更改

=====Specific example=====

=====Specific example=====

[[文件:Examples of causal decoupling and downward causation.png|居左]]

[[文件:Examples of causal decoupling and downward causation.png|居左]]

The author of the paper <ref name=":5" /> lists a specific example (as above), to demonstrate the condition under which [[causal decoupling]], [[downward causation]] and [[causal emergence]] take place. This example is a special Markov process. Here, <math>p_{X_{t + 1}|X_t}(x_{t + 1}|x_t)</math> represents the dynamic relationship, and <math>X_t=(x_t^1,…,x_t^n)\in\{0,1\}^n</math> is the microstate. The definition of this process is to determine the probability of taking different values of the state <math>x_{t + 1}</math> at the next moment by checking the values of the variables <math>x_t</math> and <math>x_{t + 1}</math> at two consecutive moments. This involves determine if the sum modulo 2 of all dimensions of <math>x_t</math> is the same as the first dimension of <math>x_{t + 1}</math>: if they are different, the probability is 0; otherwise, judge whether <math>x_t,x_{t + 1}</math> have the same sum modulo 2 value in all dimensions. If both conditions are satisfied, the value probability is <math>\gamma/2^{n - 2}</math>, otherwise the value probability is <math>(1-\gamma)/2^{n - 2}</math>. Here <math>\gamma</math> is a parameter and <math>n</math> is the total dimension of x.

Here, <math>p_{X_{t + 1}|X_t}(x_{t + 1}|x_t)</math> represents the micro-dynamics, and <math>X_t=(x_t^1,…,x_t^n)\in\{0,1\}^n</math> is the micro-state. The definition of this process is to determine the probability of taking different values of the state <math>x_{t + 1}</math> at the next moment by checking the values of the variables <math>x_t</math> and <math>x_{t + 1}</math> at two consecutive moments. This involves determining if the sum modulo 2 of all dimensions of <math>x_t</math> is the same as the first dimension of <math>x_{t + 1}</math>: if they are different, the probability is 0; otherwise, it will judge whether <math>x_t,x_{t + 1}</math> have the same sum modulo 2 value in all dimensions. If both conditions are satisfied, the value of probability is <math>\gamma/2^{n - 2}</math>, otherwise the probability is <math>(1-\gamma)/2^{n - 2}</math>. Here <math>\gamma</math> is a parameter and <math>n</math> is the total dimension of [math]x[/math].

In fact, if <math>\sum_{j = 1}^n x^j_t</math> is even or 0, then <math>\oplus^n_{j = 1} x^j_t:=1</math>, otherwise <math>\oplus^n_{j = 1} x^j_t:=0</math>. Therefore, the result of <math>\oplus^n_{j = 1} x^j_t</math> is the parity of the entire X sequence, and the first dimension can be regarded as a parity check bit. <math>\gamma</math> actually represents the probability that a mutation occurs in two bits of the X sequence. This mutation can ensure that the parity of the entire sequence remains unchanged, and the parity check bit of the sequence also conforms to the actual parity of the entire sequence.

In fact, if <math>\sum_{j = 1}^n x^j_t</math> is even or 0, then <math>\oplus^n_{j = 1} x^j_t:=1</math>, otherwise <math>\oplus^n_{j = 1} x^j_t:=0</math>. Therefore, the result of <math>\oplus^n_{j = 1} x^j_t</math> is the parity of the entire [math]X[/math] sequence, and the first dimension can be regarded as a parity check bit. <math>\gamma</math> actually represents the probability that a mutation occurs in two bits of the [math]X[/math] sequence. This mutation can ensure that the parity of the entire sequence remains unchanged, and the parity check bit of the sequence also conforms to the actual parity of the entire sequence.

Therefore, the macroscopic state of this process can be regarded as the parity of the overall parity of the sequence, and the probability distribution of this parity is the result of the XOR calculation of the microstate. <math>x_{t + 1}^1</math> is a special microstate that always remains consistent with the macroscopic state of the sequence at the previous moment. Therefore, when only the first item in the second judgment condition is satisfied, the downward causation of the system occurs. When only the second item is satisfied, the causal decoupling of the system occurs. When both items are satisfied simultaneously, it is said that causal emergence occurs in the system.

Therefore, the macroscopic state of this process can be regarded as the overall parity of the sequence, and the probability distribution of this parity is the result of the XOR calculation of the micro-state. <math>x_{t + 1}^1</math> is a special micro-state that always remains consistent with the macro-scopic state of the sequence at the previous moment. Therefore, when only the first item in the second judgment condition is satisfied, the downward causation of the system occurs. When only the second item is satisfied, the causal decoupling of the system occurs. When both items are satisfied simultaneously, it is said that causal emergence occurs in the system.

====Causal emergence theory based on singular value decomposition====

====Causal emergence theory based on singular value decomposition====