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[MUSIC PLAYING]

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PROFESSOR: Well, Hal just told
us how you build robust

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systems. The key idea was--

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I'm sure that many of you don't
really assimilate that

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yet-- but the key idea is that
in order to make a system

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that's robust, it has to be
insensitive to small changes,

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that is, a small change in the
problem should lead to only a

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small change in the solution.

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There ought to be
a continuity.

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The space of solutions ought to
be continuous in this space

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of problems.

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The way he was explaining how
to do that was instead of

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solving a particular problem
at every level of

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decomposition of the problem at
the subproblems, where you

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solve the class of problems,
which are a neighborhood of

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the particular problem that
you're trying to solve.

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The way you do that is by
producing a language at that

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level of detail in which the
solutions to that class of

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problems is representable
in that language.

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Therefore when you makes more
changes to the problem you're

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trying to solve, you generally
have to make only small local

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changes to the solution you've
constructed, because at the

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level of detail you're working,
there's a language

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where you can express the
various solutions to alternate

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problems of the same type.

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Well that's the beginning of a
very important idea, the most

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important perhaps idea that
makes computer science more

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powerful than most of the other
kinds of engineering

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disciplines we know about.

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What we've seen so far
is sort of how to use

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embedding of languages.

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And, of course, the power of
embedding languages partly

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comes from procedures
like this one that

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I showed you yesterday.

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What you see here is the
derivative program that we

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described yesterday.

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It's a procedure that takes a
procedure as an argument and

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returns a procedure
as a value.

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And using such things
is very nice.

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You can make things like push
combinators and all that sort

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of wonderful thing that
you saw last time.

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However, now I'm going to
really muddy the waters.

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See this confuses the issue of
what's the procedure and what

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is data, but not very badly.

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What we really want to do is
confuse it very badly.

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And the best way to do that is
to get involved with the

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manipulation of the algebraic
expressions that the

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procedures themselves
are expressed in.

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So at this point, I want to talk
about instead of things

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like on this slide, the
derivative procedure being a

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thing that manipulates
a procedure--

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this is a numerical method
you see here.

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And what you're seeing is
a representation of the

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numerical approximation
to the derivative.

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That's what's here.

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In fact what I'd like to talk
about is instead things that

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look like this.

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And what we have here are rules
from a calculus book.

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These are rules for finding
the derivatives of the

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expressions that one
might write in

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some algebraic language.

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It says things like a derivative
of a constant is 0.

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The derivative of the valuable
with respect to which you are

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taking the derivative is 1.

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The derivative of a constant
times the function is the

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constant times the derivative
of the function,

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and things like that.

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These are exact expressions.

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These are not numerical
approximations.

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Can we make programs?

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And, in fact, it's very easy
to make programs that

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manipulate these expressions.

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Well let's see.

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Let's look at these rules
in some detail.

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You all have seen these rules
in your elementary calculus

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class at one time or another.

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And you know from calculus
that it's easy to produce

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derivatives of arbitrary
expressions.

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You also know from your
elementary calculus that it's

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hard to produce integrals.

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Yet integrals and derivatives
are opposites of each other.

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They're inverse operations.

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And they have the same rules.

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What is special about these
rules that makes it possible

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for one to produce derivatives
easily and

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integrals why it's so hard?

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Let's think about that
very simply.

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Look at these rules.

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Every one of these rules, when
used in the direction for

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taking derivatives, which is
in the direction of this

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arrow, the left side is
matched against your

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expression, and the right side
is the thing which is the

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derivative of that expression.

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The arrow is going that way.

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In each of these rules, the
expressions on the right-hand

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side of the rule that are
contained within derivatives

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are subexpressions, are proper
subexpressions, of the

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expression on the
left-hand side.

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So here we see the derivative
of the sum, with is the

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expression on the left-hand
side is the sum of the

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derivatives of the pieces.

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So the rule of moving to the
right are reduction rules.

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The problem becomes easier.

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I turn a big complicated problem
it's lots of smaller

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problems and then combine the
results, a perfect place for

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recursion to work.

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If I'm going in the other
direction like this, if I'm

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trying to produce integrals,
well there are several

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problems you see here.

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First of all, if I try to
integrate an expression like a

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sum, more than one
rule matches.

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Here's one that matches.

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Here's one that matches.

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I don't know which
one to take.

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And they may be different.

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I may get to explore
different things.

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Also, the expressions become
larger in that direction.

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And when the expressions become
larger, then there's no

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guarantee that any particular
path I choose will terminate,

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because we will only terminate
by accidental cancellation.

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So that's why integrals
are complicated

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searches and hard to do.

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Right now I don't want to do
anything as hard as that.

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Let's work on derivatives
for a while.

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Well, these roles are
ones you know for

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the most part hopefully.

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So let's see if we can write a
program which is these rules.

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And that should be very easy.

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Just write the program.

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See, because while I showed you
is that it's a reduction

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rule, it's something appropriate
for a recursion.

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And, of course, what we have for
each of these rules is we

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have a case in some
case analysis.

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So I'm just going to write
this program down.

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Now, of course, I'm going to be
saying something you have

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to believe.

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Right?

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What you have to believe is I
can represent these algebraic

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expressions, that I can grab
their parts, that I can put

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them together.

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We've invented list structures
so that you can do that.

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But you don't want to worry
about that now.

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Right now I'm going to write the
program that encapsulates

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these rules independent of
the representation of the

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algebraic expressions.

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You have a derivative of
an expression with

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respect to a variable.

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This is a different
thing than the

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derivative of the function.

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That's what we saw last time,
that numerical approximation.

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It's something you can't
open up a function.

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It's just the answers.

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The derivative of
an expression is

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the way it's written.

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And therefore it's a syntactic
phenomenon.

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And so a lot of what we're going
to be doing today is

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worrying about syntax,
syntax of expressions

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and things like that.

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Well, there's a case analysis.

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Anytime we do anything
complicated thereby a

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recursion, we presumably
need a case analysis.

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It's the essential
way to begin.

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And that's usually
a conditional

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of some large kind.

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Well, what are their
possibilities?

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the first rule that you saw is
this something a constant?

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And what I'm asking is, is the
expression a constant with

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respect to the variable given?

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If so, the result is 0,
because the derivative

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represents the rate of
change of something.

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If, however, the expression that
I'm taking the derivative

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of is the variable I'm varying,
then this is the same

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variable, the expression var,
then the rate of change of the

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expression with respect
to the variable is 1.

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It's the same 1.

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Well now there are a couple
of other possibilities.

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It could, for example,
be a sum.

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Well, I don't know how I'm going
to express sums yet.

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Actually I do.

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But I haven't told you yet.

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But is it a sum?

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I'm imagining that there's
some way of telling.

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I'm doing a dispatch on the type
of the expression here,

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absolutely essential in
building languages.

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Languages are made out of
different expressions.

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And soon we're going to see
that in our more powerful

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methods of building languages
on languages.

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Is an expression a sum?

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If it's a sum, well, we know the
rule for derivative of the

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sum is the sum of the
derivatives of the parts.

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One of them is called
the addend and the

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other is the augend.

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But I don't have enough space
on the blackboard

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to such long names.

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So I'll call them A1 and A2.

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I want to make a sum.

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Do you remember which is the sum
for end or the menu end?

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Or was it the dividend
and the divisor or

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something like that?

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Make sum of the derivative
of the A1, I'll call it.

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It's the addend of the
expression with respect to the

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variable, and the derivative of
the A2 of the expression,

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because the two arguments,
the addition with

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respect to the variable.

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And another rule that we know is
product rule, which is, if

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the expression is a product.

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By the way, it's a good idea
when you're defining things,

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when you're defining predicates,
to give them a

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name that ends in
a question mark.

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This question mark doesn't
mean anything.

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It's for us as an agreement.

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It's a conventional interface
between humans so you can read

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my programs more easily.

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So I want you to, when you write
programs, if you define

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a predicate procedure, that's
something that rings true of

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false, it should have a name
which ends in question mark.

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The list doesn't care.

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I care.

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I want to make a sum.

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Because the derivative of a
product is the sum of the

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first times the derivative of
the second plus the second

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times the derivative of the
first. Make a sum of two

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things, a product of, well, I'm
going to say the M1 of the

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expression, and the derivative
of the M2 of the expression

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with respect to the variable,
and the product of the

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derivative of M1, the multiplier
of the expression,

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with respect to the variable.

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It's the product of that and the
multiplicand, M2, of the

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expression.

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Make that product.

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Make the sum.

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Close that case.

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And, of course, I could add as
many cases as I like here for

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a complete set of rules you
might find in a calculus book.

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So this is what it takes to
encapsulate those rules.

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And you see, you have to realize
there's a lot of

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wishful thinking here.

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I haven't told you anything
about how I'm going to make

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these representations.

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Now, once I've decided that
this is my set of rules, I

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think it's time to play with
the representation.

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Let's attack that/

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Well, first of all, I'm
going to play a pun.

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It's an important pun.

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It's a key to a sort
of powerful idea.

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If I want to represent sums, and
products, and differences,

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and quotients, and things like
that, why not use the same

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language as I'm writing
my program in?

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I write my program in algebraic
expressions that

250
00:13:23,130 --> 00:13:29,280
look like the sum of the product
on a and the product

251
00:13:29,280 --> 00:13:34,330
of x and x, and things
like that.

252
00:13:34,330 --> 00:13:39,390
And the product of b and x and
c, whatever, make that a sum

253
00:13:39,390 --> 00:13:40,960
of the product.

254
00:13:40,960 --> 00:13:43,230
Right now I don't want to have
procedures with unknown

255
00:13:43,230 --> 00:13:48,300
numbers of arguments, a product
of b and x and c.

256
00:13:51,050 --> 00:13:54,280
This is list structure.

257
00:13:54,280 --> 00:13:56,950
And the reason why this is nice,
is because any one of

258
00:13:56,950 --> 00:14:00,380
these objects has a property.

259
00:14:00,380 --> 00:14:01,970
I know where the car is.

260
00:14:01,970 --> 00:14:04,100
The car is the operator.

261
00:14:04,100 --> 00:14:08,190
And the operands are the
successive cdrs the successive

262
00:14:08,190 --> 00:14:12,276
cars of the cdrs of the
list that this is.

263
00:14:12,276 --> 00:14:14,470
It makes it very convenient.

264
00:14:14,470 --> 00:14:15,560
I have to parse it.

265
00:14:15,560 --> 00:14:17,590
It's been done for me.

266
00:14:17,590 --> 00:14:19,685
I'm using the embedding
and Lisp to advantage.

267
00:14:22,860 --> 00:14:29,340
So, for example, let's start
using list structure to write

268
00:14:29,340 --> 00:14:35,390
down the representation that I'm
implicitly assuming here.

269
00:14:35,390 --> 00:14:37,590
Well I have to define various
things that are implied in

270
00:14:37,590 --> 00:14:38,640
this representation.

271
00:14:38,640 --> 00:14:41,210
Like I have to find out
how to do a constant,

272
00:14:41,210 --> 00:14:42,520
how you do same variable.

273
00:14:42,520 --> 00:14:45,890
Let's do those first. That's
pretty easy enough.

274
00:14:45,890 --> 00:14:47,410
Now I'm going to be introducing
lots of primitives

275
00:14:47,410 --> 00:14:49,680
here, because these are
the primitives that

276
00:14:49,680 --> 00:14:51,745
come with list structure.

277
00:14:51,745 --> 00:14:53,015
OK, you define a constant.

278
00:15:02,800 --> 00:15:06,350
And what I mean by a constant,
an expression that's constant

279
00:15:06,350 --> 00:15:10,860
with respect to a veritable,
is that the expression is

280
00:15:10,860 --> 00:15:11,530
something simple.

281
00:15:11,530 --> 00:15:13,550
I can't take it into
pieces, and yet

282
00:15:13,550 --> 00:15:16,550
it isn't that variable.

283
00:15:16,550 --> 00:15:18,940
I can't break it up, and yet
it isn't that variable.

284
00:15:18,940 --> 00:15:22,840
That does not mean that there
may be other expressions that

285
00:15:22,840 --> 00:15:25,230
are more complicated
that are constants.

286
00:15:25,230 --> 00:15:26,700
It's just that I'm going to
look at the primitive

287
00:15:26,700 --> 00:15:30,510
constants in this way.

288
00:15:30,510 --> 00:15:34,080
So what this is, is it says
that's it's the and.

289
00:15:34,080 --> 00:15:37,030
I can combine predicate
expressions which return true

290
00:15:37,030 --> 00:15:38,610
or false with and.

291
00:15:38,610 --> 00:15:45,600
Something atomic, The expression
is atomic, meaning

292
00:15:45,600 --> 00:15:47,050
it cannot be broken
into parts.

293
00:15:47,050 --> 00:15:51,150
It doesn't have a car and a cdr.
It's not a list. It adds

294
00:15:51,150 --> 00:15:54,250
a special test built
into the system.

295
00:15:54,250 --> 00:16:06,950
And it's not identically
equal to that variable.

296
00:16:06,950 --> 00:16:11,810
I'm representing my variable
by things that are symbols

297
00:16:11,810 --> 00:16:16,550
which cannot be broken into
pieces, things like x, and y,

298
00:16:16,550 --> 00:16:17,800
things like this.

299
00:16:19,850 --> 00:16:21,890
Whereas, of course, something
like this can be broken up

300
00:16:21,890 --> 00:16:23,140
into pieces.

301
00:16:24,860 --> 00:16:40,560
And the same variable of an
expression with respect to a

302
00:16:40,560 --> 00:16:48,840
variable is, in fact, an
atomic expression.

303
00:16:48,840 --> 00:16:50,040
I want to have an atomic

304
00:16:50,040 --> 00:16:59,330
expression, which is identical.

305
00:17:08,030 --> 00:17:13,329
I don't want to look inside
this stuff anymore.

306
00:17:13,329 --> 00:17:16,040
These are primitive maybe.

307
00:17:16,040 --> 00:17:18,180
But it doesn't matter.

308
00:17:18,180 --> 00:17:22,569
I'm using things that are given
to me with a language.

309
00:17:22,569 --> 00:17:24,300
I'm not terribly interest
in them

310
00:17:24,300 --> 00:17:26,900
Now how do we deal with sums?

311
00:17:26,900 --> 00:17:29,100
Ah, something very interesting
will happen.

312
00:17:29,100 --> 00:17:32,380
A sum is something which is not
atomic and begins with the

313
00:17:32,380 --> 00:17:35,230
plus symbol.

314
00:17:35,230 --> 00:17:36,680
That's what it means.

315
00:17:36,680 --> 00:17:39,125
So here, I will define.

316
00:17:45,630 --> 00:18:08,380
An question is a sum if and it's
not atomic and it's head,

317
00:18:08,380 --> 00:18:15,140
it's beginning, its car of the
expression is the symbol plus.

318
00:18:19,950 --> 00:18:21,730
Now you're about to see
something you haven't seen

319
00:18:21,730 --> 00:18:23,690
before, this quotation.

320
00:18:26,240 --> 00:18:29,440
Why do I have that
quotation there?

321
00:18:29,440 --> 00:18:30,970
Say your name,

322
00:18:30,970 --> 00:18:31,410
AUDIENCE: Susanna.

323
00:18:31,410 --> 00:18:32,270
PROFESSOR: Louder.

324
00:18:32,270 --> 00:18:33,190
AUDIENCE: Susanna

325
00:18:33,190 --> 00:18:34,250
PROFESSOR: Say your name.

326
00:18:34,250 --> 00:18:35,160
AUDIENCE: Your name.

327
00:18:35,160 --> 00:18:35,910
PROFESSOR: Louder.

328
00:18:35,910 --> 00:18:36,960
AUDIENCE: Your name.

329
00:18:36,960 --> 00:18:39,100
PROFESSOR: OK.

330
00:18:39,100 --> 00:18:42,040
What I'm showing you here
is that the words

331
00:18:42,040 --> 00:18:45,520
of English are ambiguous.

332
00:18:45,520 --> 00:18:52,220
I was saying, say your name.

333
00:18:52,220 --> 00:18:57,070
I was also possibly saying
say, your name.

334
00:19:00,710 --> 00:19:04,100
But that cannot be distinguished
in speech.

335
00:19:04,100 --> 00:19:09,600
However, we do have a notation
in writing, which is quotation

336
00:19:09,600 --> 00:19:14,180
for distinguishing these
two possible meanings.

337
00:19:14,180 --> 00:19:19,490
In particular, over here, in
Lisp we have a notation for

338
00:19:19,490 --> 00:19:21,510
distinguishing these meetings.

339
00:19:21,510 --> 00:19:24,992
If I were to just write a plus
here, a plus symbol, I would

340
00:19:24,992 --> 00:19:29,400
be asking, is the first element
of the expression, is

341
00:19:29,400 --> 00:19:32,220
the operator position
of the expression,

342
00:19:32,220 --> 00:19:34,760
the addition operator?

343
00:19:34,760 --> 00:19:36,330
I don't know.

344
00:19:36,330 --> 00:19:39,550
I would have to have written the
addition operator there,

345
00:19:39,550 --> 00:19:41,330
which I can't write.

346
00:19:41,330 --> 00:19:45,470
However, this way I'm asking,
is this the symbolic object

347
00:19:45,470 --> 00:19:49,790
plus, which normally stands
for the addition operator?

348
00:19:49,790 --> 00:19:50,420
That's what I want.

349
00:19:50,420 --> 00:19:53,110
That's the question
I want to ask.

350
00:19:53,110 --> 00:19:55,430
Now before I go any further,
I want to point out the

351
00:19:55,430 --> 00:19:59,780
quotation is a very complex
concept, and adding it to a

352
00:19:59,780 --> 00:20:03,510
language causes a great
deal of troubles.

353
00:20:03,510 --> 00:20:06,370
Consider the next slide.

354
00:20:06,370 --> 00:20:11,830
Here's a deduction which we
should all agree with.

355
00:20:11,830 --> 00:20:17,530
We have, Alyssa is smart and
Alyssa is George's mother.

356
00:20:17,530 --> 00:20:22,350
This is an equality, is.

357
00:20:22,350 --> 00:20:27,470
From those two, we can deduce
that George's mother is smart.

358
00:20:27,470 --> 00:20:32,320
Because we can always substitute
equals for equals

359
00:20:32,320 --> 00:20:34,250
in expressions.

360
00:20:34,250 --> 00:20:36,420
Or can we?

361
00:20:36,420 --> 00:20:41,400
Here's a case where we have
"Chicago" has seven letters.

362
00:20:41,400 --> 00:20:45,100
The quotation means that I'm
discussing the word Chicago,

363
00:20:45,100 --> 00:20:46,365
not what the word represents.

364
00:20:49,940 --> 00:20:54,830
Here I have that Chicago is the
biggest city in Illinois.

365
00:20:54,830 --> 00:20:57,160
As a consequence of this, I
would like to deduce that the

366
00:20:57,160 --> 00:20:59,250
biggest city in Illinois
has seven letters.

367
00:20:59,250 --> 00:21:00,785
But that's manifestly false.

368
00:21:05,480 --> 00:21:09,420
Wow, it works.

369
00:21:09,420 --> 00:21:13,420
OK, so once we have things like
that, our language gets

370
00:21:13,420 --> 00:21:14,540
much more complicated.

371
00:21:14,540 --> 00:21:17,440
Because it's no longer true that
things we tend to like to

372
00:21:17,440 --> 00:21:19,750
do with languages, like
substituting equals for equals

373
00:21:19,750 --> 00:21:22,840
and getting right answers, are
going to work without being

374
00:21:22,840 --> 00:21:24,550
very careful.

375
00:21:24,550 --> 00:21:26,470
We can't substitute into what's
called referentially

376
00:21:26,470 --> 00:21:30,410
opaque contexts, of which a
quotation is the prototypical

377
00:21:30,410 --> 00:21:33,380
type of referentially
opaque context.

378
00:21:33,380 --> 00:21:35,560
If you know what that means, you
can consult a philosopher.

379
00:21:35,560 --> 00:21:38,790
Presumably there is
one in the room.

380
00:21:38,790 --> 00:21:42,280
In any case, let's continue now,
now that we at least have

381
00:21:42,280 --> 00:21:45,660
an operational understanding of
a 2000-year-old issue that

382
00:21:45,660 --> 00:21:47,350
has to do with name, and
mention, and all sorts of

383
00:21:47,350 --> 00:21:48,600
things like that.

384
00:21:52,440 --> 00:21:59,790
I have to define what I mean,
how to make a sum of two

385
00:21:59,790 --> 00:22:02,250
things, an a1 and a2.

386
00:22:02,250 --> 00:22:03,590
And I'm going to do
this very simply.

387
00:22:03,590 --> 00:22:13,600
It's a list of the symbol
plus, and a1, and a2.

388
00:22:13,600 --> 00:22:17,075
And I can determine
the first element.

389
00:22:21,600 --> 00:22:34,510
Define a1 to be cadr. I've just

390
00:22:34,510 --> 00:22:36,370
introduced another primitive.

391
00:22:36,370 --> 00:22:39,990
This is the car of the
cdr of something.

392
00:22:39,990 --> 00:22:43,950
You might want to know why car
and cdr are names of these

393
00:22:43,950 --> 00:22:46,720
primitives, and why they've
survived, even though they're

394
00:22:46,720 --> 00:22:48,970
much better ideas like
left and right.

395
00:22:48,970 --> 00:22:51,380
We could have called them
things like that.

396
00:22:51,380 --> 00:22:54,340
Well, first of all, the names
come from the fact that in the

397
00:22:54,340 --> 00:22:57,470
great past, when Lisp was
invented, I suppose in '58 or

398
00:22:57,470 --> 00:23:00,730
something, it was on a
704 or something like

399
00:23:00,730 --> 00:23:01,870
that, which had a machine.

400
00:23:01,870 --> 00:23:04,440
It was a machine that had an
address register and a

401
00:23:04,440 --> 00:23:05,340
decrement register.

402
00:23:05,340 --> 00:23:07,610
And these were the contents of
the address register and the

403
00:23:07,610 --> 00:23:08,270
decrement register.

404
00:23:08,270 --> 00:23:09,880
So it's an historical
accident.

405
00:23:09,880 --> 00:23:11,880
Now why have these
names survived?

406
00:23:11,880 --> 00:23:14,110
It's because Lisp programmers
like to talk to each other

407
00:23:14,110 --> 00:23:15,900
over the phone.

408
00:23:15,900 --> 00:23:18,460
And if you want to have a long
sequence of cars and cdrs you

409
00:23:18,460 --> 00:23:22,530
might say, cdaddedr, which
can be understood.

410
00:23:22,530 --> 00:23:26,330
But left of right or right of
left is not so clear if you

411
00:23:26,330 --> 00:23:26,970
get good at it.

412
00:23:26,970 --> 00:23:30,570
So that's why we have
these words.

413
00:23:30,570 --> 00:23:33,566
All of them up to four deep
are defined typically in a

414
00:23:33,566 --> 00:23:34,816
Lisp system.

415
00:23:38,270 --> 00:23:40,540
A2 to be--

416
00:23:43,540 --> 00:23:46,170
and, of course, you can see
that if I looked at one of

417
00:23:46,170 --> 00:23:54,630
these expressions like the sum
of 3 and 5, what that is is a

418
00:23:54,630 --> 00:24:06,100
list containing the symbol
plus, and a number 3,

419
00:24:06,100 --> 00:24:11,470
and a number 5.

420
00:24:11,470 --> 00:24:16,100
Then the car is the
symbol plus.

421
00:24:16,100 --> 00:24:19,210
The car of the cdr. Well
I take the cdr and

422
00:24:19,210 --> 00:24:20,070
then I take the car.

423
00:24:20,070 --> 00:24:21,200
And that's how I get to the 3.

424
00:24:21,200 --> 00:24:22,630
That's the first argument.

425
00:24:22,630 --> 00:24:24,370
And the car of the cdr
of the cdr gets me to

426
00:24:24,370 --> 00:24:25,640
this one, the 5.

427
00:24:28,860 --> 00:24:32,970
And similarly, of course, I
can define what's going on

428
00:24:32,970 --> 00:24:35,300
with products.

429
00:24:35,300 --> 00:24:36,550
Let's do that very quickly.

430
00:24:48,760 --> 00:24:51,130
Is the expression a product?

431
00:24:51,130 --> 00:25:01,910
Yes if and if it's true, that's
it's not atomic and

432
00:25:01,910 --> 00:25:13,260
it's EQ quote, the asterisk
symbol, which is the operator

433
00:25:13,260 --> 00:25:15,800
for multiplication.

434
00:25:15,800 --> 00:25:35,090
Make product of an M1 and an
M2 to be list, quote, the

435
00:25:35,090 --> 00:25:40,930
asterisk operation
and M1 and M2.

436
00:25:40,930 --> 00:26:00,596
and I define M1 to be cadr and
M2 to be caddr. You get to be

437
00:26:00,596 --> 00:26:02,690
a good Lisp programmer because
you start talking that way.

438
00:26:02,690 --> 00:26:06,430
I cdr down lists and console
them up and so on.

439
00:26:06,430 --> 00:26:09,480
Now, now that we have
essentially a complete program

440
00:26:09,480 --> 00:26:10,880
for finding derivatives,
you can add more

441
00:26:10,880 --> 00:26:12,360
rules if you like.

442
00:26:12,360 --> 00:26:14,800
What kind of behavior
do we get out of it?

443
00:26:14,800 --> 00:26:17,930
I'll have to clear that x.

444
00:26:17,930 --> 00:26:28,060
Well, supposing I define foo
here to be the sum of the

445
00:26:28,060 --> 00:26:30,450
product of ax square
and bx plus c.

446
00:26:30,450 --> 00:26:34,020
That's the same thing we see
here as the algebraic

447
00:26:34,020 --> 00:26:35,260
expression written in the
more conventional

448
00:26:35,260 --> 00:26:37,860
notation over there.

449
00:26:37,860 --> 00:26:40,620
Well, the derivative of foo with
respect to x, which we

450
00:26:40,620 --> 00:26:46,250
can see over here, is this
horrible, horrendous mess.

451
00:26:46,250 --> 00:26:50,760
I would like it to
be 2ax plus b.

452
00:26:50,760 --> 00:26:52,240
But it's not.

453
00:26:52,240 --> 00:26:54,620
It's equivalent to it.

454
00:26:54,620 --> 00:26:56,090
What is it?

455
00:26:56,090 --> 00:27:00,510
I have here, what do I have?

456
00:27:00,510 --> 00:27:04,550
I have the derivative of
the product of x and x.

457
00:27:04,550 --> 00:27:09,410
Over here is, of course,
the sum of x times

458
00:27:09,410 --> 00:27:12,830
1 and 1 times x.

459
00:27:12,830 --> 00:27:14,100
Now, well, it's the first times
the derivative of the

460
00:27:14,100 --> 00:27:15,330
second plus the second times the
derivative of the first.

461
00:27:15,330 --> 00:27:17,780
It's right.

462
00:27:17,780 --> 00:27:20,220
That's 2x of course.

463
00:27:20,220 --> 00:27:26,600
a times 2x is 2ax plus 0X square
doesn't count plus B

464
00:27:26,600 --> 00:27:29,100
over here plus a bunch of 0's.

465
00:27:29,100 --> 00:27:30,130
Well the answer is right.

466
00:27:30,130 --> 00:27:34,390
But I give people take off
points on an exam for that,

467
00:27:34,390 --> 00:27:35,690
sadly enough.

468
00:27:35,690 --> 00:27:37,830
Let's worry about that
in the next segment.

469
00:27:37,830 --> 00:27:39,080
Are there any questions?

470
00:27:42,970 --> 00:27:44,070
Yes?

471
00:27:44,070 --> 00:27:46,790
AUDIENCE: If you had left the
quote when you put the plus,

472
00:27:46,790 --> 00:27:51,120
then would that be referring
to the procedure plus and

473
00:27:51,120 --> 00:27:53,560
could you do a comparison
between that procedure and

474
00:27:53,560 --> 00:27:55,460
some other procedure
if you wanted to?

475
00:27:55,460 --> 00:27:56,320
PROFESSOR: Yes.

476
00:27:56,320 --> 00:27:58,960
Good question.

477
00:27:58,960 --> 00:28:05,650
If I had left this quotation
off at this point, if I had

478
00:28:05,650 --> 00:28:08,720
left that quotation off at that
point, then I would be

479
00:28:08,720 --> 00:28:12,790
referring here to the procedure
which is the thing

480
00:28:12,790 --> 00:28:15,510
that plus is defined to be.

481
00:28:15,510 --> 00:28:22,810
And indeed, I could compare
some procedures with each

482
00:28:22,810 --> 00:28:25,070
other for identity.

483
00:28:25,070 --> 00:28:28,080
Now what that means is
not clear right now.

484
00:28:28,080 --> 00:28:30,100
I don't like to think
about it.

485
00:28:30,100 --> 00:28:31,840
Because I don't know exactly
what it would need to compare

486
00:28:31,840 --> 00:28:32,450
procedures.

487
00:28:32,450 --> 00:28:35,610
There are reasons why that
may make no sense at all.

488
00:28:35,610 --> 00:28:38,890
However, the symbols,
we understand.

489
00:28:38,890 --> 00:28:41,240
And so that's why I
put that quote in.

490
00:28:41,240 --> 00:28:42,510
I want to talk about
the symbol that's

491
00:28:42,510 --> 00:28:43,760
apparent on the page.

492
00:28:46,250 --> 00:28:48,700
Any other questions?

493
00:28:48,700 --> 00:28:50,720
OK.

494
00:28:50,720 --> 00:28:51,060
Thank you.

495
00:28:51,060 --> 00:28:54,210
Let's take a break.

496
00:28:54,210 --> 00:29:30,010
[MUSIC PLAYING]

497
00:29:30,010 --> 00:29:31,580
PROFESSOR: Well, let's see.

498
00:29:31,580 --> 00:29:35,560
We've just developed a fairly
plausible program for

499
00:29:35,560 --> 00:29:38,390
computing the derivatives of
algebraic expressions.

500
00:29:38,390 --> 00:29:40,400
It's an incomplete program,
if you would

501
00:29:40,400 --> 00:29:42,330
like to add more rules.

502
00:29:42,330 --> 00:29:46,020
And perhaps you might extend
it to deal with uses of

503
00:29:46,020 --> 00:29:48,390
addition with any number of
arguments and multiplication

504
00:29:48,390 --> 00:29:49,950
with any of the number
of arguments.

505
00:29:49,950 --> 00:29:52,470
And that's all rather easy.

506
00:29:52,470 --> 00:29:57,620
However, there was a little
fly in that ointment.

507
00:29:57,620 --> 00:30:02,980
We go back to this slide.

508
00:30:02,980 --> 00:30:09,000
We see that the expressions that
we get are rather bad.

509
00:30:09,000 --> 00:30:11,620
This is a rather
bad expression.

510
00:30:11,620 --> 00:30:14,000
How do we get such
an expression?

511
00:30:14,000 --> 00:30:16,940
Why do we have that
expression?

512
00:30:16,940 --> 00:30:19,060
Let's look at this expression
in some detail.

513
00:30:19,060 --> 00:30:21,850
Let's find out where all
the pieces come from.

514
00:30:21,850 --> 00:30:24,590
As we see here, we
have a sum--

515
00:30:24,590 --> 00:30:27,012
just what I showed you at the
end of the last time--

516
00:30:27,012 --> 00:30:30,110
of X times 1 plus 1
time X. That is a

517
00:30:30,110 --> 00:30:32,590
derivative of this product.

518
00:30:32,590 --> 00:30:36,270
The produce of a times that,
where a does not depend upon

519
00:30:36,270 --> 00:30:40,430
x, and therefore is constant
with respect to x, is this

520
00:30:40,430 --> 00:30:43,910
sum, which goes from here all
the way through here and

521
00:30:43,910 --> 00:30:44,840
through here.

522
00:30:44,840 --> 00:30:48,470
Because it is the first thing
times the derivative of the

523
00:30:48,470 --> 00:30:54,900
second plus the derivative of
the first times the second as

524
00:30:54,900 --> 00:30:57,970
the program we wrote
on the blackboard

525
00:30:57,970 --> 00:31:00,740
indicated we should do.

526
00:31:00,740 --> 00:31:06,690
And, of course, the product of
bx over here manifests itself

527
00:31:06,690 --> 00:31:15,220
as B times 1 plus 0 times X
because we see that B does not

528
00:31:15,220 --> 00:31:19,085
depend upon X. And so the
derivative of B is this 0, and

529
00:31:19,085 --> 00:31:23,100
the derivative of X with respect
itself is the 1.

530
00:31:23,100 --> 00:31:26,510
And, of course, the derivative
of the sums over here turn

531
00:31:26,510 --> 00:31:29,360
into these two sums of the
derivatives of the parts.

532
00:31:29,360 --> 00:31:32,620
So what we're seeing here is
exactly the thing I was trying

533
00:31:32,620 --> 00:31:37,760
to tell you about with Fibonacci
numbers a while ago,

534
00:31:37,760 --> 00:31:44,430
that the form of the process
is expanded from the local

535
00:31:44,430 --> 00:31:48,850
rules that you see in the
procedure, that the procedure

536
00:31:48,850 --> 00:31:51,720
represents a set of local rules
for the expansion of

537
00:31:51,720 --> 00:31:53,520
this process.

538
00:31:53,520 --> 00:31:59,440
And here, the process left
behind some stuff, which is

539
00:31:59,440 --> 00:32:00,370
the answer.

540
00:32:00,370 --> 00:32:03,850
And it was constructed by the
walk it takes of the tree

541
00:32:03,850 --> 00:32:05,775
structure, which is
the expression.

542
00:32:08,390 --> 00:32:11,540
So every part in the answer we
see here derives from some

543
00:32:11,540 --> 00:32:14,670
part of the problem.

544
00:32:14,670 --> 00:32:17,360
Now, we can look at, for
example, the derivative of

545
00:32:17,360 --> 00:32:20,500
foo, which is ax square plus
bx plus c, with respect to

546
00:32:20,500 --> 00:32:25,320
other things, like here, for
example, we can see that the

547
00:32:25,320 --> 00:32:27,860
derivative of foo with
respect to a.

548
00:32:27,860 --> 00:32:29,390
And it's very similar.

549
00:32:29,390 --> 00:32:32,840
It's, in fact, the identical
algebraic expression, except

550
00:32:32,840 --> 00:32:34,320
for the fact that theses
0's and 1's are

551
00:32:34,320 --> 00:32:35,900
in different places.

552
00:32:35,900 --> 00:32:38,260
Because the only degree of
freedom we have in this tree

553
00:32:38,260 --> 00:32:42,285
walk is what's constant with
respect to the variable we're

554
00:32:42,285 --> 00:32:44,550
taking the derivative
with respect to and

555
00:32:44,550 --> 00:32:45,800
was the same variable.

556
00:32:48,310 --> 00:32:51,660
In other words, if we go back
to this blackboard and we

557
00:32:51,660 --> 00:32:55,302
look, we have no choice what
to do when we take the

558
00:32:55,302 --> 00:32:58,150
derivative of the sum
or a product.

559
00:32:58,150 --> 00:33:03,840
The only interesting place here
is, is the expression the

560
00:33:03,840 --> 00:33:07,590
variable, or is the expression
a constant with respect to

561
00:33:07,590 --> 00:33:10,130
that variable for very, very
small expressions?

562
00:33:10,130 --> 00:33:13,220
In which case we get various
1's and 0's, which if we go

563
00:33:13,220 --> 00:33:17,290
back to this slide, we can see
that the 0's that appear here,

564
00:33:17,290 --> 00:33:21,740
for example, this 1 over here
in derivative of foo with

565
00:33:21,740 --> 00:33:25,980
respect to A, which gets us an
X square, because that 1 gets

566
00:33:25,980 --> 00:33:32,770
the multiply of X and X into
the answer, that 1 is 0.

567
00:33:32,770 --> 00:33:34,173
Over here, we're not taking
the derivative of foo with

568
00:33:34,173 --> 00:33:36,690
respect to c.

569
00:33:36,690 --> 00:33:40,301
But the shapes of these
expressions are the same.

570
00:33:40,301 --> 00:33:42,561
See all those shapes.

571
00:33:42,561 --> 00:33:43,811
They're the same.

572
00:33:50,480 --> 00:33:53,750
Well is there anything
wrong with our rules?

573
00:33:53,750 --> 00:33:54,030
No.

574
00:33:54,030 --> 00:33:56,250
They're the right rules.

575
00:33:56,250 --> 00:33:58,160
We've been through
this one before.

576
00:33:58,160 --> 00:34:02,020
One of the things you're going
to begin to discover is that

577
00:34:02,020 --> 00:34:03,270
there aren't too many
good ideas.

578
00:34:06,510 --> 00:34:12,139
When we were looking at rational
numbers yesterday,

579
00:34:12,139 --> 00:34:14,909
the problem was that we got
6/8 rather then 3/4.

580
00:34:14,909 --> 00:34:17,949
The answer was unsimplified.

581
00:34:17,949 --> 00:34:21,150
The problem, of course,
is very similar.

582
00:34:21,150 --> 00:34:24,350
There are things I'd like to be
identical by simplification

583
00:34:24,350 --> 00:34:27,350
that don't become identical.

584
00:34:27,350 --> 00:34:30,690
And yet the rules for doing
addition a multiplication of

585
00:34:30,690 --> 00:34:34,000
rational numbers were correct.

586
00:34:34,000 --> 00:34:36,350
So the way we might solve this
problem is do the thing we did

587
00:34:36,350 --> 00:34:37,940
last time, which always works.

588
00:34:37,940 --> 00:34:40,690
If something worked last time
it ought to work again.

589
00:34:40,690 --> 00:34:43,120
It's changed representation.

590
00:34:43,120 --> 00:34:45,350
Perhaps in the representation
we could put in a

591
00:34:45,350 --> 00:34:48,940
simplification step that
produces a simplified

592
00:34:48,940 --> 00:34:50,199
representation.

593
00:34:50,199 --> 00:34:52,580
This may not always
work, of course.

594
00:34:52,580 --> 00:34:55,210
I'm not trying to say that
it always works.

595
00:34:55,210 --> 00:34:59,540
But it's one of the pieces of
artillery we have in our war

596
00:34:59,540 --> 00:35:01,560
against complexity.

597
00:35:01,560 --> 00:35:04,360
You see, because we solved our
problem very carefully.

598
00:35:04,360 --> 00:35:06,170
What we've done, is
we've divided the

599
00:35:06,170 --> 00:35:07,630
world in several parts.

600
00:35:07,630 --> 00:35:12,980
There are derivatives rules and
general rules for algebra

601
00:35:12,980 --> 00:35:16,420
of some sort at this
level of detail.

602
00:35:16,420 --> 00:35:17,925
and i have an abstraction
barrier.

603
00:35:21,874 --> 00:35:32,710
And i have the representation of
the algebraic expressions,

604
00:35:32,710 --> 00:35:33,960
list structure.

605
00:35:37,420 --> 00:35:43,050
And in this barrier, I have
the interface procedures.

606
00:35:43,050 --> 00:35:49,410
I have constant, and things
like same-var.

607
00:35:54,680 --> 00:35:58,095
I have things like
sum, make-sum.

608
00:36:02,310 --> 00:36:06,770
I have A1, A2.

609
00:36:06,770 --> 00:36:09,370
I have products and things
like that, all the other

610
00:36:09,370 --> 00:36:11,490
things I might need for various
kinds of algebraic

611
00:36:11,490 --> 00:36:13,000
expressions.

612
00:36:13,000 --> 00:36:18,080
Making this barrier allows me
to arbitrarily change the

613
00:36:18,080 --> 00:36:21,710
representation without changing
the rules that are

614
00:36:21,710 --> 00:36:25,060
written in terms of that
representation.

615
00:36:25,060 --> 00:36:28,220
So if I can make the problem
go away by changing

616
00:36:28,220 --> 00:36:32,320
representation, the composition
of the problem

617
00:36:32,320 --> 00:36:35,660
into these two parts has
helped me a great deal.

618
00:36:35,660 --> 00:36:38,860
So let's take a very simple
case of this.

619
00:36:38,860 --> 00:36:40,330
What was one of the problems?

620
00:36:40,330 --> 00:36:44,115
Let's go back to this
transparency again.

621
00:36:44,115 --> 00:36:48,280
And we see here, oh yes, there's
horrible things like

622
00:36:48,280 --> 00:36:53,190
here is the sum of an
expression and 0.

623
00:36:53,190 --> 00:36:55,590
Well that's no reason to think
of it as anything other than

624
00:36:55,590 --> 00:36:57,300
the expression itself.

625
00:36:57,300 --> 00:36:59,970
Why should the summation
operation have

626
00:36:59,970 --> 00:37:03,450
made up this edition?

627
00:37:03,450 --> 00:37:05,550
It can be smarter than that.

628
00:37:05,550 --> 00:37:09,080
Or here, for example, is
a multiplication of

629
00:37:09,080 --> 00:37:10,816
something by 1.

630
00:37:10,816 --> 00:37:12,990
It's another thing like that.

631
00:37:12,990 --> 00:37:14,960
Or here is a product of
something with 0, which is

632
00:37:14,960 --> 00:37:17,800
certainly 0.

633
00:37:17,800 --> 00:37:21,430
So we won't have to make
this construction.

634
00:37:21,430 --> 00:37:23,800
So why don't we just do that?

635
00:37:23,800 --> 00:37:25,340
We need to change the way
the representation

636
00:37:25,340 --> 00:37:27,990
works, almost here.

637
00:37:37,500 --> 00:37:42,060
Make-sum to be.

638
00:37:42,060 --> 00:37:44,020
Well, now it's not something
so simple.

639
00:37:44,020 --> 00:37:48,500
I'm not going to make a list
containing the symbol plus and

640
00:37:48,500 --> 00:37:51,322
things unless I need to.

641
00:37:51,322 --> 00:37:52,630
Well, what are the
possibilities?

642
00:37:57,220 --> 00:37:59,420
I have some sort
of cases here.

643
00:37:59,420 --> 00:38:09,160
If I have numbers, if
anyone is a number--

644
00:38:09,160 --> 00:38:10,930
and here's another primitive
I've just introduced, it's

645
00:38:10,930 --> 00:38:15,230
possible to tell whether
something's number--

646
00:38:15,230 --> 00:38:23,090
and if number A2, meaning
they're not symbolic

647
00:38:23,090 --> 00:38:26,280
expressions, then why not
do the addition now?

648
00:38:26,280 --> 00:38:29,540
The result is just a
plus of A1 and A2.

649
00:38:32,270 --> 00:38:34,000
I'm not asking if these
represent numbers.

650
00:38:34,000 --> 00:38:37,100
Of course all of these symbols
represent numbers.

651
00:38:37,100 --> 00:38:39,100
I'm talking about whether
the one I've got is the

652
00:38:39,100 --> 00:38:43,420
number 3 right now.

653
00:38:43,420 --> 00:38:59,070
And, for example, supposing A1
is a number, and it's equal to

654
00:38:59,070 --> 00:39:06,900
0, well then the answer
is just A2.

655
00:39:06,900 --> 00:39:10,698
There is no reason to
make anything up.

656
00:39:10,698 --> 00:39:27,630
And if A2 is a number,
and equal A20, then

657
00:39:27,630 --> 00:39:30,210
the result is A1.

658
00:39:30,210 --> 00:39:32,770
And only if I can't figure out
something better to do with

659
00:39:32,770 --> 00:39:41,160
this situation, well, I can
start a list. Otherwise I want

660
00:39:41,160 --> 00:39:45,790
the representation to be the
list containing the quoted

661
00:39:45,790 --> 00:39:51,850
symbol plus, and A1, and A2.

662
00:39:58,720 --> 00:40:00,660
And, of course, a very
similar thing

663
00:40:00,660 --> 00:40:03,020
can be done for products.

664
00:40:03,020 --> 00:40:05,650
And I think I'll avoid
boring you with them.

665
00:40:05,650 --> 00:40:07,740
I was going to write it
on the blackboard.

666
00:40:07,740 --> 00:40:09,080
I don't think it's necessary.

667
00:40:09,080 --> 00:40:10,830
You know what to do.

668
00:40:10,830 --> 00:40:12,880
It's very simple.

669
00:40:12,880 --> 00:40:17,890
But now, let's just see the kind
of results we get out of

670
00:40:17,890 --> 00:40:21,870
changing our program
in this way.

671
00:40:21,870 --> 00:40:25,920
Well, here's the derivatives
after having just changed the

672
00:40:25,920 --> 00:40:28,810
constructors for expressions.

673
00:40:28,810 --> 00:40:35,810
The same foo, aX square plus bX
plus c, and what I get is

674
00:40:35,810 --> 00:40:40,265
nothing more than the derivative
of that is 2aX plus

675
00:40:40,265 --> 00:40:40,660
B.

676
00:40:40,660 --> 00:40:42,670
Well, it's not completely
simplified.

677
00:40:42,670 --> 00:40:45,170
I would like to collect
common terms and sums.

678
00:40:45,170 --> 00:40:47,180
Well, that's more work.

679
00:40:47,180 --> 00:40:51,130
And, of course, programs to do
this sort of thing are huge

680
00:40:51,130 --> 00:40:52,440
and complicated.

681
00:40:52,440 --> 00:40:56,510
Algebraic simplification, it's
a very complicated mess.

682
00:40:56,510 --> 00:40:58,240
There's a very famous program
you may have heard of called

683
00:40:58,240 --> 00:41:02,750
Maxima developed at MIT in the
past, which is 5,000 pages of

684
00:41:02,750 --> 00:41:05,830
Lisp code, mostly the algebraic
simplification

685
00:41:05,830 --> 00:41:07,080
operations.

686
00:41:10,080 --> 00:41:12,210
There we see the derivative
of foo.

687
00:41:12,210 --> 00:41:14,390
In fact, X is at something I
wouldn't take off more than 1

688
00:41:14,390 --> 00:41:18,390
point for on an elementary
calculus class.

689
00:41:18,390 --> 00:41:20,960
And the derivative of foo with
respect to a, well it's gone

690
00:41:20,960 --> 00:41:24,730
down to X times X, which
isn't so bad.

691
00:41:24,730 --> 00:41:28,200
And the derivative of foo with
respect to b is just X itself.

692
00:41:28,200 --> 00:41:30,730
And the derivative of foo with
respect to c comes out 1.

693
00:41:30,730 --> 00:41:34,260
So I'm pretty pleased
with this.

694
00:41:34,260 --> 00:41:36,830
What you've seen is, of
course, a little bit

695
00:41:36,830 --> 00:41:41,020
contrived, carefully organized
example to show you how we can

696
00:41:41,020 --> 00:41:43,610
manipulate algebraic
expressions, how we do that

697
00:41:43,610 --> 00:41:46,890
abstractly in terms of abstract
syntax rather than

698
00:41:46,890 --> 00:41:53,910
concrete syntax and how we can
use the abstraction to control

699
00:41:53,910 --> 00:41:57,850
what goes on in building
these expressions.

700
00:41:57,850 --> 00:42:00,910
But the real story isn't just
such a simple thing as that.

701
00:42:00,910 --> 00:42:03,890
The real story is, in fact, that
I'm manipulating these

702
00:42:03,890 --> 00:42:04,450
expressions.

703
00:42:04,450 --> 00:42:06,860
And the expressions are
the same expressions--

704
00:42:06,860 --> 00:42:08,150
going back to the slide--

705
00:42:08,150 --> 00:42:12,110
as the ones that are
Lisp expressions.

706
00:42:12,110 --> 00:42:13,890
There's a pun here.

707
00:42:13,890 --> 00:42:17,020
I've chosen my representation
to be the same as the

708
00:42:17,020 --> 00:42:22,830
representation in my language
of similar things.

709
00:42:22,830 --> 00:42:26,990
By doing so, I've invoked
a necessity.

710
00:42:26,990 --> 00:42:30,890
I created the necessity to have
things like quotation

711
00:42:30,890 --> 00:42:35,690
because of the fact that my
language is capable of writing

712
00:42:35,690 --> 00:42:39,820
expressions that talk about
expressions of the language.

713
00:42:39,820 --> 00:42:42,420
I need to have something that
says, this is an expression

714
00:42:42,420 --> 00:42:45,320
I'm talking about rather than
this expression is talking

715
00:42:45,320 --> 00:42:48,400
about something, and I want
to talk about that.

716
00:42:51,290 --> 00:42:54,420
So quotation stops and says,
I'm talking about this

717
00:42:54,420 --> 00:42:55,670
expression itself.

718
00:42:58,140 --> 00:43:03,030
Now, given that power, if I can
manipulate expressions of

719
00:43:03,030 --> 00:43:07,370
the language, I can begin to
build even much more powerful

720
00:43:07,370 --> 00:43:09,860
layers upon layers
of languages.

721
00:43:09,860 --> 00:43:12,370
Because I can write languages
that not only are embedded in

722
00:43:12,370 --> 00:43:16,730
Lisp or whatever language you
start with, but languages that

723
00:43:16,730 --> 00:43:20,400
are completely different, that
are just, if we say,

724
00:43:20,400 --> 00:43:23,280
interpreted in Lisp or
something like that.

725
00:43:23,280 --> 00:43:26,232
We'll get to understand those
words more in the future.

726
00:43:26,232 --> 00:43:30,150
But right now I just want to
leave you with the fact that

727
00:43:30,150 --> 00:43:36,160
we've hit a line which gives
us tremendous power.

728
00:43:36,160 --> 00:43:37,900
And this point we've bought
a sledgehammer.

729
00:43:37,900 --> 00:43:42,250
We have to be careful to what
flies when we apply it.

730
00:43:42,250 --> 00:43:42,570
Thank you.

731
00:43:42,570 --> 00:43:43,820
[MUSIC PLAYING]