52 Things: Number 7: How does randomness help in computation, and what is the class BPP?

52 Things: Number 7: How does randomness help in computation, and what is the class BPP?

52件事：第7：随机性如何帮助计算，什么是BPP？

 

This is the latest in a series of blog posts to address the list of '52 Things Every PhD Student Should Know To Do Cryptography': a set of questions compiled to give PhD candidates a sense of what they should know by the end of their first year. This blog post concentrates on the Complexity Class BPP.
这是一系列博客文章中的最新一篇，旨在解决“每个博士生在做密码学时应该知道的52件事”：这是一组问题，旨在让博士生在第一年结束时了解他们应该知道什么。这篇博客文章主要关注复杂性类BPP。

 

So far, during this blog series Ryan has introduced us to complexity classes, and in particular to P:
到目前为止，在这个博客系列中，Ryan向我们介绍了复杂性类，特别是P：

• P  is the class of languages decidable in polynomial time by a deterministic Turing machine.
  P是一类在多项式时间内可由确定性图灵机判定的语言。

Then, Guy introduced us to complexity class NP:
然后，Guy向我们介绍了复杂性类NP： 

• NP is the class of languages decidable in polynomial time by a nondeterministic Turing machine.
  NP是一类在多项式时间内可由不确定性图灵机判定的语言。

This week I am going to introduce the complexity class BPP:
本周，我将介绍复杂性类BPP：

• BPP is the class of languages that are recognised by a probabilistic polynomial time Turing machine with an error probability of 13.
  BPP是一类被概率多项式时间图灵机识别的语言，其错误概率为 13 。

Probabilistic Turing Machine
概率图灵机

A probabilistic Turing Machine[1] is a type of nondeterministic Turing Machine which randomly chooses between the available transitions at each point according to a probability distribution. What this means is that a probabilistic Turing machine can have stochastic results. On the same input, it could have different run times, accept it on one occasion and reject it on another. It could also never stop. This Turing Machine gives rise to other complexity classes such as RP, ZPP and, the one we're discussing in this post, BPP.
概率图灵机[1]是一种不确定性图灵机，它根据概率分布在每个点的可用转换之间随机选择。这意味着概率图灵机可以有随机结果。对于相同的输入，它可能有不同的运行时间，在一个场合接受它，在另一个场合拒绝它。它也永远不会停止。这个图灵机产生了其他复杂度类，如RP、ZPP和我们在这篇文章中讨论的BPP。

A little about the complexity class BPP
关于复杂性类BPP的一些信息

So as we have seen from the definition, the class BPP (Bounded-Error probabilistic polynomial time) contains the decision problems that are solvable in polynomial time by a probabilistic Turing machine with error probability 13. Note that this error probability can be chosen to be any value strictly between 0 and 12 due to a result named the amplification lemma which I will not discuss further here. The class BPP contains P, the class of problems solvable in polynomial time by a deterministic Turing Machine, this is due to the fact that a deterministic Turing Machine is a special case of the probabilistic Turing Machine (taking the only possible path with probability 1). As talked about in Guy's post, there is an open (Millennium) problem conjecturing as to whether P = NP. There is a similar question with BPP, being P = BPP?  The number of problems known to be in BPP but not known to be in P is decreasing.
因此，正如我们从定义中看到的，类BPP（有界误差概率多项式时间）包含可由误差概率为 13 的概率图灵机在多项式时间内解决的决策问题。注意，由于被命名为放大引理的结果，该误差概率可以被选择为严格地在 0 和 12 之间的任何值，我将在这里不再进一步讨论。类BPP包含P，这是一类可由确定性图灵机在多项式时间内解决的问题，这是由于确定性图灵机是概率图灵机的特例（采用概率为1的唯一可能路径）。正如盖伊的帖子中所说，关于P是否为NP，存在一个开放的（千禧年）问题。关于BPP，也有一个类似的问题，即P是否为BPP？已知存在于BPP中但未知存在于P中的问题数量正在减少。

 

An example of a BPP Problem
BPP问题的一个例子

One of the most famous problems known to be in BPP  but not known to be in P was determining whether a number was prime. However, recently (2002) a deterministic polynomial time algorithm was found[2] for this problem meaning that it is indeed in P. Another problem known to be in BPP and currently still not known to be in P is polynomial identity testing[3], the problem of determining whether a polynomial is identically equal to the zero polynomial.
已知存在于BPP中但不存在于P中的最著名的问题之一是确定一个数是否为素数。然而，最近（2002）发现了这个问题的确定多项式时间算法[2]，这意味着它确实在P中。另一个已知在BPP中但目前还不知道在P中的问题是多项式恒等式测试[3]，即确定多项式是否等同于零多项式的问题。

 

There are still many very important unanswered questions on the topic of Complexity Classes. Some of which, if answered, could have a large impact on shaping the future of Cryptography and Computer Science.
关于复杂性类的主题，仍然有许多非常重要的问题没有得到解答。其中一些问题，如果得到回答，可能会对密码学和计算机科学的未来产生重大影响。