Lecture 1A: Overview and Introduction to Lisp

Lecture 1A: Overview and Introduction to Lisp

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

14 thoughts to “Lecture 1A: Overview and Introduction to Lisp”

  1. very nice ! some of the video uploads of the lectures indeed had some audio problems in them. hopefully this is fixed now.

  2. geez im sure glad he was not my teacher. omg about 30 minutes of unrelated drivel. if i was his student i would quit and do something silly like gender studies.

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