Emergence is a pregnant property in various fields. It is the fact for a
phenomenon to appear surprisingly and to be such that it seems at first sight
that it is not possible to predict its apparition. That is the reason why it
has often been said that emergence is a subjective property relative to the
observer. Some mathematical systems having very simple and deterministic rules
nevertheless show emergent behavior. Studying these systems shed a new light on
the subject and allows to define a new concept, computational irreducibility,
which deals with behaviors that even though they are totally deterministic
cannot be predicted without simulating them. Computational irreducibility is
then a key for understanding emergent phenomena from an objective point of view
that does not need the mention of any observer.

出现是各个领域中的一个重要属性，可以突如其来地出现，初看并不能预测其出现。因此被认为是相对于观察者的主观属性。一些具有简单和确定性规则的数学系统却展现出突出的行为。研究这些系统为对主观和客观不需要观察者的理解提供了新思路，使得描述计算不可简化成为理解出现现象的关键。

解释出现

Explaining Emergence

We explore several concepts for analyzing the intuitive notion of
computational irreducibility and we propose a robust formal definition, first
in the field of cellular automata and then in the general field of any
computable function f from N to N. We prove that, through a robust definition
of what means "to be unable to compute the nth step without having to follow
the same path than simulating the automaton or the function", this implies
genuinely, as intuitively expected, that if the behavior of an object is
computationally irreducible, no computation of its nth state can be faster than
the simulation itself.

本文从细胞自动机领域到任何可计算函数 f 的通用领域探讨了分析计算不可约性的几个概念，并提出了一个稳健的形式化定义；通过定义 “在没有遵循模拟自动机或函数的相同路径的情况下无法计算第 n 步骤” 这一概念，我们证明了如果一个对象的行为是计算不可约的，则其第 n 个状态的计算速度不可能比模拟本身更快。