First We need to define our Turing machine rules.

首先我们需要定义我们的图灵机规则。

That's what made the Turing machine so useful in answering Hilbert's question on decidability.

这就是图灵机在回答希尔伯特关于可判定性的问题时如此有用的原因。

When a Turing machine halts, the program has finished running and the tape is the output.

当图灵机停止时，程序已经完成运行，磁带是输出。

But sometimes a Turing machine never halts.

但有时图灵机永远不会停止。

He proposed a hypothetical computing machine, which we now call a Turing Machine.

他提出了一种假想的计算机器，我们现在称之为图灵机。

Here's a tiny fragment of it shown in terms of Turing machines.

这是以图灵机形式显示的一小部分。

To find one Turing designed another Turing machine that built on top of H.

为了找到一个图灵设计了另一个构建在 H.

In fact in terms of what it can and cannot compute, there's no computer more powerful than a Turing machine.

事实上，就它能计算什么和不能计算什么而言，没有比图灵机更强大的计算机了。

To this day, the best computational systems are those that can do everything a Turing machine can.

直到今天，最好的计算系统是那些可以做图灵机能做的一切的系统。

That's how Turing machines work pretty simple right, so you might be wondering why there's such a big deal.

这就是图灵机的工作方式，非常简单，所以您可能想知道为什么会有这么大的问题。

There's no way to tell, in general, if a Turing machine will halt or not on a given input.

一般来说，没有办法判断图灵机是否会在给定的输入下停止。

You could imagine exporting these instructions to any other Turing machine, which would then perform in exactly the same way as the first.

你可以想象将这些指令导出到任何其他图灵机，其运行方式与第一台图灵机完全相同。

Our Turing machine is all ready to go so let's start it.

我们的图灵机已经准备就绪，让我们开始吧。

Remember Turing proved that Turing machines could implement any computation.

请记住，图灵证明了图灵机可以执行任何计算。

For example, we know our Turing machine will halt when given the input 1 1 0.

例如，我们知道当给定输入 1 1 0 时，我们的图灵机将停止。

To answer Hilbert's decision problem, Turing applied these new Turing machines to an intriguing computational puzzle: the halting problem.

为了回答希尔伯特的决策问题，图灵将这些新的图灵机应用于一个有趣的计算难题：停机问题。

Although this sounds simple, a Turing machine's arbitrarily large memory and program mean it can execute any computable algorithm if given enough time.

虽然这听起来很简单，但图灵机任意大的内存和程序意味着如果有足够的时间，它可以执行任何可计算的算法。

It's a paradox and this paradox means that the halting problem cannot be solved with Turing machines.

这是一个悖论，这个悖论意味着图灵机无法解决停机问题。

Imagine we have a hypothetical Turing machine that takes a description of a program and some input for his tape and always outputs.

想象一下，我们有一个假设的图灵机，它接受一个程序的描述和他的磁带的一些输入，并且总是输出。

So Turing said, let's assume we can make a machine H that can determine whether any Turing machine will halt or not on a particular input.

所以图灵说，让我们假设我们可以制造一台机器 H，它可以确定任何图灵机是否会在特定输入时停止。