
Chinese: 
有人可能会问这些方法有多么基本; 也许人们可以提出
一个更好的衡量随机性的标准。
算法复杂性中最令人惊讶的结果之一是数学中称为
“收敛的定义”。
这种现象类似于定义中其它相似的类型 ，例如20世纪30年代人们，KurtGödel
(哥德尔)，Alonzo Church(丘奇)，Alan Turing(图灵)，
Emile Post，Stephen Kleene，RózsaPéter
以及许多其他人试图通过不同的独立方法来表征算法的概念，
这些方法在计算能力方面都是等价的，在计算能力方面，

English: 
One may ask how fundamental these measures
may be; perhaps one can come up with a better
measure for randomness.
One of the most surprising results in algorithmic
complexity is what is known in mathematics
as the "convergence in definitions".
This is a phenomenon similar to other types
of convergence in definitions such as in the
notion of an algorithm in the 1930s when people
such as Kurt Gödel, Alonzo Church, Alan Turing,
Emile Post, Stephen Kleene, Rózsa Péter,
among many others tried to characterise the
notion of an algorithm by different independent
approaches that turned out all to be equivalent
in computational power, giving the sense that
the concept of algorithm had been mathematically

Chinese: 
让人感觉到算法概念的所有这些特征，今天已经被所知的丘奇 - 图灵论文在数学上捕获。
坚信算法的任何实际定义都会陷入与
图灵或丘奇的定义等价。
算法随机性也发生了类似的事情，
我的前博士论文导师让-保罗德拉哈伊与丘奇-图灵论文相似。
所有随机性的定义
与以前的特征都是等价的。
事实上，像Per Martin-Löf，Greg Chaitin，Andrei
Kolmogorov，Leonid Levin，Clauss-Peter

English: 
captured by all these characterisations leading
to what is known today as the Church-Turing
thesis, that is the strong believe that any
practical definition of algorithm will collapse
into an equivalent definition to the one that
Turing or Church provided.
Something similar has happened with algorithmic
randomness leading to what Jean-Paul Delahaye,
my former PhD thesis advisor, calls the Martin-Löf-Chaitin
thesis similar to the Church-Turing thesis.
The Martin-Löf-Chaitin thesis is the thesis
that all definitions of randomness will be
equivalent to one of the previous characterisations.
And indeed, when people such as Per Martin-Löf,
Greg Chaitin, Andrei Kolmogorov, Leonid Levin,
Clauss-Peter Schnorr, among many others independently
proposed characterisations of randomness they

English: 
also found that all these definitions were
essentially the same as they were equivalent
among each other when the Church-Turing thesis
was assumed as almost everybody does in the
field also because the definitions of computation
appear very robust, and so not only each of
those definitions was able to characterise
intuitive notions of randomness such as compression,
predictability and typicality as discussed
before, but they also do so in a very general
and comprehensive way.
Notice, however, that statistical randomness
is not in the list of equivalent definitions
of randomness because surprisingly, even when
it is pervasive its use, misuse an abuse in
science, statistical randomness and measures
such as Shannon entropy are approaches that
do not provide the accepted mathematical characterisation
of randomness.

Chinese: 
Schnorr， 以及其他许多人独立提出随机性的特征，
他们发现所有这些定义基本上与Church-Turing论文相同,
假设为几乎所有人都在该领域时，因为计算的定义看起来非常强大，
因此不仅每个定义都能够表征直观的随机性概念，如压缩，
可预测性和典型性，但他们也以非常一般
和全面的方式这样做。
然而，请注意，统计随机性不在随机性的等效定义列表中，
因为令人惊讶的是， 它的普遍使用，
科学中的滥用，统计随机性和诸如香农熵等措施
也是随机性数学表征不能接受的方法。

English: 
So, we have seen how, according to algorithmic
complexity, if an object is random then it
is impossible to compress it.
We have also seen how compressibility is a
sufficient test for non-randomness, that is,
if you find a short computer program for some
data then you know that the data is not algorithmic
random.
On the other hand, we also briefly mentioned
the concept of lack of particular or special
property that we call 'typicality', so we
don't call random something that is atypical
because it can be described by using that
lack of typicality.
It turns out that this intuitive concept is
thus also related to other intuitive properties
of randomness, in particular you can see how
something being atypical can be used to compress
an object.
The basic idea is that if something is not
typical then the non-typical feature gives
you some sort of handle to pick that object
among more typical objects, which contradicts

Chinese: 
因此，我们已经看到，根据算法的复杂性，如果一个对象是随机的，
那么就不可能压缩它。
我们还看到可压缩对非随机性是一个充分测试，也就是说，
如果找到一些针对某些数据的简短计算机程序，
那么您就知道数据不是随机算法的。
另一方面，我们还简要地提到了缺乏特殊属性的概念，
我们将其称为“典型性”，因此我们不会将随机性称为非典型的东西，
因为它可以通过使用缺乏典型性来描述。
事实证明直观的概念因此也与随机性的其它直观的属性有关，
特别是您可以看到非典型的东西
如何用于压缩对象。
基本思想是，如果某些东西不典型，那么非典型特征
会提供某种处理方式，以便在更典型的对象中选择该对象，

Chinese: 
这与其随机化并将其与压缩概念相关联的直观概念相矛盾。
人们还可以为这些特征设计统计测试，
但首先要形式化允许的特征，例如所谓的递归特征，
这些特征可以通过计算机程序来表示，
这是可以通过传统统计来表征的特征，
因为计算机程序可以容易地捕获任何统计规律性，甚至低计算能力的
例如常规语言，但统计数据无法表征递归特征。
但它是瑞典数学家 Per Martin-Löf 和学生 Kolmogorov ，
设计了一个通用测试来测试任何递归或可计算特征的序列，

English: 
the intuitive idea that it is random and related
it to the concept of compression.
One can also devise statistical tests for
these kind of properties but one first thing
is to formalise the kind of allowed properties,
such as the so-called recursive properties,
those are properties that can be characterised
by computer program, which is a generalisation
of the properties that can be characterised
by traditional statistics given that computer
programs can easily capture any statistical
regularity with even low computational power
such as regular languages but statistics cannot
characterise recursive properties.
But it was Per Martin-Löf, the Swedish mathematician
and student of Kolmogorov himself, that devised
a universal test to test a sequence for any
recursive or computable property thereby technically

English: 
achieving another formal characterisation
of randomness.
As an example of a recursive property it can
be whether a sequence has an even number of
1s or whether the digits of a sequence are
the digits of a mathematical constant such
as the mathematical constant pi that comes
from a short computer program implementing
one of the many formulas that can generate
the digits of pi.
So random sequences can then be characterised
by failing to meet any property that a computer
program can code.
Finally, another characterisation from which
we started at the beginning in the list of
intuitive properties of randomness was that
of the unpredictability of a random sequence
similar to the characterisation of Shannon
entropy.
What Klaus Peter Schnorr and others mathematically
proved is that it is impossible to make money

Chinese: 
从技术上实现随机性的另一种形式表达。
作为递归属性的示例，可以是序列是否具有偶数个1
或者序列的数字是否是数学常量的数字，
例如来自一个短计算机程序的计算数学常数pi。
许多公式可以生成数字pi。
因此随机序列的特征在于
不能满足任何计算机程序可以编码的属性。
最后，我们从随机性的直观属性列表
开始的另一个特征是随机序列的不可预测性，
类似于香农熵的表征。
Klaus Peter Schnorr 和其它人在数学上证明，

English: 
by guessing the next digits in a truly random
sequence when using a recursive or computable
betting strategy, and that is something to
be expected but if you are using Shannon entropy
in practice it will fail because you can simply
produce a random sequence with no statistical
patterns but generated by a pseudo-random
generator and you can predict every digit
yet Shannon entropy would suggest is random.
However, it is when there are no predictable
patterns, statistical or computable, that
a sequence can truly be deemed random.
This convergence in definitions of mathematical
randomness removed from Shannon entropy means
that each definition assigns exactly the same
randomness as each other.
In other words, the extension of each definition
is the same; the set that each definition
characterises contain exactly the same objects,
thereby strongly suggesting that each definition

Chinese: 
当使用递归或可计算的投注策略时，通过猜测真正随机序列中的下一个数字是不可能赚钱的，
但是如果你使用的是 Shannon (香农) 熵, 这是可以预期的。
在实践中它会失败，因为你可以简单地生成一个没有统计模式
但是由伪随机生成器生成的随机序列，
尽管香农熵建议是随机的，你可以预测每一个数字位。
然而，当没有可预测的模式，统计或可计算的时，
序列可以真正被认为是随机的。
从香农熵中去除的数学随机性定义的这种收敛
意味着每个定义分配彼此完全相同的随机性。
换句话说，每个定义的扩展是相同的;
每个定义所表征的集合包含完全相同的对象，

English: 
has proven itself to be fundamental in a mathematical
way.
We can write this elegant result in a compact
manner as a causal chain:
incompressibility \[LeftRightArrow] unpredictability
\[LeftRightArrow] typicality
A series of universal results both in the
sense of being general and in the sense of
Turing-universality leads to the conclusion
that the definition of algorithmic randomness
is mathematically objective.
In summary:
- Martin-Löf proves that there is a universal
statistical test that can test for all computable

Chinese: 
从而强烈建议每个定义已经证明是数学方法的基础。
我们有一个因果链：
不可压缩性, 不可预测性的典型性
一系列普遍的结果,
在图灵 - 普遍性意义上导致算法随机性的定义
在数学上是客观的。
总结：
Martin-Löf 证明有一种普遍的统计测试,

Chinese: 
可以测试对象的可计算特征，但是不可计算或半可计算。
因此，他对随机性的定义是普遍的
足够包含所有有效的随机性测试。
Schnorr表示基于投注策略的可预测性方法引出
随机性的另一个特征，这反过来相当于Martin-Löf随机性。

English: 
properties of an object but is uncomputable
or semi-computable.
His definition of randomness is therefore
general enough to encompass all effective
tests for randomness.
- Schnorr shows that a predictability approach
based in betting strategies leads to another
characterisation of randomness, which in turn
is equivalent to Martin-Löf randomness.
