Turing realized this halting problem was very similar to the decidability problem.

图灵意识到这个停机问题与可判定性问题非常相似。

Their halting problem is whether or not they'll tile the plane.

他们的停机问题在于他们是否会平铺平面。

Its halting problem is literally whether or not it halts.

它的停机问题实际上就是它是否停机。

Complex quantum systems are Turing-complete, and their halting problem is the spectral gap question.

复杂量子系统是图灵完备的，其停机问题就是谱隙问题。

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。

So this solution to the halting problem proves that not all problems can be solved by computation.

所以这个停机问题的解决方案证明了并不是所有的问题都可以通过计算来解决。

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。

Unfortunately, Turing came up with a proof that shows the halting problem was in fact unsolvable, through a clever logical contradiction.

不幸的是，图灵通过巧妙的逻辑矛盾得出了一个证明，表明停机问题实际上是无法解决的。

So if you could solve the halting problem, you could solve the twin prime conjecture and all sorts of other unsolved questions.

因此， 如果你能解决停机问题，你就能解决孪生素数猜想和所有其他未解决的问题。

Turing reasoned if there existed a program, whose halting behavior was not decidable by age, it would mean the halting problem is unsolvable.

图灵推断，如果存在一个程序，其停机行为不能由年龄决定，则意味着停机问题无法解决。

Although they are powerful, every Turing-complete system comes with a catch, its own analog of the halting problem, some undecidable property of the system.

尽管功能强大， 但是每个图灵完备系统都存在缺陷，即其自身的停机问题类似物，以及系统的一些不可判定性质。