Donald Knuth: Algorithms, Complexity, and The Art of Computer Programming | Lex Fridman Podcast #62

The following is a conversation with Donald Knuth, one of the greatest and most impactful computer scientists and mathematicians ever. He's the recipient of the 1974 Turing Award, considered the Nobel Prize of computing. He's the author of the multi-volume work, the magnum opus, *The Art of Computer Programming*. He made several key contributions to the rigorous analysis of computational complexity of algorithms, including the popularization of asymptotic notation that we all affectionately know as the Big O notation. He also created the TeX typesetting system, which most computer scientists, physicists, mathematicians, and scientists and engineers in general use to write technical papers and make them look beautiful. I can imagine no better guest to in 2019 with than Don, one of the kindest, most brilliant people in our field.

This podcast was recorded many months ago. It's one I avoided because, perhaps counter-intuitively, the conversation meant so much to me. If you can believe it, I knew even less about recording back then, so the camera angle is a bit off. I hope that's okay with you. The office space was a bit cramped for filming, but it was a magical space where Don does most of his work. It meant a lot to me that he would welcome me into his home. It was quite a journey to get there, as many people know, he doesn't check email, so I had to get creative. The effort was worth it.

I've been doing this podcast on the side for just over a year. Sometimes I had to sacrifice a bit of sleep, but always happy to do it and to be part of an amazing community of curious minds. Thank you for your kind words, support for the interesting discussions, and I look forward to many more of those in 2020. This is the Artificial Intelligence Podcast. If you enjoy it, subscribe on YouTube, give it five stars on Apple Podcasts, follow on Spotify, support on Patreon, or simply connect with me on Twitter at @lexfridman (spelled F-R-I-D-M-A-N).

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And now, here's my conversation with Donald Knuth.

In 1957, at Case Tech, you were once allowed to spend several evenings with an IBM 650 computer, as you've talked about in the past. Then you fell in love with computing. Then, can you take me back to that moment with the IBM 650? What was it that grabbed you about that computer?

So, the IBM 650 was this, this machine that, well, it didn't fill a room, but it, it was, it was big and noisy. But when I first saw it, it was through a window, and there were just a lot of lights flashing on it. And I was a freshman. I had a job with the statistics group, and I was supposed to punch cards and pour data and then sort them on another machine. But then they got this new computer. It came in, and I, and it had interesting, like, you know, lights. Okay, so, well, but I had the key to the building, so I could, you know, like, I could get in and look at it and got a manual for it. And, and my first experience was based on the fact that I could punch cards. Basically, would you, a big thing for though, deal with thick? But the 650 was, you know, big in size, but, but incredibly small in power, in memory. It had, it had 2,000 words of memory, and a word of memory was 10 decimal digits plus a sign. And it, it would do to add two numbers together, you could probably expect that it would take, oh, say, three milliseconds. So that's pretty fast. It's the memories that constraint. The memories, the problem. That was why it was three milliseconds? Because it took five milliseconds for the drum to go around, and you had to wait, I don't know, five cycle times. If you have an instruction, one position on the drum, then it would be ready to read the data for the instruction, and three notches. The drum is 50 cycles around, and you go three cycles, and you can get the data, and then you can go another three cycles and get, and get the next instruction if the instruction is there. Otherwise, otherwise, you spin until you get to there. Play. And, and we had no random-access memory whatsoever until my senior year. You see, here we got 50 words of random access memory, which were, which were priceless. And we would, and we would move stuff up to the up to the random access memory in 60-word chunks, and then we would start again. So it's separating when to go up there.

And could you have predicted the future, 60 years later, of computing from then?

You know, in fact, the hardest question I was ever asked was, what could I have predicted? In other words, the interviewer asked me, she said, you know, what about computing has surprised you? You know, and immediately I ran, I rattled off a couple dozen things. And she said, okay, so what didn't surprise you? And I was, I tried for five minutes to think of something that I thought I would have predicted, and I, and I, and I couldn't. But I, let me say that this machine, I didn't know. Well, it, there wasn't, there wasn't much else in the world at that time. The 650 was the first machine that was, that there were more than a thousand of. Ever before that, there were, you know, there was each machine, there might be a half a dozen examples, maybe. My first mass-market, mass-produced, the first one, yeah, done in quantity. And, and IBM, I didn't sell them, they, they rented them. But, but they, they rented them to universities at great, you know, I had a great deal. And, and so that's why a lot of students learned about computers at that time.

So you refer to people, including yourself, who gravitate toward a kind of computational thinking as geeks. For at least, I've heard you use that terminology. Is it true that you think there's something that happened to me as I was growing up that made my brain structure in a certain way that resonates with computers? So there's the space of people, it's 2% of the population, you empirically estimate that's a prick. It's been proven fairly constant over most of my career. However, it might be different now because kids have different experiences when they're young. So what does the world look like to a geek? What is, what is this aspect of thinking that is unique to their makes it, yeah, that makes a geek?

This is the important question. In the 50s, IBM noticed that there were geeks and non-geeks, and so they tried to hire geeks. And they put out advertisements saying, you know, if you play chess, come to Madison Avenue for an interview, or something like this. They were, they were trying for some things. So what, what is it that I find easy, and other people tend to find harder? And, and I think there's two main things. One is this, this ability to jump, jump levels of abstraction. So you see something in the large, and you see something in the small, and, and can you pass between those unconsciously? So, you know, that in order to solve some big problem, what you need to do is add one to A into a certain register, or anything that gets you to another step. And you can, and below the, yeah, I mean, I don't go down to the electron level, but I knew what those milliseconds were, what the drum was like on the 650. I knew how I was gonna factor a number or or find a root of an equation or something. Because of what was doing. And, and as I'm debugging, I'm going through, you know, did I make a key punch error? Did I, did I write the wrong instruction? Do I have the wrong, wrong thing in a register? And each level, at each level, it is different. And so this idea of being able to see something at all at lots of levels and fluently go between them, it seems to me to be more pronounced, much more pronounced in in the people that with computers. Like I got. So in my books, I also don't stick after the high level, but, but I, but I mix low-level stuff with high-level. And this means that some people think, you know, that I, that I should write better books, and it's probably true. But, but other people say, well, but that's, if you think like, like that, then that's the way to train yourself, like to keep mixing the levels and, and learn more and more how to jump between. So that, that's the one thing. The other, the other thing is that it's more of a talent to be able to deal with non-uniformity, where there's case one, case two, case three, instead of instead of having one or two rules that govern everything. So if, so it doesn't bother me if I need, like, an algorithm has ten steps to it, you know, each step is does something else. That doesn't bother me. But a lot of, a lot of pure mathematics is based on one or two rules which, which are universal. And, and, and so this means that people like me sometimes work with systems that are more complicated than necessary because it doesn't bother us that we don't, that we didn't figure out the simple rule.

And you mentioned that while Jacobi, Boole, Abel, and all the mathematicians in the 19th century may have had symptoms of geek, the first 100% legit geek was Turing. Alan Turing. You, I, I think he had, yeah, a lot more of this quality than anyone could from reading the kind of stuff he didn't so hot. As Turing, what influence has Turing had on you?

Well, well, your way. And so I didn't know that aspect of him until after I graduated some years. I, at undergraduate, we had a class that talked about computability theory and Turing machines, and, and that was all. It sounded like a very specific kind of purely theoretical approach to stuff. So when, how old was I when I, when I learned that he thought he had, you know, designed machines, and that he wrote the, you know, you wrote a wonderful manual for for Manchester machines, and, and he invented all the subroutines, and, and he was a real hacker, that he had his hands dirty. I thought for many years that he had only done purely formal work. As I started reading his own publications, I could, yeah, you know, I could feel this kinship. And, and of course, he had a lot of peculiarities, like he wrote numbers backwards. Because I mean, left to right to the right to left, because that's the, that's it was easier for computers to process them that way. What do you mean, left to right? He would write pi as, you know, 9.5143. I mean, okay, right. Forget it. 3.14159. On the blackboard. I mean, when he, he, we had trained himself to, to do that because the computers he was working with worked that way inside. Trained himself to think like a computer. Well, there you go. That's nuts. Geek thinking.

You've practiced some of the most elegant formalism in computer science, and yet you're the creator of a concept like literate programming, which seems to move closer to natural language type of description of programming. Yep. Yeah, absolutely. So how do you see those two as conflicting, as the formalism of theory and the idea of literate programming?

So there we are in a non-uniform system. Well, I don't think one-size-fits-all, and I don't, and I don't think all truth lies in one, in one kind of expertise. And so somehow, in a way, you'd say my, what my life is a convex combination of English and mathematics. And you're okay with that? And not only that, I think thriving. I wish, you know, I want my kids to be that way. I want, etc. Not left-brain, right-brain at the same time. You got a lot more done. That was part of the.

And I've heard that you didn't really read for pleasure until into your 30s. Literature. True?

You know more about me than I do. But I'll try to be consistent with what you're really. Yeah, just believe me. Yeah, just go with whatever story I tell you, it'll be easier that way.

The conversation I've heard mentioned a Philip Roth's *American Pastoral*, which I love as a book. I don't know if it was, it was mentioned as something that I think was meaningful to you as well. In either case, what literary books had a lasting impact on you?

What, okay, good. So I, so I met Roth already. Well, we both got doctors from Harvard on the same day. So I, so we were, yeah, we had lunch together and stuff like that. But he knew that, you know, computer books would never sell well. Well, all right. So you say you, you, you, you're a teenager when you left Russia. So I, I have to say that Tolstoy was one of the big influences on me. I especially like *Anna Karenina*, not because of a particular area of the plot of the story, where, but because there's this character who, you know, did the philosophical discussions. It's all, it's a whole way of life is worked out there. It's among the characters. Until in, and so it, that I thought was was especially beautiful. On the other hand, Dostoevsky, I, I didn't like at all because I, I felt that he, his genius was mostly because he kept forgetting what he, what he had started out to do, and he was just sloppy. I didn't think that that he polished his stuff at all. And, and I tend to admire somebody who, who dots the i's and crosses the t's.

So the music of the prose, this way, you admire more?

And I certainly do admire the music of the language, which I couldn't appreciate in the Russian original. But, but I can. And Victor Hugo. Glenn's close friendships, much his closer. But, but Tolstoy, I like the same reason I like Herman Wouk as a novelist. That I think I like his book *Marjorie Morningstar* has a similar character in who, who developed his own personal philosophy and export, and it called goes in, and was consistent. Yeah, right. And it's worth, worth pondering.

So, like Nietzsche? And like what you don't like? Friedrich Nietzsche? Or age? Yeah. No, no. You like this. Like, I keep seeing quotations for Nietzsche, and you never tempt me to read any further. Please. Full of contradictions. We will certainly not appreciate him. But Schiller, you know, I'm trying to get the cross. What I appreciate in literature, and part of it is the, is, is as you say, the music of the language, of the way it flows. Take Raymond Chandler versus Dashiell Hammett. Dashiell Hammett's sentences are awful, and Raymond Chandler's are beautiful. They just flow. So I, I don't, I don't read literature because it's supposed to be good for me or because somebody said it's great. But, but I could find things that I like. I mean, you mentioned you address like James Bond. So like, I love Ian Fleming. I think he's got a, he had a really great gift for if he has a golf game or game of bridge or something, and this comes into a story, it'll, it'll be the most exciting golf game or, you know, the absolute best possible hands at bridge that that exists. And, and he exploits it and tells it beautifully as well.

So in connecting some things here, looking at literate programming and being able to convey code algorithms to a computer in a way that mimics how humans speak, how what do you think about natural language in general and the messiness of our human world about trying to express, yeah, difficult things?

So the idea of literate programming is to, is really to try to understand something better by seeing it from these two perspectives, the formal and the informal. If we try to understand a complicated thing, if we can look at it in different ways. And so this is, in fact, the key to technical writing. A good technical writer tries not to be obvious about it, but says everything twice, formally and informally, or maybe three times. But you try to give the reader a way to put the concept into his own brain or her own brain.

Is that better for the writer or the reader, or both?

Well, the writer just tries to understand the reader. That's the goal of a writer is to have a good mental image of the reader and to say what the reader expects next, and to impress the reader with what has impressed the writer. Why something is interesting. So when you have a computer program, we try to, instead of looking at it as something that we're just trying to give an instruction to the computer, what we really want to be is giving giving insight to the person who's who's gonna be maintaining this program, or to the programmer himself when he's debugging it, as to why this stuff is being done. And so all the techniques of exposition that a teacher uses or book writers make you a better programmer, if your, if your program is going to be not just a one-shot deal.

So how difficult is that? Do you see hope for the combination of informal and formal for the programming task?

Yeah, I, I'm the wrong person to ask, I guess, because I'm a geek. But I think for a geek, it's easy. I don't know. I don't know. See, not some people have difficulty writing, and that might be because there's something in their brain structure that makes it hard for them to write, or, or it might be something just that they haven't had enough practice. I'm not the right one to, to, uh, to judge. But I don't think you teach any person any particular skill. Like, I do think that that writing is, is half of my life, and so I put it together. And let program. He won't even when I'm writing a one-shot program, I, I write it in a literate way because I get it right faster. Though now, does it get compiled automatically?

So I guess on the technical side, my question was, how difficult is it to design a system where much of the programming is done informally?

Informally? Yeah, informally. I think whatever works to make it understandable is good. But then you have to also understand how informal is. You have to know the limitations. You have to connect. So, by putting the formal and informal together, this, this is where this is where it gets locked into your, into your brain. Now, you can, you can say informally, well, I'm working on a problem right now. So let's go there.

I get that. Can you give me an example of connecting the informal and the formal?

Well, it's a little too complicated an example. There's a puzzle that's self-referential. It's called a Japanese arrow puzzle. And, and you're given a bunch of boxes, each one points north, east, south, or west. And at the end, you're supposed to fill in each box with the number of distinct numbers that it points to. So if I put a three in a box, that means that, and it's pointing to five other boxes, that means that there's going to be three different numbers in those five boxes. And, and those boxes are pointing. What I might be pointing to me, one of my might be pointing the other way. But anyway, I kind of defined a set of numbers that obeys this complicated condition that each number counts how many distinct numbers it points to. Well, and still, a guy sent me his solution to this problem where he, where he presents formal statements that that say either this is true, or this is true, this is true. And, and, and so I try to render that formal statement informally. And I try to say, I contain a three, and, and the guys I'm pointing to contain the numbers one, two, and six. So by putting it informally, and also I converted into a dialogue statement that helps me understand the logical statement that he's written down as a string of numbers in terms of some abstract variables.

Eddie, yeah, that's really interesting. So maybe an extension of that, there has been a resurgence in computer science and machine learning and neural networks, so using data to construct algorithms. So it's another way to construct algorithms, really? Yes, you can think of it that way. So as opposed to natural language to construct algorithms, use data to construct others. So what, what's the view of this branch of computer science where data is almost more important than the mechanism of the algorithm? It seems to be suited to a certain kind of non-geek, and would you know, which is probably why it's, it's like it's taken off, that it has its own community that I thought really, that really resonates with that. But it's hard to, you know, to trust something like that because nobody, even the people who, who work with it, they have no idea what has been learned. That's a really interesting thought that it's, it makes algorithms more accessible to a different community, a different type of brain.

Yep. And that's really interesting because, just like literate programming, perhaps could make programming more accessible to a certain kind of brain. There are people who think it's just a matter of education, and anybody can learn to be a great programmer, or anybody can to be a great skier.

Uh, yeah, you know, I, I wish that were true. But, but I know that there's a lot of things that I've tried to do, and I, and like I was well-motivated, and I kept trying to build myself up, and I never got past a certain level. I can't, for example, I can't view three-dimensional objects in my, in my head. I have to, I have to make a model and look at it and study it from all points of view, and then I start to get some idea. But other people are good at four dimensions, I mean, physicists. Yeah.

So let's go to *The Art of Computer Programming*. In 1962, you set the table of contents for this magnum opus, right? Yeah. It was supposed to be a single book for 12 chapters. Now, today, what is it? 57 years later, you're in the middle of volume 4 of 7, and in the middle of going for B? Is 4B precisely? Can I ask you for an impossible task, which is, try to summarize the book so far, maybe by giving a little examples? From the sorting and the search in the combinatorial algorithms, if you were to give a summary, a quick elevator summary?

Yeah, right. What, depending how many floors that are in the building. Yes. The first volume, called Fundamental Algorithms, talks about something that you can't, the stuff you can't do without, I guess. That you have to know the basic concepts of what is a program, now, what is it, what is it algorithm, and, and, and it also talks about a low-level machine, so you can have some, some kind of an idea what's going on. And it has basic concepts of input/output and subroutines, induction, induction, writes mathematical. So, so the thing that makes my book different from a lot of others is that all that I try to not only present the algorithm, but I try to analyze them. And which means to quantitatively say, not only does it work, but it works this fast. Okay? And so I need math for them. And then there's the standard way to structure data inside and represent information in the computer. So that's all volume 1. Volume 2 talks, it's called Semi-Numerical Algorithms. And here we're, here we're writing programs, but we're also dealing with numbers. Algorithms deal with with any kinds of objects, but but specific when there's objects or numbers, well, then then we have certain special paradigms that apply to things that have numbers. And so there's, what, there's like, there's arithmetic on numbers, and, and there's matrices full of numbers, there's random numbers, and there's power series full of numbers. There's different algebraic concepts that have numbers in structured ways. And the arithmetic in the way a computer would think about arithmetic is a floating point, floating point arithmetic, high precision arithmetic, not only addition, subtraction, multiplication, but also comparison of numbers. So then, check. Then volume three talks about, I like that one. Sort, insert, sorting, a circle of sorting, right? So, so here, you know, we're not getting necessarily with numbers, because you slipped, you saw it, letters and other objects. And searching, we're doing all the time. We Google nowadays, but I mean, we have to find stuff. So again, algorithms that that underlie all kinds of applications. Like, you know, none of these volumes, it's about a particular application, but the applications are examples of of why people want to know about sorting, why people want to know about random numbers. So then volume 4 goes into combinatorial algorithms. Again, this is where we have zillions of things to deal with. And we, and here we keep finding cases where one good idea can make something go more than a million times faster. And, and, and we're dealing with problems that are probably never going to be solved efficiently, but that doesn't mean we give up on them. And, and, and we have this chance to have good ideas and, and go much, much faster on them. So, so that's combinatorial algorithms. And those are the ones that are, yeah.

I'm using charting. Most fun for you?

Well, how many torial algorithms are the ones that I always, that I always enjoyed the most, because that's when my skill at programming had most payoff. You know, the difference between an obvious algorithm that you think up first thing, and, you know, and a good, you know, an interesting, subtle algorithm that's not so obvious, but but runs circles around the other one. That's, that's where computer science 3D comes comes in. And, and a lot of these combinatorial methods were found first in applications to artificial intelligence or cryptography. And in my case, I, I just liked him. And it was associated more with puzzles that you like the most. In the domain of graphs and graph theory. Graphs are great because they're terrific models of so many things in the real world. And, and, and you, you throw numbers on a graph, you got a network. And so there you're right, there you have, but many more things. So, but combinatorial in general is any arrangement of objects that that has some kind of a higher structure, non-non-random structure. And it's okay. It is possible to put something together satisfying all these conditions. Like I mentioned arrows a minute ago, you know, is there a way to to put these numbers on a bunch of boxes that that are pointing to each other? Is that going to be possible at all? That's volume four. That's volume four. What is a sage of Hawaiian for A? Was part one? And what happened was in 1962, when I started writing down a table of contents, it wasn't going to be a book about computer programming in general. It was going to be a book about how to write compilers. And I was asked to write a book explaining how to how to write a compiler. And at that time, there were only a few dozen people in the world who had written compilers, and I happened to be one of them. So, and I also had some experience for writing for, like, the campus newspaper and things like that. So, so I said, okay, great. I'm the only person I know who who's written a compiler but hasn't invented any new techniques for writing compilers. And all the other people I knew had super ideas, but I couldn't see that they would be able to write a book that wouldn't that would describe anybody else's ideas with their own. So I could be the, I could be the journalist, and I could explain what all these cool ideas about compiler writing that were. And, and then I, I started pretty well. Yeah, let me, you need, and have a chapter about data structures. You need to have some introductory material. I want to talk about searching, because a compiler writer has to, has to look up the variables in a symbol table and find out, you know, which, when you, when you write the name of a variable in one place, it's supposed to be the same as the one you put somewhere else. So you need all these basic techniques. And I, and I, you know, kind of know some arithmetic stuff. So I threw, I threw in these chapters. And I threw in a chapter on combinatorics, because that was what I really enjoyed programming the most. But there weren't many algorithms and known about combinatorial methods in 1962. So that was a kind of a short chapter, but it was sort of thrown in just for fun. And chapter 12 was going to be actual compilers, applying all the stuff in chapters 1 to 11 to make compilers. Well, okay. So that was my table of contents from 1962. And during the 70s, the whole field of combinatorics went through a huge explosion. People talk about it, combinatorial explosion, and they usually mean by that that the number of cases goes up, you know, you change N to N+1, and all of a sudden you, your problem has gotten more than 10 times harder. But there was an explosion of ideas about combinatorics in the 70s, and to the point that, by, like, 1975, I bet you more than half of all the journals of computer science were about combinatorial methods.

And what kind of problems were occupying people's minds? What kind of problems in combinatorics was it? It's that gravity? Graph theory?

Yeah, gravity was was quite dominant. I mean, no, but all of the NP-hard problems that you have, like Hamiltonian path, or Traveling Salesperson, going beyond, yeah, yeah, going beyond graphs. You had operations research. Whenever it was a small class of problems that had efficient solutions, and they were associated with some special mathematical construction. But once we went to things that involve three things at a time, instead of instead of two, all of a sudden the things got harder. So we had satisfiability problems, or if you have, if you have clauses, every clause has two logical elements in it, then we can satisfy it, linear time, we can test for satisfiability in linear time. But if you allow yourself three variables in the clause, then nobody knows how to do it. So these articles were about trying to find better or better ways to to solve cryptography problems and graph theory problems where the we have lots of data, but we didn't know how to find the best subset. So the data, like with sorting, we could get the answer, didn't take long.

So how did they continue to change from the 70s to today?

Yeah, so now there may be half a dozen conferences whose topic is combinatorics, different kind. But fortunately, I don't have to rewrite my book every month, you know, like I had to in in the 70s. But still, there's huge amount of work being done, and people getting better ideas on these problems that don't seem to have really efficient solutions, but we can still get into a lot more with them. And so this book that I'm finishing now is, I've got a whole bunch of brand new methods that the authors, I know there's no other, there's no other book that covers that covers this particular approach. And, and so I'm trying to do my best of exploring the tip of the iceberg, and, and, and I try out lots of things and, and keep, keep rewriting, finding as I find better, better methods.

So what's your writing process like? What's your thinking and writing process like every day? So what's your routine, even?

Yeah, I guess it's actually the best question because I spent seven days a week doing it. The most prepares to answer it. Yeah, yeah. But okay, so the chair I'm sitting in is where I do that's where the magic happens. Well, reading and writing. That many chairs usually sitting over there where I have other books, some reference books. But, but I found this chair, which was designed by a Swedish guy. Anyway, it turns out this was the only chair I can really sit in for hours and hours and not know that I'm in a chair. But then I have the stand-up desk right next to us. And, and so after I write something with pencil and eraser, I get up and I type it and revise and rewrite. The kernel, the idea is first put on paper. Yep. That's worth writing. And I call, write maybe five programs a week, of course, literate programming. And these are before I describe something in my book, I always program it to see how it's working. And I, and I tried a lot. So for example, I learned at the end of January, I learned of a breakthrough by four Japanese people who had extended one of my methods in in a new direction. And so I, I spent the next five days writing a program to implement what they did. And then I, you know, but they had only generalized part of what I had done. So that I had to see if I could generalize more parts of it. And then I had to take their approach, and I had to, I had to try it out on a couple of dozen of the other problems I had already worked out with that with my old methods. And so that took another couple of weeks. And then I would, you know, then I, then I started to see the light nicely. And, and I started writing the final draft. And, and then I would, you know, type it up. Involves some new mathematical questions. And so I wrote to my friends and might be good at solving those problems. And they solved some of them. So I put that in as exercises. And, and so a month later, I had absorbed one new idea that I, that I learned. And, you know, I'm glad I heard about it in time, otherwise my, I wouldn't put my book out before I heard about the idea. On the other hand, this book was supposed to come in at 300 pages, and I'm up to 350 now. That added 10 pages to the book. But if I learn about another one, I probably first gonna shoot me.

Well, so in the process, in that one month process, are some days harder than others? Are some days harder than others?

Well, yeah, my work is fun, but I also work hard. And every big job has parts that are a lot more fun than others. And so many days I'll say, why do I have to have such high standards? Like, why couldn't I just be sloppy and not try this out, and, you know, just just report the answer? But I, but I know that people are counting on me to do this. And so, okay, so, okay, Donald, grit my teeth and do it. And, and, and then the joy comes out when I see that actually, you know, I'm getting good results. And, and I get, and I even more when I see that somebody has actually read and understood what I wrote and told me how to make it even better. I did want to mention something about the about the method. So I got this tablet here where I do the first, you know, the first writing of concepts. Okay. So, and what language I didn't write. So, hey, take a look at. But, you know, here, random say, explain how to draw such skewed pixel diagrams. Okay. So I got this paper about 40 years ago when I was visiting my sister in Canada, and they make tablets of paper with this nice large size and just the right very small space between. Like, oh, yeah, yeah, particularly also just, yeah, you know, I've got these manuscripts going back to the 60s. And, and, and those are when I get my ideas on paper. Okay. But I'm a good typist. In fact, I went to typing school when I was when I was in high school. And so I can type faster than I think. So then when I do the editing, you know, stand up and type, then I, then I revise this, and it comes out a lot different than what you look for. Style and rhythm and things like that come out at the at the typing stage. And you type in TeX. And I type in TeX.

And can you, can you think in TeX?

No. So to a certain extent, I have, I have only a small number of idioms that I use. Like, you know, a beginning or theorem, I do something for displayed equation, I do something, and so on. But, but I, I have to see it. And in the way that it's on here. Yeah, right. For example, Turing wrote what the other direction. You don't write macros. You don't think in macros, particularly. But when I need a macro, I'll go ahead and, and do it. But, but the thing is, they, I also write to fit. I mean, I'll, I'll change something if I can, if I can save a line. I've got, you know, it's like haiku. I'll figure out a way to rewrite the sentence so that it'll look better on the page. And I shouldn't be wasting my time on that. But, but I can't resist because I know it's only another three percent of the time or something like that.

And it could also be argued that that is what life is about.

Ah, yes. In fact, that's true. Like, like I worked in the garden one day a week, and that's that's kind of a description of my life is getting rid of weeds, you know, removing bugs for programs. And so, you know, a lot of writers talk about, you know, basically suffering the writing process. Yeah, having, you know, it's extremely difficult. And I think of programming, especially the or technical writing that you're doing, can be like that. Do you find yourself methodologically, how do you, every day sit down to do the work? Is it a challenge? You kind of say it's, you know, oh yeah, it's fun, but it'd be interesting to hear if there are non-fun parts that you really struggle with.

Yes. The fun comes with when I'm able to put together ideas of two people who didn't know about each other. And, and, and so I might be the first person that saw both of their ideas. And so then, you know, then I get to make the synthesis. And that gives me a chance to be creative. But the dredge work is where I act. I've got to chase everything down to its root. This leads me into really interesting stuff. I mean, like I learned about Sanskrit. Nice. Yeah. And again, you know, I try to give credit to all the authors. And so I write, like, so I write to people who know that the people thought as if they're dead. I communicate this way. And I gotta get the math right. And I got to tag all my programs, try to find holes in them. And I rewrite the programs over after I get a better idea.

Is there ever dead ends? Data?

So yeah, I throw stuff out. Yeah. Look, one of the things that I spent a lot of time preparing a major example based on the game of baseball. And I know a lot of people who, for whom baseball is the most important thing in the world. You know, yes. But it's, but I also know a lot of people from cricket is the most important in the world, or soccer, or something, you know. And, and I realized that if, if I had a big sample, I mean, it was gonna have a fold-out illustration and everything. I was saying, well, what, what am I really teaching about algorithms here? Where I had this, this baseball example. And if I was a person who knew only cricket, what would they think about this? And, and so I ripped the whole thing out. But I, you know, I had, I had something that would really appeal to people who grew up with baseball as as a major theme in their life, which is a lot of people. But, yeah, so I said on minority, the small minority. I took out bowling to even a smaller minority.

What is the art in *The Art of Programming*? Why, why is there of the few words in the title? Why is art one of them?

Well, that's, that's what I wrote my Turing lecture about. And so when people talk about art, it really, I mean, what the word means is something that's not nature. So when you have artificial intelligence, that that art comes from the same root, saying that this is something that was created by, by human beings. And then it's gotten a further meaning, often fine art, which has this beauty to the to the mix. And says, you know, we have things that are artistically done. And, and this means not only done by humans, but also done in a way that's elegant and brings joy and has, has I guess what Tolstoy calls dusky. But anyway, it's that part that that says that it's done well, as well as not only different from nature in general, then art is what human beings are specifically good at. And when they say, hey, like artificial intelligence, well, they're trying to mimic human beings. But there's an element of fine art and beauty.

You are. Well, that's what I, that's what I try to also say, that you can write a program and make a work of art.

So now, in terms of surprising, you know, what ideas in writing from sort and search to the combinatorial algorithms, what ideas have you come across that were particularly surprising to you, that that changed the way you see a space of?

I get a surprise every time I have a bug in my program. But that isn't really what your transformational surprises. For example, in volume 4A, I was especially surprised when I learned about a data structure called BDD, Boolean Decision Diagram. Because I sort of had the feeling that as an old-timer, and, you know, I've been programming since this since the 50s, and BDDs weren't invented until 1986. And here comes a brand new idea that revolutionized the way to represent a Boolean function. And Boolean functions are so basic to all kinds of things. In it, I mean, logically underlies it everything we can describe all of what we know in terms of logic, somehow. And, and propositional logic, I thought that was cutting edge. Everything was known. But, but he, but here comes Randy Bryant and, oh, and discovers that BDDs are incredibly powerful. Then, then that's all. So I, that means I have a whole new section to the book that I never would have thought of until 1986. Not until the 1990s when I went, when people started to got to use it for, you know, billion-dollar applications, and it was, it was the standard way to design computers for a long time until until SAT solvers came along in the year 2000. So that's another great big surprise. So, uh, a lot of these things have have totally changed the structure of my book. And the middle third of volume 4B is about that solvers. And that's 300 plus pages, which is all about material, mostly about material that was discovered in this century. And I had to start from scratch and meet all the people in the field and write. I have 15 different SAT solvers that I wrote while preparing that. Seven of them are described in the book. Others were for my own experience. So newly invented data structures or ways to represent a whole new class of algorithm. Calling you classified? Yeah. And the interesting thing about the BDDs was that the theoretician started looking at it and started to.

describe all the things you couldn't do with BD DS. And so they were getting a bad, they were getting a bad name because, you know, okay, they were, they were useful, but they didn't solve everything. I'm sure that the theoreticians, in the next 10 years, are gonna show why machine learning doesn't solve everything. But I not only worried about the worst case, I get a huge delight when I can actually solve a problem that I couldn't solve before. Yeah, even though I can't solve the problem that's that it suggests as a further problem, like I know that I'm way better than I was before. And so I found out that BD DS could do all kinds of miraculous things, and so I had been quite a few years learning about that territory. So in general, what brings you more pleasure: in proving or showing a worst-case analysis of an algorithm, or showing a good average case, or just showing a good case that, you know, something good pragmatically can be done with this algorithm?

Yeah, I like a good case that that is maybe only a million times faster than I was able to do before, but and not worried about the fact that, and that is still, that is still gonna take too long if I double the size of the problem. So that said, you popularized the asymptotic notation for describing running time. Obviously, in the analysis of algorithms, worst cases are such an important part. Do you see any aspects of that kind of analysis is lacking? So, in notation, well, the main purpose you have notations that that help us for the problems we want to solve, and so that they match our, they match our intuitions. And people who worked in number theory had used asymptotic notation in what, in a certain way, but it was only known to a small group of people. And and I realized that, in fact, it was very useful to be able to have a notation for something that we don't know exactly what it is, but we only know partial about it. And so on.

Stick. So, for example, instead of Big O notation, let's just, let's just take a much simpler notation where I say 0 or 1, or 0 1 or 2. And suppose that, suppose that when I had been in high school, we would be allowed to put in the middle of our formula, X plus 0 1 or 2 equals Y. Okay? And then, then we would learn how to multiply two such expressions together and, and you know, deal with them. Well, the same thing, Big O notation says, here's something that's, I'm not sure what it is, but I know it's not too big. I know it's not bigger than some constant times N squared or something like that. Fine. So I write Big O of N squared. And now I learned how to add Big O of N squared to Big O of N cubed, and I know how to add Big O of N squared to 2 plus 1 and square that, and how to take logarithms and exponentials to have Big O's in the middle of them. And that turned out to be hugely valuable in all of the work that I was trying to do.

Is I'm trying to figure out how good? So, have there been algorithms in your journey that perform very differently in practice than they do in theory? Well, the worst case of a comet, our logarithm is almost always horrible. But but we have SAT solvers that are solving, where one of the, one of the last exercises in that part of my book was to figure out a problem that has a hundred variables that's that's difficult for a SAT solver. But uh, but you would think that a problem with a hundred boolean variables has required to do 2 to the 100th operations, because that's the number of possibilities. When you have 200 boolean variables, and 2 to the 100th, to the 100th is way bigger than then we can handle. 10 to the 17th is a lot.

You've mentioned over the past few years that you believe P may be equal to NP, but that it's not really, you know, somebody does prove that P equals NP, it will not directly lead to an actual algorithm to solve difficult problems. Can you explain your intuition here? Has it been changed? And in general, on the difference between easy and difficult problems of P and NP, and so on? Yes. So the popular idea is, if an algorithm exists, then somebody will find it, and it's just a matter of writing it down. One point. Well, but many more algorithms exist than anybody can understand or ever make you discover. Yeah, because they're just way beyond human comprehension. The total number of algorithms is more than mind-boggling. So, so we have situations now where we know that an algorithm exists, but we don't know, we don't the foggiest idea what the algorithms are. There's there are simple examples based on on game playing where you have where you say, well, there must be an algorithm that exists to win in the game of hex, because for the first player to win in the game of hex, because hex is always either a win for the first player or the second player. Well, what's the game of hex? There's a game of hex, which is which is based on putting pebbles onto a hexagonal board, and and the white player tries to get a white path from left to right, and the black player tries to get a black path from bottom to top. And how does capture occur? Just so. And and and there's no capture. You just put pebbles down, one at a time. But there's no draws, because they, after all the white and black are played, there's either going to be a white path across from east to west, or a black path from from bottom to top. So there's always, you know, it's a perfect information game, and people, people play take turns, like like tic-tac-toe. And hex, or it can be different sizes. But we, there's no possibility of a draw, and player to move one at a time. And so it's got to be either a first player win or a second player win. Mathematically, you follow out all the trees, and either, either there's always the win for the first player or the second player. Okay? And it's finite. The game is finite, so there's an algorithm that will decide. You can show it has to be one or the other, because the second player could mimic the first player with kind of a pairing strategy. And so you can show that it has to be, what it has to be one or that. But we don't know any algorithm. No way.

There's a case where you can prove the existence of the solution, but we, but nobody knows anyway how to find it. But more like the algorithm question, there's a very powerful theorem in graph theory by Robertson and Seymour that says that every class of graphs that is closed under taking minors has a polynomial time algorithm to determine whether it's in this class or not. Now, a class of graphs, for example, planar graphs. These are graphs that you can draw in a plane without crossing lines. And and a planar graph is closed under taking minors, means that you can shrink an edge into a point, or you can delete an edge. And so you start with a planar graph, and shrink any edge to a point, it's still planar. Deleting edges to a planar. Okay? Now, but there are millions of different ways to describe a family of graphs that still is remains the same. Undertaking minor. And Robertson and Seymour proved that any such family of graphs, there is a finite number of minimal graphs that are obstructions. So that if it's not in the family, then then it has to contain, then there has to be a way to shrink it down and until you get one of these bad minimal graphs that's not in the family. For in plate, case for planar graph, the minimal graph is a is a five-pointed star where there, everything pointed to another. And the minimal graph consisting of trying to connect three utilities to three houses without crossing lines. And so there are two, there are two bad graphs that are not planar. And every, every non-planar graph contains one of these two bad graphs by by shrinking. And he said again, so he proved that there's a finite number of these bad guys. Always a finite. No, somebody says, here's a family, it's hard to believe, and they present its sequence of 20 papers. I mean, in there, it's deep work. But it, you know, it's because that's for any arbitrary class. So it's for any arbitrary class that's closed under taking minors, that's closed under. Maybe I'm not understanding, because it seems like a lot of them are closed under taking minors. Almost all the important classes of graphs are. There are tons of of such graphs, but also hundreds of them that arise in applications. Like I have a book over here called Classes of Graphs, and then and it, it's amazing how many different classes people have looked at. So why do you bring up this theorem? Lower this proof? So, you know, there are lots of algorithms that that are known for special classes of graphs. For example, if I have a certain, if I have a chordal graph, then I can color it efficiently. If I have some kinds of graphs, it'll make a great network very soon. Like you'd like to test, you somebody gives you a graph that's always in this family of graphs. If so, then I hope, then I can, I can go to the library and find an algorithm that's gonna solve my problem on that graph. Okay? So we, we have, we want to have a graph that says, number than that says, give me a graph, I'll tell you whether it's and whether it's in this family or not. Okay? And so all I have to do is test whether or not that does this given graph have a minor that's one of the bad ones. A minor is is everything you can get by shrinking and removing edges. And given any minor, there's a polynomial time algorithm saying, I can tell whether this is a minor of you. And there's a finite number of bad cases. So I just try, you know, does it have this bad case by polynomial time? I got the answer. Does he have this bad case? Probably time I got the answer. Total polynomial time. And so I've solved the problem. However, all we know is that the number of minors is finite. We don't know what. We might only know one or two of those minors, but we don't know that if we got 20 of them, we don't know there might be 20, 25. The Halloween. All we know is that is that it's finite. So here we have a polynomial time algorithm that we don't know. Mm-hmm. That's a really great example of what you worry about, or why you think P equals NP won't be useful. But still, why do you hold the intuition that P equals NP? Because you have to rule out so many possible algorithms have been not working. You know, you can, you can take the graph and you can represent it as in terms of certain prime numbers, and then you can multiply those together, and then you can, then you can take the bitwise AND and and, you know, and construct some certain constant in polynomial time. And then that's, you know, perfectly valid algorithm. And that there's so many algorithms of that kind. A lot of times we see random, you take data and and and we get coincidences that that that some fairly random looking number actually is useful because because it god, it happens to it happens to solve a problem just because, you know, there's there's so many hairs on your head. But it seems like unlikely that two people are going to have the same number of hairs on their head. But but they're obvious. But you can count how many people there are and how many hairs on there. So there must be people walking around in the country to have the same number of hairs on their head. Well, that's the kind of a coincidence that you might say also, you know, this this particular combination of operations just happens to prove that a graph is has a Hamiltonian path. And I see lots of cases where unexpected things happen when you have enough, enough possibilities. But because the space of possibility is so huge, I have to rule them all out. And so that's the reason for my intuition. It's good by no means a proof. I mean, some people say, you know, well, P can't equal NP because you've had all these smart people, you know, the smartest designers of algorithms that have been wrecking their brains for years and years, and and there's million-dollar prizes out there, and you know, none of them, nobody has thought of the algorithm. So it must, must be no such job. On the other hand, I can use exactly the same logic and I can say, well, P must be equal to NP because there's so many smart people out here been trying to prove it unequal to NP, and they've all failed. You know, this kind of reminds me of the discussion about the search for aliens. They've been trying to look for them, and we haven't found them yet, therefore they don't exist. Yeah, but you can show that there's so many planets out there that they very possibly could exist. Yeah, and right. And then there's also the possibility that that they exist, but they, they all discovered machine learning or something, and and then blew each other up.

Well, on that small, quick danger, let me ask, do you think there's intelligent life out there in the universe? I have no idea. Do you hope so? Do you think about it? It, I, I don't, I don't spend my time thinking about things that I could never know, really. And yet, you do enjoy the fact that there are many things you don't know. You do enjoy the mystery of things. I enjoy the fact that there, that I have limits. Yeah, but I don't, but I don't take time to answer unsolvable questions. I got it. Well, because you've taken on some tough questions that may seem unsolvable. You have taken on some tough questions and you seem unsolvable. If there is, because we are thrilled when I can get further than I ever thought I could. Right? Yeah, but but I don't, what much like was religion, these, I'm glad the dirt, that that there are no proof that God exists or not. I mean, I think it would spoil the mystery. It would be too dull. Yeah.

So to quickly talk about the other art of artificial intelligence, what is, if you, what's your view? You know, the artificial intelligence community has developed as part of computer science and in parallel with computer science since the 60s. What's your view of the AI community from the 60s to now? So all the way through, it was the people who were inspired by trying to mimic intelligence or to do things that that were somehow the greatest achievements of intelligence that had been inspiration to people who have pushed the envelope of computer science, maybe more than any other group of people. So it's all the way through, it's been a great source of of good problems to to sink teeth into and and getting, getting partial answers and then more and more successful answers over the years. So this has, this has been the inspiration for lots of the great discoveries of computer science. Are you yourself captivated by the possibility of creating algorithms having echoes of intelligence in them? Not as much as most of the people in the field, I guess I would say. But but that's not to say that they're wrong, or that it's just, you asked about my own personal preferences. And yeah, but but the thing that I, that I worry about is when people start believing that they've actually succeeded. And because the, it seems to me, this huge gap between really understanding something and being able to pretend to understand something and give these, give the illusion of understanding something. Do you think it's possible to create without understanding? Yeah. So to, uh, I do that all the time. To run, I mean, that's why I use random numbers. I like. Yeah, but I, but there's, there's still what this great gap. I don't know, certainly it's impossible, but I'm like, but I don't see a, anything coming any closer to really the, the kind of stuff that I would consider intelligence. Say, you've mentioned something that on that line of thinking, which I very much agree with. So, the Art of Computer Programming, as the book is focused on single processor algorithms and for the most part. And you mentioned that's only because I set the table of contents in 1962. You have to remember for sure. There's no, I'm glad I didn't wait until 1965 or one book maybe will touch in the Bible. But one book can't always cover the entirety of everything. So I'm glad, yeah, I'm glad the, the table of contents for The Art of Computer Programming is what it is. But you did mention that that you thought that an understanding of the way ant colonies are able to perform incredibly organized tasks might well be the key to understanding human cognition. So these fundamentally distributed systems. So what do you think is the difference between the way Don Knuth would sort a list and an ant colony would sort a list or performing algorithm sorting a list isn't same as cognition, though? But but I know what you're getting at. Is well, the advantage of ant colony, at least we can see what they're doing. We, we know which ant has talked to which other ant, and and and and it's much harder with the quick brains to just to know how to, what extent of neurons are passing signals. So I understand that ant colony might be a, if they have the secret of cognition, think of an ant colony as a cognitive single being rather than as a colony of lots of different ants. I mean, just like the cells of our brain are, and and the microbiome, and all that is interacting entities. But but somehow I consider myself to be a single person. Well, you know, ant colony, you can say might be cognitive, is somehow. And it's, yeah, I mean, you know, I, okay, I like, I smash a certain ant, and mmm, that's stung. What was that, right? You know, but if we're going to crack the, the, the secret of cognition, it might be that we could do so by, but my psyche, note how ants do it, because we have a better chance to measure, and they're communicating by pheromones and by touching each other and sight. But but not by much more subtle phenomena, like electric currents going through. But even a simpler version of that. What are your thoughts of maybe Conway's Game of Life? Okay, so Conway's Game of Life is is able to simulate any, any computable process, and any deterministic process. Is like how you went there? I mean, that's not its most powerful thing, I would say. I mean, you can simulate it, but the magic is that the individual units are distributed. Yes. And extremely simple. Yes. We can understand exactly what the primitives are. The primitives is just like with the ant colony, even simple. But if we, but still, it doesn't say that I understand, I understand life. I mean, I understand it. It gives me an, it gives me a better insight into what does it mean to to have a deterministic universe? What does it mean to to have free choice, for example?

Do you think God plays dice? Yes. I don't see any reason why God should be forbidden from using the most efficient ways to to, I mean, we, we know that dice are extremely important and inefficient algorithms. There are things like that couldn't be done well without randomness. And so I don't see any reason why my God should be prohibited. But when the, when the algorithm requires it, you don't see why the, no, the physics should constrain it. Yeah.

So in 2001, you gave a series of lectures at MIT about religion and science. Well, that was 1999. But you published the book came out in Cooper. So in 1999, you spent a little bit of time in Boston, enough to give those lectures. Yeah, and I read in the 2001 version that most of it, it's quite a fascinating read. I recommend people, it's a transcription of your lectures. So what did you learn about how ideas get started and grow from studying the history of the Bible? You've rigorously studied a very particular part of the Bible. What did you learn from this process about the way us human beings, as a society, develop and grow ideas, share ideas? And by those ideas, I, I tried to summarize that. I wouldn't say that I, that I learned a great deal of really definite things, like right where I could make conclusions. But I learned more about what I don't know. You have a complex subject, which is really beyond human understanding. So, so we give up on saying, I'm never going to get to the end of the road, and I'm never going to understand it. But you say, but but maybe it might be good for me to to get closer and closer and learn more about more and more about something. And so, you know, oh, how can I do that efficiently? And the answer is, well, use randomness. And so to try a random subset of the, that is within my grasp, and and and and study that in detail, instead of just studying parts that somebody tells me to study, or instead of studying nothing because it's too hard. So I, I, I decided for my own amusement that one ones that I would, I would take a subset of the, of the verses of the Bible, and I would try to find out what the best thinkers have said about that small subset. And I had had about, let's say, 660 verses out of out of 3,000. I think it's one out of 500 or something like this. And so then I went to the libraries, which which are well indexed. Uh, you can, you, you know, I spent, for example, at at Boston Public Library, I, I would go once a week for a year. And I went to, I have done time stuff and over Harvard Library to look at this. Yes, that weren't in the Boston Public, where they where scholars had looked at. And you can call in the eight, and you can go down the shelves, and and you can pretty, you can look at the index and say, oh, there it is. This verse, I mentioned anywhere in this book? If so, look at page 105. So I was like, I could learn not only about the Bible, but about the secondary literature about the Bible, the things that scholars have written about it. And so that that gave me a way to, uh, to zoom in on parts of the things so that I could get more, more insight. And and so I look at it as a way of giving me some firm pegs, which I can, which I could hang pieces of information. But not as as things where I would say, and therefore, this is true. In this random approach of sampling the Bible, what did you learn about the, the most, you know, central, oh, one of the biggest accumulation of ideas? You know, to me, that, that the main thrust was not the one that most people think of as saying, you know, you know, don't have sex or something like this, but that the main thrust was to try to, to try to figure out how to live in harmony with God's wishes. I'm assuming that God exists. And I say, I'm glad that I, that there's no way to prove this, because that would, that would, I would run through the proof once and then I'd forget it. And and it would, and I would never just speculate about spiritual things and mysteries otherwise. And I think my life would be very incomplete. So I, so I'm assuming that God exists. But if, but a lot of things, people say God doesn't exist, but that's still important to them. And so in a way, in a way that might still be other God is there or not, in some sense. So it, guys, important to them. It's one of the, one of the verses I studied, acts is you can interpret as saying, you know, it's much better to be an atheist than not to care at all. So I would say it's, yeah, it's similar to the P equals NP discussion. Yeah.

You mentioned a mental exercise that I'd love it if you could partake in yourself, a mental exercise of being God. And so how would you, if you were God, Don Knuth, how would you present yourself to the people of Earth? You mentioned your love of literature, and there was this book that would, that really, uh, I can recommend to you if I can't think, yeah, the title I think is Blasphemy. It talks about God revealing himself through a computer in in Los Alamos. And it, it's the only book that I've ever read where the punchline was really the very last word of the book, and it explained the whole idea of the book. And so I don't want to give that away. But it, but it's really very much about this question that that she raised. But but suppose God said, okay, that my previous means of communication with the world are not the best for the 21st century. So what should I do now? And and it's conceivable that that it would that that God would choose the way that's described in this book. And another way to look at this exercise is looking at the human mind, looking at the human spirit, the human life in a systematic way. I think it mostly you want to learn humility. You want to realize that once we solve one problem, that doesn't mean it worked at all. So no other problems are going to drop out. And and and and we have to realize that that there are there are things beyond our, beyond our ability. I see hubris all around. Yeah. Well said.

If you were to run program analysis on your own life, how did you do in terms of correctness, running time, resource use, asymptotically speaking, of course? Okay, yeah. Well, I would say that question has not been asked me before. And I, I started out with library subroutines and and learning how to be an automaton that was obedient. And I had the great advantage that I didn't have anybody to blame for my failures. If I started getting not understanding something, I, I knew that I should stop playing ping pong. And that was that into it. It was my fault that I was that I wasn't studying hard enough or something, rather than that somebody was discriminating against me in some way. And I don't know how to avoid this, the existence of biases in the world. But I, but I, but I know that that's an extra burden that I didn't have to suffer from. And and then I, I found from from parents, I learned the idea of of altruism to other people as being more important than then when I get out of stuff myself. I, you know, that I need to, I need to be happy enough, enough in order to be able to speed up service. But I thought, but I, you know, but I, I came to a philosophy for finally that that I phrased as point eight is enough. There was a TV show once called Hate is Enough, which was about a, you know, somebody had eight kids. But but I, I say point eight is enough, which means if I can have a way of rating happiness, I think it's good design that to have to have an organism that's happy about eighty percent of the time. And if it was a hundred percent of the time, it would be like everybody's on drugs and never and and and and everything collapses, nothing works because everybody's just too happy. Do you think you've achieved that point eight optimal work? There are times when I, when I'm down, and I, you know, and I think, I mean, I know that I'm chemically right. I know that I've actually been programmed to be, I to be depressed a certain amount of time. And and if that gets out of kilter, and I'm more depressed, and you know, sometimes like, like I find myself trying to say, now, who should I be mad at today? There must be a reason why. But but then I realize, you know, it's just my, it's just my chemistry telling me that I'm supposed to be mad at somebody. And so and so I triggered up, say, okay, go to sleep and get better. But but if I'm, but if I'm not a hundred percent happy, that doesn't mean that I should find somebody that that's screaming and try to size them up. But I'd be like, I'm saying, you know, okay, I'm not 100% happy, but but I'm happy enough to death to be a, you know, part of a sustainable situation. So, so that's kind of the numerical analysis I do. You invert stores the human life is a point eight. Yeah. I hope it's okay to talk about.

As you talked about previously, in 2006, you were diagnosed with prostate cancer. Has that encounter with mortality changed you in some way or the way you see the world? The first encounter with mortality was when my dad died. And I, I went through a month when I sort of came to kink, you know, be comfortable with the fact that I was going to die someday. And during that month, I don't know, I, I felt okay, but I couldn't sing. And, you know, and I, and I couldn't do original research either, like tighten. Right. I sort of remember after three or four weeks, the first time I started having a technical thought that made sense and was maybe slightly creative. I could sort of feel the, you know, that and that something was starting to move again. But that was, you know, so I felt very empty for until I came to grips with the, I, yes, I learned that this is a sort of a standard grief process that people go through. Okay. So then now I'm at a point in my life, even more so than in 2006, where where all of my goals have been fulfilled, except for finishing The Art of Computer Programming. I, I had one made unfulfilled goal that I'd wanted all my life to write a piece of a piece, piece of music that, and I had an idea for for a certain kind of music that I thought ought to be written, at least somebody ought to try to do it. And I, and I felt that it was a, that it wasn't going to be easy, but I wanted, I wanted proof of concept. I wanted to know if it was going to work or not. And so I spent a lot of time, and finally I finished that piece. And we had the, we had the world premiere last year on my 80th birthday. And we had another premiere in Canada, and there's talk of concerts in Europe and various things. So that, but that's done. It's part of the world's music now, and it's either good or bad, but I did what I was hoping to do. So the only thing that I know that that I have on my agenda is to is to try to do as well as I can with The Art of Computer Programming until I go.

See now, do you think there's an element of point eight? Point eight? Yeah. Well, I look at it more that I got actually took 21.0 with when that concert was over with. I mean, I, you know, so in 2006, I was at point eight. Um, so when I was diagnosed with prostate cancer, then I said, okay, well, maybe this is yet, you know, I've, I've had all kinds of good luck all my life, and there's no, I'm nothing to complain about. So I might die now, and we'll see what happened. And so, so it's quite seriously, I went and I had no expectation that I deserved better. I didn't make any plans for the future. I had my surgery, I came out of the surgery, and spent some time learning how to walk again, and so on. It was painful for a while, but I got home and I realized I hadn't really thought about what what to do next. I hadn't, I hadn't any expectation. And I'm still alive. Okay, now I can write some more books. But it, but I didn't come with the attitude that, you know, I, you know, this was this was terribly unfair. And I just said, okay, I was accepting whatever it turned out. You know, I look like I've gotten, I got more than my shirt already. So why should I? And I didn't, and I really, when I got home, I read, I realized that I had really not thought about the next step, what I would do after I would doubt, after I would be able to work. And I had sort of thought of it as if as this might, you know, I was comfortable with with the fact that it was at the end. But but I was hoping that I would still, you know, be able to learn about satisfiability, and also someday even write music. I didn't start, I didn't started seriously on the music project until 2012. So I'm gonna be in huge trouble if I don't talk to you about this.

In the 70s, you created the TeX typesetting system together with Metafont language for font description and Computer Modern family of typefaces that has basically defined the methodology and the aesthetic of the countless research fields, right? Math, physics, well beyond design and so on. Okay, well, first of all, thank you. I think I speak for a lot of people in saying that. But question, in terms of beauty, there's a beauty to typography that you've created. And yet beauty is hard to find, right? How does one create beautiful letters and beautiful equations? Like what, what, so I mean, perhaps there are no words to be describing, you know, be described in the process. But so the great Harvard mathematician George Birkhoff wrote a book in the 30s called Aesthetic Measure, where he would have pictures of vases, and underneath would be a number, and this was how beautiful the vase was. And he had a formula for this. And he actually also wrote about music. And so he could, he could, you know, so I thought maybe I would, part of my musical composition, I would try to program his algorithms and, you know, so that I would, I would write something that had the highest number by his score. Well, it wasn't quite rigorous enough work for a computer to to do. But anyway, people have tried to put numerical value on beauty. But and he did probably the most serious attempt. And and George Gershwin's teacher also wrote two volumes where he talked about his method of of composing music. But but you're talking about another kind of beauty, and beauty in letters and letter forms and whatever that overture is, right? So so and so that's the beholder, as they say. But kinder striving for excellence in whatever definition you want to give to beauty, then you try to get as close to that as you can, somehow with it. I guess I guess I'm trying to ask, and there may not be a good answer, what loose definitions were you operating under with the community of people that you're working on? Oh, the loose definition, I wanted it to appeal to me. To me. I knew you personally. Yeah, that's a good start. Yeah. No, and it failed that test when I got volume two came out with this with the new printing, and I was expecting to be the happiest day of my life, and I felt like burning, like how angry I was that I opened the book, and it was in the same beige covers, and but it didn't look right on the page. The number two was particularly ugly. I couldn't stand any page that had a two in its page number. And I was expecting that it was, you know, I spent all this time making measurements, and I, and I had Kent had looked at dolphins in different different ways, and I hate, I had great technology, but but it did, you know, but I, but I wasn't done. I had, I had to retune the whole thing. After 1961, has it ever made you happy finally? Oh, oh, yes. Or is it appointing? Oh, no, no. And so many books have come out that would never have been written without this. I just didn't just draw it's just it's a joy. But I could, but now I, I mean, all these pages that are sitting up there, I don't have a, if I didn't like him, I would change him like that. That's my, nobody else has this ability. They have to stick with what I gave them. Yes.

So in terms of the other side of it, there's the typography, so the look of the top of the type, and the curves and the lines. What about the spacing? But what about the spacing? Because you know, the white space, you know, it seems like you could be a little bit more systematic about the layout. Oh, yeah, you can always go further. I, I didn't, I didn't stop at point eight. I stopped, I stopped about point nine eight. Seems like you're not following your own rule for happiness. Or is no, no, no. I, there's, okay, the course, there's just, what is the Japanese word wabi-sabi, or something where the most beautiful works of art are those that have flaws, because then the person who perceives them as their own appreciation and that gives the viewer more satisfaction, or so on. But but I, but no, no, with typography, I wanted it to look as good as I could in in the vast majority of cases. And then when it doesn't, then I, I say, okay, that's 2% more work for the wrote for the author. But but I didn't want to, I didn't want to say that my job was to get 200% with and take all the work away from the author. That's what I meant by that.

So if you were to venture a guess, how much of the nature of reality do you think we humans understand? So you mentioned you appreciate mystery. How much of the world about us is shrouded in mystery? Are we, are we, if you were to put a number on it, what percent of it all do we understand? Oh, we totally. How many leading zeros? Any point zero, point zero zero. I don't know. Now, I think it's infinitesimal. How do we think about that? What do we do about that? We continue one step at a time. Yeah. We muddle through. I mean, we do our best. We realize that one, that nobody's perfect. Then we, and we try to keep advancing. But we don't spend time saying, we're not there, we're not all the way to the end. Some, some mathematicians that that would be in the office next to me when I was in the math department, they would never think about anything smaller than countable infinity. And I never, you know, we intersect that countable infinity because I really got up to countable infinity. I was always talking about finite stuff. But but even, even limiting to finite stuff, which was, which is, which the universe might be, there's no way to really know what whether the universe is in isn't just made out of capital N, whenever you want to call them quarks or whatever, where capital N is some fun, a number. All of the numbers that are comprehensible are still way smaller than most, almost all finite numbers. I, I got this one paper called supernatural numbers, where I, what I guess you've probably ran into something called Knuth arrow notation, did you ever run into that? Where anyway, so you take the number, I think it's like, and I called it super K, but I named it after myself. But it's, but in arrow notation is something like ten and then four arrows and a three or something. Might not. Okay. No, the arrow notation, if you have, if you have no arrows, that means multiplication. XY means X times X times X times X Y times. If you have one arrow, that means exponentiation. So X one arrow Y means X to the X to the X to the X to the X Y times. So I find out, by the way, that this is notation was invented by a guy in 1830, and he was like, he was a, a, one of the English nobility who who spent his time thinking about stuff like this. And it was exactly the same concept that I, that I'm, I used arrows, and he used a slightly different notation. But anyway, this, and then this Ackerman's function is is based on the same kind of ideas. But Ackerman was 1920s. But anyway, you got this number 10 quadruple arrow 3. So that's that says, well, we take, you know, we take 10 to the 10 to the 10 to the 10 to the 10 to the 10th. Anyway, how many times do we do that? Oh, Ken double arrow two times or something. I mean, how tall is that stack? But but then we do that again, because that was the only 10 triple quadruple arrow two. We take quadruple three large number. It gets way beyond comprehension. Okay, yeah. And and and so, but it's so small compared to what finite numbers really are, because I want to using four arrows and, you know, in ten and a three. I mean, let's have that, let's have that many number arrows. I mean, the boundary between infinite and finite is incomprehensible for us humans. Anyway, infinity is a good, is a useful way for us to think about extremely large, extremely large things. And and and we, we can manipulate it. But but we can never know that the universe is actually and we're near that. So it just, so I realize how little we know. But but but what we, we found an awful lot of things that are too hard for any one person to know, even with, even in our small universe. Yeah, and we did pretty good.

So when you go up to heaven and meet God and get to ask one question that would get answered, what question would you ask? What kind of browser do you have up here? [Laughter] [Music] Okay, and then oh, that's beautiful, actually. Don, thank you so much. It was a huge honor to talk to you. I really. Well, thanks for the gamut of questions. Yeah, it was fun. Thanks for listening to this conversation with Donald Knuth. Thank you to our presenting sponsor Cash App. Download it, use code Lucks Podcast, you'll get ten dollars, and ten dollars will go to First, a STEM education nonprofit that inspires hundreds of thousands of young minds to learn and to dream of engineering our future. If you enjoy this podcast, subscribe on YouTube, give it five stars on Apple Podcasts, support it on Patreon, or connect with me on Twitter. And now, let me leave you with some words of wisdom from Donald Knuth. We should continually be striving to transform every art into a science, and in the process, we advance the art. Thank you for listening and hope to see you next time.