;; Done Twice as a "Dojo" at Villiers Park on Thursday 19th March 2015 ;; To groups of about 15 ultra-clever teenagers who were thinking about doing Computer Science at university ;; The first group got as far as higher order functions in an hour. ;; The second group went a bit faster, and we had a bit more time, about an hour and a half, ;; and so we got right to iterative-improve and finding square roots of anything using it. ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; Environment DrRacket (version 6.1) ;; Language R5RS (Revised Revised Revised Revised Revised Report on the Algorithmic Language Scheme) ;; One person sits at the computer, one person helps them, the rest tell them what to do ;; Every time they achieve something significant, rotate audience->copilot->pilot->audience ;; Notes on back of hand: (define crib '( 2 3 + (+ 2 3) lambda define square < #t #f if abs Heron average improve make-improver error good-enough good-enough-guess iterative-improve )) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; Introduction to the Lambda Calculus ;; More precisely, an introduction to the algorithmic language Scheme, which is what you get if you start with ;; the lambda calculus and you trick it out with some extra stuff that often comes in handy, true and false and if ;; and define and also some types of numbers, like integers and fractions, and adding, and multiplying. ;; You can build all that stuff starting from scratch with just lambda, and it's a nice thing to do if you want ;; to understand how it all works, but I reckon you're already ok at that sort of thing. ;; So we'll start from something that can do basic arithmetic, and we'll learn how to find square roots of things. ;; This is an evaluator. You can ask it the values of things. 2 3 + ;; We can apply the procedure to the two numbers (+ 2 3) ;; Can you tell me the square of 333? (* 333 333) ;; The brackets mean (work out the value of the things in the brackets, and then do the first thing to the other things) ;; So what do you get if you add the squares of 3 and 4? (+ (* 3 3) (* 4 4)) ;; We have procedures for * and + , but if we ask the evaluator what & means, or what square means ;; it will just say 'I have no clue'. ;; It might be nice if we had a procedure for squaring things ;; How you make a procedure is with this thing called lambda, which is sort of a rewriting sort of thing. ;; Try (lambda (x) (* x x)), which means 'make me a thing which, when I give the thing x, gives me the value of (* x x) instead' (lambda (x) (* x x)) ;; #<procedure>, it says, which is very like what you get when you type in +, and it says #<procedure:+>. ;; So we hope we've made a procedure like + or * ;; How shall we use it to get the square of 333? ((lambda (x) (* x x)) 333) ;; Now obviously, typing out (lambda (x) (* x x)) every time you mean square is not brilliant, ;; so we want to give our little squaring-thing a name. (define square (lambda (x) (* x x))) ;; Now how do we find the square of 333? (square 333) ; 110889 ;; So lambda is allowing us to make new things, to turn complicated procedures into simple things ;; and define is allowing us to give things names ;; So now let's make a procedure that takes two things, and squares them both, ;; and adds the squares together, and let's call it pythag (define pythag (lambda (x y) (+ (square x) (square y)))) (pythag 3 4) ;; OK, great, now can you figure out how the procedure < works? ( < 3 4) ( < 4 3) ( < 3 4 6) ( < 3 4 2) ;; Notice that these #t and #f things are things that the evaluator knows the value of: ;; They're called true and false. #f #t ;; So now the last piece of the puzzle: ;; if takes three things: (if #t 1 2) ;1 (if #f 1 2) ;2 ;; So we've got numbers and *,+,-,/, and we've got #t #f and if, and we've got lambda, and define ;; And so all the stuff we've got above, we can think of it as a reference manual for a little language. ;; We can build the whole world out of this little language. ;; That's what God used to build the universe, and any other universes that might have come to His mind. ;; And we can use it too. ;; Here's the manual 2 * (* 2 3) (define forty-four 44) forty-four (lambda (x) (* x x)) ((lambda (x) (* x x)) 3) (if (< 2 3) 2 3) ;; And if we understand these few lines, then we understand the whole thing, and we can fit the little pieces together like this: (define square (lambda (x) (* x x))) (square 2) ;; So now I want you to use the bits to make me a function, call it absolute-value, which if you give it a number gives you back ;; the number, if it's positive, and minus the number, if it's negative. (define absolute-value (lambda (x) (if (> x 0) x (- x)))) (absolute-value 1) (absolute-value -3) (absolute-value 0) ;; So I've taught you most of the rules for Scheme, which is a sort of super-advanced lambda calculus, and so if you understand ;; the bits above, then you've got the hang of the lambda calculus plus some more stuff. ;; And it's a bit like chess. The rules of chess are super-simple, you can explain them to babies, ;; like Dr Polgar did to Judit and her sisters. ;; But that doesn't make the babies into good chess players yet. They have to practise. ;; How are we doing for time? We've done the whole of the lambda calculus, plus some extra bits. We should feel pretty smug. ;; (In both cases, this had taken about 35 minutes) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; Let's do a little practice exercise. Like a very short game of chess, now I've explained most of the rules. ;; So once upon a time there was this guy, believe it, called 'Hero of Alexandria'. ;; Or sometimes he seems to have been called 'Heron of Alexandria', like Hero was the short version, ;; like he was sometimes called Jack and sometimes called John. ;; Whatever, Hero invented the syringe, and the vending machine, and the steam engine, and the windmill, and the rocket, ;; and the shortest path theory of reflection of light, and did some theatre stuff, ;; and he was like Professor of War at the big library in Alexandria. ;; You get the impression that if Alexandria had lasted just a little bit longer, ;; the whole industrial revolution would have kicked off right there, and the Romans would have walked on the moon in about AD400. ;; And we'd all be immortal, and live amongst the stars. So you should take the fall of the Roman Empire *very* personally. ;; And one of his things was a way of finding the square roots of numbers, ;; which is so good that it was how people found square roots right up until the invention of the computer. ;; So I'm going to explain that method to you, and you're going to explain it to this computer, and then you can get the computer ;; to calculate square roots for you, really fast. And after that you're only a couple of steps away from cracking the ;; Enigma codes and winning the second world war and inventing the internet and creating an artificial intelligence ;; that will kill us all just 'cos it's got better things to do with our atoms. I'm not joking. ;; So careful.... What I've just given you is the first step on the path that leads to becoming a mighty and powerful wizard. ;; And with great power comes great something or other, you'll find it on the internet, so remember that. ;; PAUSE ;; So imagine you want to find the square root of 9. And you're a bit stuck, so you say to your friend, "What's the square root of nine?", and he says it's three. ;; How do you check? (* 3 3) ;; Bingo. There's another way to check (/ 9 3) ;; That's what it means to be the square root of something. If you divide the something by the square root, you get the square root back. ;; But what if your friend had said "err,.. 2 or something?" (/ 9 2) ;; Notice that the number you put in is too low, but the number you got back is too high. ;; So Heron says, let's take the average. ;; So we need an average function (define average (lambda (a b) (/ (+ a b) 2))) (average 2 (/ 9 2)) ; 3 1/4 ;; three and a quarter, that's like a much better guess, it's like you'd found a cleverer friend. ;; so try again. (average 3.25 (/ 9 3.25)) ; 3.009615... ;; and again (average 3.0096 (/ 9 3.0096)) ; 3.0000153.. (average 3.0000153 (/ 9 3.0000153)) ; 3.000000000039015 ;; So you see this little method makes guesses at the square root of nine into much better guesses. ;; We see that this is kind of a repetitive type thing, and if you see one of those, your first thought should be, ;; I wonder if I can get the computer to do that for me? ;; Can you make a function which takes a guess at the square root of nine, and gives back a better guess? (define improve-guess (lambda (guess) (average guess (/ 9 guess)))) ;; I'd better show you how to format these little functions so that they're easier to read (define improve-guess (lambda (guess) (average guess (/ 9 guess)))) ;; The evaluator doesn't notice the formatting, and it makes it a bit more obvious what's getting replaced by what. (improve-guess 4) ; 3 1/8 (improve-guess (improve-guess 4)) ; 3 1/400 (improve-guess (improve-guess (improve-guess 4))) ; 3 1/960800 ;; We all know what the square root of nine is, let's look at a more interesting number, two. ;; It's a bit of an open question whether 'the square root of two' is a number, or whether it's just a noise ;; that people make with their mouths shortly after you show them a square and tell them about Pythagoras' theorem. ;; Pythagoras used to have people killed for pointing out that you couldn't write down the square root of two. ;; I've got a bit of a confession to make. ;; Someone's already explained to this computer how to find square roots (sqrt 9) ; so far so good! (sqrt 2) ; 1.4142135623730951 hmmm. let's check. (square (sqrt 2)) ; 2.0000000000000004 ;; So it turns out that this guy's just said, if you can't come up with the square root of two, just lie, and come up with something ;; that works, close as dammit. ;; Which is like, bad practice, and tends to lead to Skynet-type behaviour in the long run. ;; So let's see what Hero would have said about it. ;; We need a new function that makes guesses better at being square roots of two. ;; It's a bit dirty, but let's just call that improve-guess as well. ;; That's called redefinition, or 'mutation', and it's ok when you're playing around, ;; but it's a thing you should avoid when writing real programs, because, you know, Skynet issues. ;; Hell, no-one ever got more powerful by refraining from things. (define improve-guess (lambda (guess) (average guess (/ 2 guess)))) ;; Anyone make a guess? (improve-guess 1) ; 1 1/2 ;; Any good? (square (improve-guess 1)) ; 2 1/4 ;; How wrong? (- (square (improve-guess 1)) 2) ; 1/4 ;; OK, I want you to notice that we've just done the same thing twice (define improve-guess-9 (lambda (guess) (average guess (/ 9 guess)))) (define improve-guess-2 (lambda (guess) (average guess (/ 2 guess)))) ;; Now whenever you see that you've done the same thing twice, and there's this sort of grim inevitability ;; about having to do it a third time someday, you should think: ;; Hey, this looks like exactly the sort of repetitive and easily automated task that computers are good at. ;; And so now I want you to make me (and this is probably the hard bit of the talk...) a function which ;; I give it a number and it gives me back a function which makes guesses at square roots of the number better. (define make-improve-guess (lambda (n) (lambda (guess) (average guess (/ n guess))))) ;; And now we can use that to make square root improvers for whatever numbers we like (define i9 (make-improve-guess 9)) (i9 (i9 (i9 (i9 1)))) ; 3 2/21845 (define i2 (make-improve-guess 2)) (i2 (i2 (i2 (i2 1)))) ; 1 195025/470832 ;; The first group got this far in about an hour, which was all we had time for, and then we stopped and I waffled for a bit. ;; Now how good are our guesses, exactly? (- 2 (square (i2 (i2 (i2 (i2 1)))))) ;; We could totally make a function out of that: (define wrongness (lambda (guess) (- 2 (square guess)))) (wrongness (improve-guess 1)) ; -1/4 (wrongness (improve-guess (improve-guess 1))) ; -1/144 (wrongness (improve-guess (improve-guess (improve-guess 1)))) ; -1/166464 (wrongness (improve-guess (improve-guess (improve-guess (improve-guess 1))))) ; -1/221682772224 ;; So we're getting closer! When should we stop? Let's say when we're within 0.00000001 (define good-enough? (lambda (guess) (< (absolute-value (wrongness guess)) 0.00000001))) (good-enough? (improve-guess (improve-guess 1))) ; #f (good-enough? (improve-guess (improve-guess (improve-guess (improve-guess 1))))) ; #t ;; Now, we're doing a bit too much typing for my taste. ;; What we want to do is to say: ;; I'll give you a guess. If it's good enough, just give it back. If it's not good enough, make it better AND TRY AGAIN. ;; This is the hard bit. We need to make a function that calls itself. ;; Go on, have a go (define good-enough-guess (lambda (guess) (if (good-enough? guess) guess (good-enough-guess (improve-guess guess))))) (good-enough-guess 1) ; 1 195025/470832 ;; YAY VICTORY! ;; The second group got this far in about an 1hr 10 mins, but they all still seemed keen and we didn't have to stop, so: ;; Now this is as much of the talk as I'd written, ;; but actually we've got the time to go a little bit further, if your brains haven't totally exploded, and you might like the next bit, ;; because it makes a nice punchline to the whole thing: ;; There's a pattern here, and it's called iterative-improve ;; And iterative improvement is everywhere in the world, for instance you probably got shown the Newton-Raphson solver at school, ;; which is a thing which can find roots of all sorts of equations very fast, and it works like this, you have an initial guess, and ;; Newton Raphson is a way of making a guess into a better guess, and you need to know when the answer is good enough so you can stop. ;; Or this morning I had a shower, and I got in the shower and I turned the water on to just a random position and it was too hot, so I turned the handle ;; a bit the other way and it was a bit too cold, so I turned it back just a bit and then it was ok so I stopped. ;; And that's the same pattern, and you see this sort of thing all over, it is how you solve big matrices and so on and so forth. ;; And we have just discovered this pattern, which is kind of a fundamental building block when you're writing programs, like a for loop is another basic pattern. ;; So let's see if we can make a function that takes a guess and a way of improving guesses and a way to tell if we're done yet, and gives us back an answer. (define iterative-improve (lambda (guess improve good-enough?) (if (good-enough? guess) guess (iterative-improve (improve guess) improve good-enough?)))) (iterative-improve 1 (make-improve-guess 2) good-enough?) ; 1 195025/470832 ;; This was where we stopped the second session. Here's some waffle: ;; And I think now you can see that we've abstracted a pattern here that will come in handy for the sorts of things that we're trying to do. ;; That's what this talk has really been about, how to build a language which allows you to solve the problems that you're interested in. ;; So I'd like to tidy up the program that we've just written, and put it into the sort of form that I'd have written it in, if I'd been solving this problem ;; and I'd played around for a bit and found what I thought was a nice expression of the ideas that we've been talking about. (define square (lambda (x) (* x x))) (define absolute-value (lambda (x) (if (> x 0) x (- x)))) (define make-improve-guess (lambda (n) (lambda (guess) (average guess (/ n guess))))) ; this bit is Heron's method (define make-good-enough? (lambda (n tolerance) (lambda (guess) (> tolerance (absolute-value (- n (square guess))))))) (define iterative-improve (lambda (guess improve good-enough?) (if (good-enough? guess) guess (iterative-improve (improve guess) improve good-enough?)))) (define make-square-root (lambda (guess tolerance) (lambda (n) (iterative-improve guess (make-improve-guess n) (make-good-enough? n tolerance))))) ;; We can use these bits to make the sort of square root we usually find provided: (define engineer-sqrt (make-square-root 1.0 0.00000000000001 )) (engineer-sqrt 2) ;; And here's what we might use, if we needed really good square roots for some reason: (define over-cautious-engineer-square-root (make-square-root 1 1/1000000000000000000000000000000000000000000000000000000000000000000)) (over-cautious-engineer-square-root 2) ;; And I hope you can see this this program is actually built of lots of tiny simple pieces, ;; all of which you can understand, and most of which you'll be able to reuse in other contexts. ;; In particular, iterative-improve is a terribly general thing which you can use in lots of ways. ;; And it might have taken us a while to write, although we wrote it as part of a learning-the-language finger-exercise, ;; but we never have to write it again. It works and it will keep working and we've got in the bank now. ;; Here's the take-home message: ;; If you've got a problem, build yourself a language to solve the problem in. ;; To do that, you need to start with a language that allows you to abstract what you do into simple pieces ;; which are easy to understand, so that you can see that they're right and they aren't too snarled up with ;; the little details of the problem you're working on at the moment. ;; And you need a language that allows you to combine the little pieces easily ;; to make new pieces that solve the problem that you're trying to deal with. ;; And there's a sense in which all computer languages are just this lambda calculus. ;; We've done all this in Scheme, which is lambda calculus plus some extra stuff. ;; There's nothing we've done here that can't be done in python, or in ruby, or in perl or in haskell or in lisp. ;; What distinguishes these languages is what extra stuff has already been added to them. ;; But if a language is good enough, and none of the usual features have actually been taken away, ;; which does happen sometimes, then if there's anything missing that you need you can always add it yourself. ;; And then you can use the language that you have to build the language that you need. ;; In a sense, making languages is itself an iterative improvement process. ;; And the big questions are always: ;; How do we make things better? What's good enough? When are we done? ;; Postscript ;; I'll show you a trick now. We've been using it all along and nobody noticed, ;; but it's the sort of thing that looks like magic, and I don't like magic unless I can cast the spells myself. (good-enough-guess 1) ; 1 195025/470832 (good-enough-guess 1.0) ; 1.4142135623746899 ;; This is called 'contagion'. There are really two types of numbers. ;; Numbers that look like 432/123 are called 'exact', or 'vulgar fractions' ;; Numbers that look like 1.4142 are called 'inexact', or 'approximate', or 'floating point', or 'decimal fractions' ;; The first type are the sort of numbers that children learn about in school, and that mathematicians use. ;; And the second type are the sort of numbers that engineers use, and they're actually quite a lot more complicated and fuzzy ;; than the exact type. They just sort of work like 'if it's very close, then it's good enough'. ;; The way most computers think about them, they keep about sixteen digits around, and if you want more than that, tough luck. ;; But for some purposes they're better, for instance they're easier to read, and it's a bit of a matter of taste. ;; If you multiply or add an inexact number to an exact number, the answer is always inexact. ;; You can't unapproximate something. (/ 1 3) ; 1/3 (/ 1.0 3) ; 0.3333333333333333 ;; We all know that 1/3 isn't really 0.33333333333333 ;; Mathematicians worry about that sort of thing. Engineers don't. Sometimes aeroplanes crash. Mostly they don't.

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## Sunday, November 22, 2015

### An Introduction to the Lambda Calculus

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