ฝึกพูดภาษาอังกฤษด้วยเทคนิค Shadowing จากวิดีโอ: Turing & The Halting Problem - Computerphile

Today we're going to be talking about a problem in logic, and how in solving that problem,

Alan Turing almost inadvertently invented the modern digital computer.

So we start back at the beginning of the 20th century, where mathematicians had posed this problem.

- Thank you.

In logic, we're interested in finding, do these premises entail this conclusion?

So premises are the bits you start off with in an argument.

They are the bits you know at the beginning or your assumptions.

And the conclusion is the bit you want to establish, the bit that you reason to with your argument.

And we want to know, is there a test that will tell us, yes, for sure, these premises do or don't entail this conclusion?

Is there an automatic way of finding out whether they do or whether they don't. So that's the problem.

It's called the decision problem.

and mathematicians wanted to find out Is there an answer to the decision problem for first order logic?

That's the kind of logic you learn in philosophy or mathematics at university.

So lots of mathematicians were trying to work out, is first order logic decidable?

That is, can we automatically test whether the premises entail the conclusion?

Alan Shuring was one of the first to discover that first order logic isn't decidable.

I To prove this, It's really difficult conceptually,

because you have to be able to show no possible program can give you the answer.

But how do you do that?

How do you show something about every possible program?

You can't run through every program one by one.

But Turing came up with a brilliant solution.

Yeah.

His idea goes something like this.

Suppose we have a program and let's just draw it as a black box.

It's going to take some inputs, and it's going to give us some outputs.

our program is going to solve some problem.

a problem like, do the premises entail the conclusion?

We ask it a question and it will give us an answer: yes, or no?

Now here's another question we can ask: Thank you.

Let's look at all of those possible programs.

And we're just thinking of them as black boxes at the moment.

We might want to know, is this program Given a certain input, gonna give us an answer.

Or is it going to trundle on forever and never give us an answer?

That is, is it going to halt or is it not going to halt eventually?

So think about your computer running.

You want it to give you an answer.

And whether it's a good answer or a bad answer, it's better than no answer.

No answer would mean the computer trundles round forever and ever in a loop.

And you would just never know whether it's going to finish today, tomorrow or never.

So halting is good.

So there's another question we can ask: Given some program and some inputs, will it ever halt?

Now it turns out that our logical problem Do these premises entail this conclusion?

is very similar to this halting problem.

In fact, if we can solve the logical problem, then we can solve the halting problem.

Will this program halt on this input?

So the clever part of Turing's proof is to show that it's impossible for any machine, however clever it is, to solve the halting problem.

That is to tell us whether a given machine with a given input will halt or not.

And here's how we did it.

Let's suppose we've got a machine or a program that solves the problem for us it solves the holding problem.

Don't worry about how it works, let's just think of it as a black box, taking a description of a machine and an input, and giving us an answer.

Yes it will halt or no it won't halt.

Just suppose that's possible.

Call that machine H for the halting problem.

If you give me that machine, I can transform it into a different machine like this.

I stick some extra bits on it so that If it gives me a yes answer, I make it loop forever without ever stopping.

If it gives me a no answer on the other hand, it's gonna halt straight away.

Let's call that big machine the whole thing.

H plus.

Now here's another question we can ask: what happens if I feed the whole machine into itself.

So I'm going to put H plus Thank you in here and H + in here.

So the question I'm now asking is, I'm feeding H + into itself.

So I'm asking the question: Does H + halt?

give an input H+.

And here's where it all goes wrong.

because If H + does halt, We get a yes answer.

But then it loops forever, so it doesn't halt.

On the other hand, if it doesn't halt, We get a no answer, but then it halts.

So if it does halt, then it doesn't halt.

But if it doesn't halt, then it does halt.

Either way, we get a contradiction.

It's a paradox.

And what that shows is we started off assuming that we can solve the problem.

we've ended up with a paradox.

So our assumption was bad.

It turns out there's no possible machine, no possible program that solves the halting problem.

The really clever bit about Turing's idea is it doesn't matter what kind of program or machine it is.

It doesn't matter whether it's An abstract algorithm, whether it's a real computer, a physical computer, it doesn't matter what it is.

We've proved that no such program is possible.

you Turing, as part of his argument, had to say a little bit about what's going on in these black boxes.

And the idea is that every card represents an instruction in this Turing machine.

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