Name: 什麼是單體？- 電腦愛好者 (What is a Monad? - Computerphile)
Uploaded: 2021-01-14T10:37:19.000Z
Duration: 21 min 50 s
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So, monads are a concept that was invented in mathematics in the 1960s, and

then it was rediscovered in computer science in the 1990s. And what it gives

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you, is a new way of thinking about programming with effects. And for me, this

is one of the most important new ideas in programming languages in the last 25

years. So that's what we're going to be looking at today - programming with monads.

We're going to come at this using a simple example, and the example that

we're going to look at is the idea of writing a function that evaluates simple

expressions. And I'm going to use Haskell for this, but it doesn't matter if you don't

know anything about Haskell, because we're going to use it in a very simple

way, and I'm going to explain everything as we're going along. So, what we're going

to start with, is by defining a simple datatype for the kind of expressions

that we're going to be evaluating. So, we'll use the data keyword in Haskell,

which introduces a new data type, and then we're going to define a new data

type for expressions. And then there's two things that an expression can be. It

can either be an integer value, so we'll write that down - we have Val of an Int.

Or, it can be the division of two sub-expressions. So we've got two

constructors here in our data type - we've got Val,

which builds expressions from integers, and we've got Div, which builds expressions

from two sub-expressions. So, just to reiterate what what's actually going on

here, we're declaring a new data type called Expr, and it's got two new

constructors - one called Val, which takes an integer parameter, and one

called Div, which takes two sub-expressions as parameters as well.

So basically what we're working with is expressions that are built up from

integer values using a simple division operator. So, many of you may not be

familiar of this kind of syntax, so let's have a couple of examples of values of

this data type, so that we make sure everyone's on the same page. So, what I'm

going to do here, is draw a little table. So, on one side, on the left-hand side, I'm

going to have what we would normally write down in mathematics. And then on

the right-hand side, we'll think how would you translate this into a value in

this Haskell data type? So, let's have three simple examples here - we'll have

one, and we'll have six divided by two, and

let's do one more example, we'll have six divided by three divided by one. So, these are

simple expressions built up from integers using a division operator. But, we're

writing Haskell programs today, so let's think how do these things actually get

represented as values of our expression data type? So, the first one is very

simple - if we want to represent the value one, we just need to use the Val tag, so we

write Val of one. If we want to have an expression like six divided by two, well it's

a division, so we have a Div at the top level, and then we have two values - we

have Val six, and Val two. And actually, I'll leave the last one is a little exercise

for you here, so you can try this one for yourself - how do you represent this as a

value in Haskell? Well, you're gonna need two divisions, you're going to need three

Val constructors, and then a bunch of brackets. So, this is the basic idea -

we've got simple expressions built up from integers using division, and we want

to think about how do we write a program to evaluate these expressions? Let's

write a program to do that. So, we're going to write an evaluator, and it's

going to be a program, or a function in this case, that takes an expression as

input, and what it's going to give back is the integer value of that expression.

And there's going to be two cases here, because we have two kinds of expressions.

We have a case for values, and we need to figure out what to do with that, which

we'll do in a moment. And then we have a case for division, and we need to think

what to do with that. So, we've got the skeleton here of a program, and then we

just need to fill in the details. So, how do you evaluate an integer value? Well,

that's very simple, you just give back the number - so if I had Val of one,

it's value is just one. And then how do I evaluate a division? Well, these two

expressions here, x and y, these could be as complicated as you wish. So, we need to

evaluate these recursively. So what we would do, is evaluate the first one, x, and

that will give us an integer. And then we'll evaluate the second one, y, and that

will give us another integer. And then, all we need to do is divide one by the

other. So, this is a nice simple program that evaluates these kind of expressions

built up from integers using division - we just have a simple recursive program, two

cases, and everything looks fine. But there's a

problem with this program, and the problem is that it may crash - because if

you divide a number by zero, then that's undefined, so this program will just

crash. So, in particular, if the value of the expression y here was zero, then this

division operator would crash, and you get some kind of runtime error. So we

don't want our programs to crash, so we think, what do we do to fix this problem?

First of all, what we're going to do is we're going to define a safe version of

the division operator, which doesn't crash anymore. Because that's basically

the root of the problem here - division by zero gives an undefined result, and the

program is going to crash. So, let's define a safe version of the division

operator. We're going to define a function called safediv, and it's going

to take a couple of integers, and it's going to give back Maybe an integer. And

Maybe is the way that we deal with things that can possibly fail in Haskell.

So, the type here is not Int to Int to Int, it's Int to Int to Maybe Int, because

division may fail. And we'll see how this Maybe type works in a moment. So, how do

we actually define safediv? We take two integers, n and m, and then what we'll do

is check - is the second of these zero? Because that's the case when things

would go wrong. So, if m happened to be zero, then we will give back the result

Nothing. Okay, so Nothing is one of the constructors in the Maybe type. If m is

not zero, what we're going to do is Just give back the result of dividing. So, Just

is another constructor in the Maybe type - Maybe only has two constructors, Nothing,

which represents things that have gone wrong, or in our case division by zero,

and Just, which represent things that have gone fine. In this case, we actually

just get back the result of dividing one number by the other. So, what we have here

now is a safe version of the division operator, which is explicitly checking

for the case when the program would have crashed. So this doesn't crash anymore, it

returns one of two values - either Nothing if things go wrong, or Just of the

division if things have gone fine. So, what we can do then, with this safe

division operator, is rewrite our little evaluator program to make sure that it

doesn't crash. So, our new evaluator is going to have a slightly different type

than before. So before, the original program just took an expression

as input, and then it gave back an integer. But that program could crash. The

new evaluator takes an expression as input as before, but now it Maybe gives

you an integer, because it could fail, it could have division by zero. So, how do we

rewrite this evaluator? So, we'll do the two cases again - write down the skeleton,

and then we'll fill in the details. So, in the base case, we can't just return n

this time, because we've got to return a Maybe value. And there's only two things

we could return, either Nothing or Just, and in this case the right thing to do

is to return Just of n, because if you evaluate a value that's always going to

succeed, so we use a success tag, which is Just, and then we have the integer

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什麼是單體？- 電腦愛好者 (What is a Monad? - Computerphile)

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