320
tail of the tail.

1073
00:58:39,320 --> 00:58:41,840
That's the kind of thing that
would happen as I go CONSing

1074
00:58:41,840 --> 00:58:43,860
down a stream all the way.

1075
00:58:43,860 --> 00:58:47,170
And if each time I'm doing that,
each time to compute a

1076
00:58:47,170 --> 00:58:51,830
tail, I evaluate a procedure
which then has to go

1077
00:58:51,830 --> 00:58:54,270
re-compute its tail, and
re-compute its tail and

1078
00:58:54,270 --> 00:58:56,380
recompute its tail each time,
you can see that's very

1079
00:58:56,380 --> 00:58:59,610
inefficient compared to just
having a list where the

1080
00:58:59,610 --> 00:59:02,510
elements are all there, and I
don't have to re-compute each

1081
00:59:02,510 --> 00:59:05,290
tail every time I get
the next tail.

1082
00:59:05,290 --> 00:59:15,030
So there's one little hack to
slightly change what delay is,

1083
00:59:15,030 --> 00:59:17,380
and make it a thing which is--

1084
00:59:17,380 --> 00:59:20,390
I'll write it this way.

1085
00:59:20,390 --> 00:59:27,360
The actual implementation, delay
is an abbreviation for

1086
00:59:27,360 --> 00:59:31,000
this thing, memo-proc
of a procedure.

1087
00:59:31,000 --> 00:59:35,150
Memo-proc is a special thing
that transforms a procedure.

1088
00:59:35,150 --> 00:59:39,250
What it does is it takes a
procedure of no arguments and

1089
00:59:39,250 --> 00:59:42,190
it transforms it into a
procedure that'll only have to

1090
00:59:42,190 --> 00:59:44,806
do its computation once.

1091
00:59:44,806 --> 00:59:48,700
And what I mean by that is,
you give it a procedure.

1092
00:59:48,700 --> 00:59:51,950
The result of memo-proc will be
a new procedure, which the

1093
00:59:51,950 --> 00:59:55,370
first time you call it, will
run the original procedure,

1094
00:59:55,370 --> 01:00:00,040
remember what result it got, and
then from ever on after,

1095
01:00:00,040 --> 01:00:01,610
when you call it, it just
won't have to do the

1096
01:00:01,610 --> 01:00:02,360
computation.

1097
01:00:02,360 --> 01:00:05,200
It will have cached that
result someplace.

1098
01:00:05,200 --> 01:00:06,550
And here's an implementation
of memo-proc.

1099
01:00:06,550 --> 01:00:11,210


1100
01:00:11,210 --> 01:00:12,710
Once you have the idea, it's
easy to implement.

1101
01:00:12,710 --> 01:00:15,830
Memo-proc is this little
thing that has two

1102
01:00:15,830 --> 01:00:17,390
little flags in there.

1103
01:00:17,390 --> 01:00:20,320
It says, have I already
been run?

1104
01:00:20,320 --> 01:00:23,620
And initially it says, no, I
haven't already been run.

1105
01:00:23,620 --> 01:00:29,070
And what was the result I got
the last time I was run?

1106
01:00:29,070 --> 01:00:32,200
So memo-proc takes a procedure
called proc, and it returns a

1107
01:00:32,200 --> 01:00:34,360
new procedure of no arguments.

1108
01:00:34,360 --> 01:00:38,610
Proc is supposed to be a
procedure of no arguments.

1109
01:00:38,610 --> 01:00:42,970
And it says, oh, if I'm not
already run, then I'm going to

1110
01:00:42,970 --> 01:00:44,430
do a sequence of things.

1111
01:00:44,430 --> 01:00:48,450
I'm going to compute proc,
I'm going to save that.

1112
01:00:48,450 --> 01:00:51,140
I'm going to stash that in
the variable result.

1113
01:00:51,140 --> 01:00:53,510
I'm going to make a note to
myself that I've already been

1114
01:00:53,510 --> 01:00:56,610
run, and then I'll return
the result.

1115
01:00:56,610 --> 01:00:59,010
So that's if you compute it
if it's not already run.

1116
01:00:59,010 --> 01:01:01,040
If you call it and it's already
been run, it just

1117
01:01:01,040 --> 01:01:03,420
returns the result.

1118
01:01:03,420 --> 01:01:08,400
So that's a little clever
hack called memoization.

1119
01:01:08,400 --> 01:01:12,100
And in this case, it short
circuits having to re-compute

1120
01:01:12,100 --> 01:01:15,270
the tail of the tail of the tail
of the tail of the tail.

1121
01:01:15,270 --> 01:01:17,810
So there isn't even that
kind of inefficiency.

1122
01:01:17,810 --> 01:01:20,590
And in fact, the streams will
run with pretty much the same

1123
01:01:20,590 --> 01:01:24,210
efficiency as the other
programs precisely.

1124
01:01:24,210 --> 01:01:28,110
And remember, again, the whole
idea of this is that we've

1125
01:01:28,110 --> 01:01:32,390
used the fact that there's no
really good dividing line

1126
01:01:32,390 --> 01:01:33,610
between procedures and data.

1127
01:01:33,610 --> 01:01:36,510
We've written data structures
that, in fact, are sort of

1128
01:01:36,510 --> 01:01:38,760
like procedures.

1129
01:01:38,760 --> 01:01:45,280
And what that's allowed us to
do is take an example of a

1130
01:01:45,280 --> 01:01:49,620
common control structure,
in this place iteration.

1131
01:01:49,620 --> 01:01:52,460
And we've built a data structure
which, since itself

1132
01:01:52,460 --> 01:01:54,530
is a procedure, kind of has
this iteration control

1133
01:01:54,530 --> 01:01:55,496
structure in it.

1134
01:01:55,496 --> 01:01:58,650
And that's really what
streams are.

1135
01:01:58,650 --> 01:01:59,900
OK, questions?

1136
01:01:59,900 --> 01:02:03,950


1137
01:02:03,950 --> 01:02:06,110
AUDIENCE: Your description
of tail-tail-tail, if I

1138
01:02:06,110 --> 01:02:10,050
understand it correctly, force
is actually execution of a

1139
01:02:10,050 --> 01:02:13,052
procedure, if it's done without
this memo-proc thing.

1140
01:02:13,052 --> 01:02:16,380
And you implied that memo-proc
gets around that problem.

1141
01:02:16,380 --> 01:02:20,580
Doesn't it only get around it
if tail-tail-tail is always

1142
01:02:20,580 --> 01:02:22,550
executing exactly the same--

1143
01:02:22,550 --> 01:02:23,500
PROFESSOR: Oh, that's--

1144
01:02:23,500 --> 01:02:23,910
sure.

1145
01:02:23,910 --> 01:02:26,050
AUDIENCE: I guess I
missed that point.

1146
01:02:26,050 --> 01:02:26,540
PROFESSOR: Oh, sure.

1147
01:02:26,540 --> 01:02:27,790
I mean the point is--

1148
01:02:27,790 --> 01:02:31,160


1149
01:02:31,160 --> 01:02:31,290
yeah.

1150
01:02:31,290 --> 01:02:34,160
I mean I have to do a
computation to get the answer.

1151
01:02:34,160 --> 01:02:37,590
But the point is, once I've
found the tail of the stream,

1152
01:02:37,590 --> 01:02:39,530
to get the tail of the tail,
I shouldn't have had to

1153
01:02:39,530 --> 01:02:42,980
re-compute the first tail.

1154
01:02:42,980 --> 01:02:45,370
See, and if I didn't use
memo-proc, that re-computation

1155
01:02:45,370 --> 01:02:46,460
would have been done.

1156
01:02:46,460 --> 01:02:47,710
AUDIENCE: I understand now.

1157
01:02:47,710 --> 01:02:50,830


1158
01:02:50,830 --> 01:02:52,550
AUDIENCE: In one of your
examples, you mentioned that

1159
01:02:52,550 --> 01:02:55,010
we were able to use the
substitution model because

1160
01:02:55,010 --> 01:02:56,830
there are no side effects.

1161
01:02:56,830 --> 01:03:01,040
What if we had a single
processing unit--

1162
01:03:01,040 --> 01:03:03,620
if we had a side effect,
if we had a state?

1163
01:03:03,620 --> 01:03:09,120
Could we still practically
build the stream model?

1164
01:03:09,120 --> 01:03:09,530
PROFESSOR: Maybe.

1165
01:03:09,530 --> 01:03:10,540
That's a hard question.

1166
01:03:10,540 --> 01:03:15,540
I'm going to talk a little bit
later about the places where

1167
01:03:15,540 --> 01:03:18,960
substitution and side effects
don't really mix very well.

1168
01:03:18,960 --> 01:03:21,170
But in general, I think the
answer is unless you're very

1169
01:03:21,170 --> 01:03:23,920
careful, any amount of side
effect is going to mess up

1170
01:03:23,920 --> 01:03:25,170
everything.

1171
01:03:25,170 --> 01:03:35,490


1172
01:03:35,490 --> 01:03:36,150
AUDIENCE: Sorry, I didn't
quite understand

1173
01:03:36,150 --> 01:03:39,410
the memo-proc operation.

1174
01:03:39,410 --> 01:03:41,990
When do you execute
the lambda?

1175
01:03:41,990 --> 01:03:46,270
In other words, when memo-proc
is executed, just this lambda

1176
01:03:46,270 --> 01:03:47,600
expression is being generated.

1177
01:03:47,600 --> 01:03:50,390
But it's not clear to me
when it's executed.

1178
01:03:50,390 --> 01:03:51,350
PROFESSOR: Right.

1179
01:03:51,350 --> 01:03:53,890
What memo-proc does-- remember,
the thing that's

1180
01:03:53,890 --> 01:03:57,290
going into memo-proc, the thing
proc, is a procedure of

1181
01:03:57,290 --> 01:03:57,930
no arguments.

1182
01:03:57,930 --> 01:04:00,390
And someday, you're
going to call it.

1183
01:04:00,390 --> 01:04:03,350
Memo-proc translates that
procedure into another

1184
01:04:03,350 --> 01:04:05,110
procedure of no arguments,
which someday

1185
01:04:05,110 --> 01:04:06,620
you're going to call.

1186
01:04:06,620 --> 01:04:09,890
That's that lambda.

1187
01:04:09,890 --> 01:04:17,370
So here, where I initially
built as my tail of the

1188
01:04:17,370 --> 01:04:20,680
stream, say, this procedure
of no arguments, which

1189
01:04:20,680 --> 01:04:24,100
someday I'll call.

1190
01:04:24,100 --> 01:04:27,130
Instead, I'm going to have
the tail of the stream be

1191
01:04:27,130 --> 01:04:30,650
memo-proc of it, which
someday I'll call.

1192
01:04:30,650 --> 01:04:35,340
So that lambda of nil, that gets
called when you call the

1193
01:04:35,340 --> 01:04:40,990
memo-proc, when you call the
result of that memo-proc,

1194
01:04:40,990 --> 01:04:44,400
which would be ordinarily when
you would have called the

1195
01:04:44,400 --> 01:04:47,642
original thing that
you set it.

1196
01:04:47,642 --> 01:04:49,690
AUDIENCE: OK, the reason I ask
is I had a feeling that when

1197
01:04:49,690 --> 01:04:52,610
you call memo-proc, you just
return this lambda.

1198
01:04:52,610 --> 01:04:53,770
PROFESSOR: That's right.

1199
01:04:53,770 --> 01:04:58,100
When you call memo-proc,
you return the lambda.

1200
01:04:58,100 --> 01:05:00,090
You never evaluate the
expression at all, until the

1201
01:05:00,090 --> 01:05:02,270
first time that you would
have evaluated it.

1202
01:05:02,270 --> 01:05:07,590


1203
01:05:07,590 --> 01:05:10,000
AUDIENCE: Do I understand it
right that you actually have

1204
01:05:10,000 --> 01:05:12,980
to build the list up, but
the elements of the

1205
01:05:12,980 --> 01:05:14,240
list don't get evaluated?

1206
01:05:14,240 --> 01:05:15,630
The expressions don't
get evaluated?

1207
01:05:15,630 --> 01:05:18,540
But at each stage, you actually
are building a list.

1208
01:05:18,540 --> 01:05:19,750
PROFESSOR: That's--

1209
01:05:19,750 --> 01:05:20,700
I really should have
said this.

1210
01:05:20,700 --> 01:05:22,270
That's a really good point.

1211
01:05:22,270 --> 01:05:23,660
No, it's not quite right.

1212
01:05:23,660 --> 01:05:25,080
Because what happens is this.

1213
01:05:25,080 --> 01:05:26,890
Let me draw this as pairs.

1214
01:05:26,890 --> 01:05:29,710
Suppose I'm going to make a
big stream, like enumerate

1215
01:05:29,710 --> 01:05:32,740
interval, 1 through 1 billion.

1216
01:05:32,740 --> 01:05:43,045
What that is, is a pair with
a 1 and a promise.

1217
01:05:43,045 --> 01:05:46,520


1218
01:05:46,520 --> 01:05:47,890
That's exactly what it is.

1219
01:05:47,890 --> 01:05:49,140
Nothing got built up.

1220
01:05:49,140 --> 01:05:51,600


1221
01:05:51,600 --> 01:05:56,370
When I go and force this,
and say, what happens?

1222
01:05:56,370 --> 01:06:00,530
Well, this thing is now also
recursively a CONS.

1223
01:06:00,530 --> 01:06:07,770
So that this promise now is the
next thing, which is a 2

1224
01:06:07,770 --> 01:06:11,350
and a promise to do more.

1225
01:06:11,350 --> 01:06:14,470
And so on and so on and so on.

1226
01:06:14,470 --> 01:06:18,200
So nothing gets built up until
you walk down the stream.

1227
01:06:18,200 --> 01:06:20,790
Because what's sitting here is
not the list, but a promise to

1228
01:06:20,790 --> 01:06:24,250
generate the list.
And by promise,

1229
01:06:24,250 --> 01:06:25,500
technically I mean procedure.

1230
01:06:25,500 --> 01:06:28,050


1231
01:06:28,050 --> 01:06:30,485
So it doesn't get built up.

1232
01:06:30,485 --> 01:06:34,280
Yeah, I should have said
that before this point.

1233
01:06:34,280 --> 01:06:34,490
OK.

1234
01:06:34,490 --> 01:06:34,790
Thank you.

1235
01:06:34,790 --> 01:06:36,340
Let's take a break.

1236
01:06:36,340 --> 01:06:55,828