Lecture 1A: Overview and Introduction to Lisp

[Applause]. I'd like to welcome you to this course on computer science. Actually, that's a terrible way to start. Computer science is a terrible name for this business. First of all, it's not a science. It, uh, might be engineering or it might be art. Well, we'll actually see that computer so-called science actually has a lot in common with magic. You'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 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 set 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 were, 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. 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 genomics. Well, what's what I mean by the essence of computer science? What 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 serve all of modern mathematics, figuring out a way to talk precisely about so-called declarative knowledge, what is true. Well, similarly, think in the future people will look back and say, yes, those, those primitives in the 20th century, we're fiddling around with these gadgets called computers. But really what they were doing is starting to learn how to formalize, formalize intuitions about process, how to do things, starting to, to develop a way to talk precisely about, about how to knowledge, as opposed to geometry that talks about what is true. We give you an example of that Take a look. Here is a piece of a piece of mathematics, right, that says what a square root is and if the square root of the x is the number Y, such that Y squared is equal to x and Y is greater than zero. Now that's a fine piece of mathematics. But just telling you what a square root is doesn't really say anything about about how you might go out and find one, right? So let's contrast that with a piece of imperative knowledge, right, how you might go out and find a square root. This, in fact, also comes from Egypt, not, not a, not ancient, ancient Egypt, this isn't an algorithm due to here on earth. Alexandria called how to find a square root by successive averaging, and what it says is that in Show more

order to find a square root, 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 gas, and we'll talk a little bit later about what, why. That's a reasonable thing, and you keep improving to guess until it's good enough. But that's a method. That's how to do something, as opposed to declarative knowledge that says what you're looking for, right? That's a process. Well, what's the process? In general, it's kind of hard to say. You can think of it as like a magical spirit, that sort of listen the computer and does something, and the thing that directs the process Show more

is a pattern of rules called a procedure. So procedures are the, are the spells if you like, that control these magical spirits. That are the processes. And well, I guess you know everyone needs a magical language and sorcerers. But real sorcerer is used- ancient Akkadian or or Sumerian or Babylonian or whatever. We're gonna conjure up 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 prophesies. Now, it's very easy to learn Lisp. In fact in a few minutes I'm going to teach you essentially all of Lisp and teach you essentially all of the rules, and you shouldn't find that, uh, 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, listen, it's the same way. We're going to state the rules in a few minutes and we'll be very easy to see. But what's really hard is going to be the implications of those rules, right, 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. Okay, so in computer science we're in the business of formalizing is sort of have to imperative knowledge like how to do stuff and the real issues of computer science are of course not. You know, it's telling people how to do square is so that that was all it was, there wouldn't be no big deal. The real problems come when we try to build very, very large system. The computer programs that are that are thousands of pages long, so wrong 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 there, techniques for controlling the complexity of these large systems and these techniques for component controlling, complexity or what this course is really that? And in some sense that's really what computer science is about. Now that may seem like a very strange thing to say because they- after all, a lot of people besides computer scientists- deal with controlling complexity. Are they? A large airliner is an extremely complex system, and the aeronautical engineers who design that, or you know, 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 that computer science and sometimes isn't real. You see, when an engineer is designing a physical system that's made out of real parts, like, the engineer to worry about that has 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 by the long before the million 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 right. So there's, 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. As much as I want, I know about them as precisely as I like. So, as opposed to other kinds of engineering where the constraints on which 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 Show more

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. Ok, well, what are, what are some of these techniques? They're not special to computer science. First technique which is used in all of engineering is kind of abstraction, called blackbox abstraction. It takes something and, uh, they'll build a box of doubt. Yet let's see, for example, look at that square root method. I might want to take that and build a box that 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 6. And the important thing is that I'd like to design that so that if George comes along and we'd like to compute the square root of a plus the square root of B, you can take this thing and use it as a, as a module, without having to look inside and build something that looks like this: I mean a and B 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, 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 ay or or anything at all, but that's the fundamental notion of putting something in a box using blackbox abstraction to suppress detail. And the reason for that is: you want to go off and build your bigger boxes. Now there's another reason for doing blackbox 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 their 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 to do something, I make a guess and I improve that guess and I sort of keep improving that. Yes, very nice, reduce. So there's a general strategy of are 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. See, a fixed point of the function. A fixed point of the function is something. This is the ru6 point of the function. F is a value Y, such that that f of y equals y. And the way I might do that: you start with a guess and if I want something that doesn't change when I keep applying FSI, I'll keep applying up over and over until that result doesn't change very much. That's 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 over Y, Show more

and the idea of that is that if I really had y equal to the square root of x, then y and x over Y would be the same value. They'd both be the square root of x. Right, x over the square root of x is the square root of x, and so the average if Y were equal to the square root of x, then the average wouldn't change right. 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 six 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. All right, 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 six points. And in fact, let's go back and look at that slide again- she not only is- 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 is one way to compute square is is to apply this general fixed point method. So I'd like to also be able to express that in period of 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 over Y, then what should come out of that fixed point box is a method for finding square is. So in these boxes we're building, we're not only building boxes that you input numbers and output numbers. We can be building in boxes that, Show more

in effect, compute methods like finding square roots, and my take is they're in point inputs. Functions like Y goes to the average of y and x over y, like we can. You want to do that? See, 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 mystech talks about, another procedure by the general strategy that itself talks about general strategy. Okay, well, our first topic in this course- there'll be three major major topics- will be blackbox abstraction. Let's look at that in a little bit more detail. What we're going to do is we will we'll start out talking about how list is built up at a primitive object, like what is 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. That means of combinations, and what we'll see is that there 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 compound things, how do you abstract them? How do you put those black boxes around Show more

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 rule to add as 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, they'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 a. That's our first subject: blackbox abstraction. Let's look at the second topic. He introduced it like this, he suppose I. I want to express the idea and remember we're we're talking about ideas. I don't 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 add one entry and multiply that by 2, I Show more

get a. 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. Very easy when I think about it for numbers, but suppose I I also want to use that same idea to think about I could add two vectors, a 1 and a 2, and then scale them by some, say after X, and get another vector. Or I might say I want to think about a 1 and a 2 as being polynomials and I might want to add those two polynomials and then multiply them by 2 to get a more complicated one, or a 1 and H. You might be electrical signals, and I might want to think of that, summing those two electrical signals and then amp, putting the whole thing to an amplifier, multiplying it by by some factor of 2 or something. The idea is I want to think about the general notion of that. Now, if our language is going to be a good language for expressing those kind of general ideas, I really, really can do Show more

that. So I'd like to be able to say I'm going to multiply by X the sum of a 1 and a 2 and I'd like that to express the general idea of all different kinds of things that a 1 and a 2 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 finally adding two polynomials, or if I'm adding the representation of two electrical signals or waveforms. Somewhere there has to be the knowledge of the kinds of various things that you can add and the ways of adding them, 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 here? But well, that's going to be the 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 like that's going to be our second large topic: conventional interfaces. What we're going to see is when. First we're going to talk about the problem of generic operations, which is the one I alluded to, things like plus, that that have to work with all different kinds of data. We talked 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 Show more

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. 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'll suppress some kinds of details and emphasize other kinds of detail. This is going to be the sort of the most magical part of the course. We're going to start out by actually looking at the technology for building new computer languages and will be the first thing we're going to do is actually build in Lisp a problem we're going to express in Show more

list the process of interpreting list itself, and that's going to be a very sort of self circular thing. There's a little mystical symbol that have to do with that will see itself. The process of interpreting lists is sort of a, a giant wheel of two processes: apply an eval which sort of constantly reduce expressions to each other. Then we're gonna see all sorts of other magical things. Here's another, another magical system symbol. This is kind of the sort of the why operator, which is in some sense the expression of infinity inside our Sidra language. We'll take a look at that. In any case, this section of the course is called metalinguistic abstraction- talking abstract by talking about how you construct new languages. As I said, we're going to start out by looking at the process of Show more

interpretation. We're going to look at this, apply eval loop and build lists. 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, 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 Show more

simplest kind of machines, even though we'll see something like this, which is: this is a picture of the of the chip, which is the Lisp interpreter that we will be talking about then in hardware, okay. Well, there's, there's an outline of the course. Three big topics: black out box abstraction, conventional interfaces, middle linguistic abstraction. Now, let's take a break now, and then we'll get started. [Music], [Applause]. Show more

Well, let's, let's actually start in learning list now. Actually, we'll start out by learning something much more important, maybe the very most important thing in this course, which is not list in particular, of course, but rather a Show more

general framework for thinking about languages they already alluded to. When somebody tells you they're going to show you a language, what you should say is all right. What I'd like you to tell me is: what are the? What are the primitive elements? What is 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? That it's irrelevant, right. What you say is how, if the language did not come with matrices built-in or with something else, then 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 needs of the abstraction which allow me then to use? Those are the elements in making more complicated things. Yet. Well, we're going to see that list has some primitive data and some primitive procedures. In fact, let's, let's really start, and here's a piece of primitive data and Lisp let's say: right, number three. Actually, from being very pedantic, that's not the number three, that's some symbol that represents no Plato's concept of the number three, right. And here's another. Here's some more primitive data in Lisp, right, it's seventeen, point four, actually some representation of seventeen point four, and here's another one, five. Here's another primitive object that's built in Lisp addition, actually, 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, how you, add things. But sure, those are some primitive, the elements. I can put them together. I can say: gee, watch the sum of three and seventeen, point four and five. The way I do that is to say, let's apply the Sun operator to these three numbers and I should get what: eight, seventeen, twenty five, point four. I should be able to ask Lisp what the value of this is and we'll return: twenty five, point four. Let's introduce some names. This thing that I typed, it's called a combination, and a combination consistent, 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 3 and the product of five and six and eight and two, and I should get, let's say, 30, 40, 43. So this should tell me that that's 43 forming combinations. Well, is that basic means the combination that we'll be looking at? And then, well, you see some, some syntax here. Lisp uses what's called prefix notation, which means that the operator, that the operator, is written to the left of the operands. It's just a convention and notice, it's fully parenthesized and the parenthesis make it completely unambiguous. So by looking at this I can see that there's the operator and there 1, 2, 3, 4 operands, right, and I can see that the second operand here is itself some combination that has one Operator and two operands. Parentheses in list they're a little bit or are very unlike parentheses and conventional mathematics. 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. That, because that maybe makes the grouping more distinct. List is not like that: enlist. You cannot leave out parentheses and you Show more

cannot put in extra parentheses, right, because putting in parentheses always mean exactly and precisely. This is a combination which has meaning apply an operator to operands. And if I lift this out, let 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 3, and then there's something else and an 8 and a 2, and then there's something else. Here is itself little subtree that has a star and a five and a six, and the way to think of Show more

that is really what's going on. Or 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 one whisper started and people had teletypes or punch cards or whatever. This was more convenient. They couldn't. Maybe if lists started today we would. The syntax of Lisp would look like that. Well, let's, let's look at what that actually looks like on the computer. Right here I have a lisp interaction setup. There's an 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 3, and I asked to evaluate it. And there you see, list is returned on the bottom and said: oh yeah, that's 3. Or I can say what's the sum of 3 & 4 & 8, what's that combination? And ask list to evaluate it, and that's 15. Well, 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 nineteen and a half, and you'll notice here that list has something built in that helped me keep track of all these parentheses. Watch as I type the next closed parenthesis which is going to close the combination, starting with the star, the opening. One will flash there. I'll rub Show more

this out and do it again, type closed and you see that closed as the plus closed. Again. That quizzes the star. Right now I'm back to the sum and maybe I'm going to add that all to, for that closes the plus. Now I have a complete combination and I can ask this for the value of that? That kind of paren balancing is something that's 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, 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 3 & 5 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: what's the sum of that and the product of 47? And let's say the product of 47 with the difference of 20 and 6.8. That means subtract 6.8 from 20. You see the parentheses close, close the -, close the store. And now let's get another operator. You see the list editor here. It's indenting to the right position automatically. Help me keep track. Hey, do that again. I'll close that last parenthesis again. Is he? It balances the plus Right now I can say: what's the value of that? All right, so, uh, those two things- indenting to the right level, which is called pretty printing, and flashing parentheses- are two things that a lot of systems have have built in to help you keep track, and you should learn how to use them, okay, well, those are the primitives. There's a means of combination. Now let's go up to the means of abstraction. Right, 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 could say, for example: define a to be the product of five and five. And now I could say so, say, for example, two lists: what is the product of a and a? This should be 25 and there should be six, twenty five. And then, crucial thing, I can now use a here. I've used it in the combination, but I could use that in other more complicated things that I need in turn. So I could say: define B to be the sum of, will say a and the product of five and a. You close Show more

the plus. Let's take a look at that on the computer, see how that looks. It's all. I'll just type what I wrote on the board. I can say: define a to be the product of five and five. I'll tell that to lift. And notice what Lisp responded there with: with an A in the bottom. In general, when you type in a definition that lifts, it responds with the knee, the symbol being defined, and I could say: the list, what is the product of a and a? It says that's 625. I can define B to be the sum of a and the product of five and a close. A friend closes the store close, the press close. The define- this says, okay, be there on the bottom and now I can tell you, list, 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 divided, another Show more

primitive operator. I've divided you by far: the entity to a. This is okay, that's 55, all right, so there's what it looks like. There's the basic means of defining some something. It's the simplest kind of naming, but it's not really very powerful. See what I'd really like to name. We're talking about general methods. I'd like to name, or the general idea that, for example, I could multiply five by five or six by six. Oh, thousand and one by a thousand and one a thousand, 1.7 by a thousand and 1.7, right, I'd like to be able to name the general idea of multiplying something by itself. We know what that is. That's called squaring. The way I can do that- and Lisp- is, I can say: you're fine, just square something X, multiply X by itself, and then, having having done that, I could say so, list, for example, what's the square of 10, and this will say a hundred. Yeah, Show more

let's actually look at that a little more closely. Yeah, right, there's the definition of square. To square something, right, multiply it by itself. All right, you see this X here. Right, that X is kind of a pronoun, which is the something that I'm going to square, and then what I do with it is: I multiply X by, multiply it by itself. Okay, right, so there's a notation for a for defining a procedure. Actually, this is a little bit confusing because sort of how I might use square and I say square of X or square 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 a little bit more clear that I'm naming something. I'll say: define Square to be lambda of X times X, X. Here I'm meaning something square, but just like over, here I'm naming something Show more

a. The thing that I'm naming square here I named 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 lisps way of saying: make up 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 Show more

what it does is return the result of multiplying X by itself. Now, in general, we're going to be using- we're going to be using this top form of defining this because it's a little bit more convenient, but don't lose sight of the Show more

fact that it's really this: in fact, as far as the Lisp interpreter is concerned, there's no difference between typing this to it and typing this to it, and there's a word for that. We're 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 weak 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. Ok, well, let's, let's look at that on the computer too. So I'll come in. I'll say: define square of X, give you times X, X, and I can you know, I'll tell this fat to square. You have named something square. And now, having done that, I can ask list for what's the square of a thousand and one? Or, in general, I could say what's the square of the sum of five and seven? All right, square twelves, 144. Or I can use square itself as an element in some combination. I can say what's the sum of the square of 3 and the square of 4? Hey, 1916 is 25. Or I can use square as an element in some much more complicated thing. I can say what's the? What's the square of the square, of the square of Show more

1,001? There's the square of the square of the square of 1,001. Or I can say: the list, what is square itself, what's the value of that? And Lisp returns some conventional way of telling me that that's a procedure, does compound procedure, Square and what the value of square is. This procedure and the thing with the Stars and the brackets are just lisps conventional way of describing that. Let's look at two more examples of defining this right here there are two more procedures. I can define the average of x and y to be the sum of x and y / - we're having had average and mean square. Having had average in 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 2 + 3? And I should get the average of 4 + 9, which is 6 and 1/2. The key thing here is that having defined square, I can use it as if it were primitive, right? So if we look here on the slide I look at mean square, write, 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 key thing in Lisp: that you not make Show more

arbitrary distinctions between things that happen to be primitive in the language and things that happen to be built in person. You think 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 can drive that home by looking on the computer one more time. We Show more

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 typed that 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 Show more

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. Okay, 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, 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 the X for X less than 0, it's 0 for X equal to 0 and it's X for X greater than 0. And let's pass a way of Show more

making case analyses. Let me define for you absolute value. They define: the absolute value of X is conditional, right? This means case analysis, cond. Okay, if X is less than 0, the answer is negate X, right? What I've written here is a clause. This is a. This whole thing is a conditional clause and it has two parts. Show more

This part here is a is a predicate or a condition. That's a condition, and the condition is expressed by something called a predicate, and a predicate and Lisp is some sort of thing that returns either true or false. And you see, Lisp has a primitive procedure, less then, 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, well, the minus sign, and Lisp is a little bit funny- if there are more than if there are two or more arguments. If there's two arguments and subtract the second one from the first we saw that if there's one argument, it negates it. All. Right, so this corresponds to that. And then there's another Cronk laws. It says: in the case where X is equal to 0, the answer is 0, and the case where X is greater than 0, the answer is X. Close that Clause, close the cond, close the definition. There's a definition of absolute value and you see, Show more

it's a case analysis that looks very much like the case analysis you use in mathematics. Ok, 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 the absolute value, which looks almost the same, which says: if X is less than 0, 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 Condor, or whichever way you like it. You can think of con, distinct tactic sugar for if, or you can think of if as syntactic sugar for con, and it doesn't make any difference. Person implementing a list system will pick one and implement the other in terms of that, and it doesn't matter which one you pick. Okay, why don't we break now and then take some questions? How come sometimes when I write to, fine, 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? Okay, 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 the symbol square to be there. So way you should think about it is what to find us, is you right to find? 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 the 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]. [Applause] All right, 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, essentially any other language. You probably think that's a that's not believable, right? Because you know that these languages have things like four statements, and and do until a while or something. But uh, we don't really need any of that fact, we're not gonna use any of that in this course. Let me. Let me show you why. Again, looking back at that 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. And already alluded to the idea. The idea is that if the initial guess that she took, if that initial guess was actually equal to the square root of x, then G here would be equal to x over 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 over G will be smaller than the square root of x, so that when you average G and x over G, you get something in between. All right. So if you pick a G that's 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 over G will be smaller than the square root of Show more

x. So averaging always gives you something in between. And then it's not quite trivial but it's. 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 over 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 always converge to the square root of the axe. And so that's this method of success of the averaging due to Heron of Alexandria. Let's- uh, let's write it and 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: fine, to try a guess for the Show more

square root of x. What do we do? We'll say: if the guess is good enough, 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 the 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 it guess good enough and how do we improve a guess? So let's look at that, the algorithm to improve the guess. Right, to improve a guess for the square root of x, we average. That was the algorithm. We average to guess with the Show more

quotient of dividing X by the guess. All right, that's how we improve a guess. And to tell whether he guess is good enough. Well, we have to decide something. Let's, 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, for example? So what? That 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. Right, okay, so there's a, 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. All right, have the whole thing. I have the notion of how to, how to compute a square root. That's some kind of module, right? That's some kind of black box. It's defined in terms of right. It's defined in terms of how to try a guess for the square root of x. Try is to find in terms of well, telling whether something is good enough and telling how to improve something. So, good enough, try us to find in terms of good enough and improve, and let's see what else. I still in Wolfe, you'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 operators. So try- square roots defined in terms of try, tries to find in terms of good enough and improve, Show more

but also try itself. So try is also defined in terms of how to try itself. Well, that may give you, give you some problem right here. Your high school geometry teacher probably told you that it's it's naughty to try and define things in terms of themselves because it doesn't make sense. But that's false, all right. Sometimes it makes perfect sense to defy the things in terms of themselves, and this is a case and we can look at that, we can write down what this means. You'd say, suppose I ask 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 this square root of Show more