### 数学代考|计算复杂性理论代写computational complexity theory代考|Transient Lengths and Cycle Periods

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## 数学代考|计算复杂性理论代写computational complexity theory代考|Transient Lengths and Cycle Periods

For any cellular automata acting on a finite state space, every state eventually maps to a fixed point or cycle. If a rule is injective, it is reversible and every state is a fixed point, or is on a cycle. If not injective, there will be states without predecessors, Garden-of-Eden states. As indicate, however, if a rule is additive its Garden-of-Eden states are spurious in the sense that they do have predecessors if the state space is enlarged.

The following theorem lists several significant properties of cellular automata rules acting on $\mathcal{E}(\mathcal{A}, \mathcal{Z})$ or $\mathcal{E}\left(\mathcal{A}, \mathcal{Z}^{+}\right)$with left justified neighborhoods.

Theorem 3 ([81]) Let $\mathcal{X}$ be a $k$-site cellular automata rule acting on $\mathcal{E}(\mathcal{A}, \mathcal{Z})$ or on $\mathcal{E}\left(\mathcal{A}, Z^{+}\right)$with left justified neighborhoods. Then the following statements are equivalent: (a) $X$ is surjective, (b) $\chi$ has an empty Garden-ofEden, (c) Every finite sequence $\mu_{0} \ldots \mu_{n-1}$ has exactly $p^{k-1}$ pre-images and every state $\mu$ has at most $p^{k-1}$ predecessors, (d) $\mathcal{X}$ maps eventually periodic states to eventually periodic states and non-periodic states to non-periodic states, (e) as a map of the interval $[0,1] X$ maps rationals to rationals and irrationals to irrationals.

If $\mathcal{X}: \mathcal{E}\left(\mathcal{A}, \mathcal{Z}{n}\right) \mapsto \mathcal{E}\left(\mathcal{A}, \mathcal{Z}{n}\right)$ is a $k$-site rule with $|\mathcal{A}|=p$ and either periodic or null boundary conditions, the state transition diagram, $\operatorname{STD}(\mathcal{X})$ is a graph with $p^{n}$ vertices labeled by the set of $p$-adic numbers $\left{i_{0}, \ldots, i_{n-1} \mid 0 \leq i_{r} \leq p-1\right}$. An edge is directed from the vertex $i_{0}, \ldots, i_{n-1}$ to the vertex $j_{0}, \ldots, j_{n-1}$ if and only if $\chi\left(i_{0}, \ldots, i_{n-1}\right)=j_{0}, \ldots, j_{n-1}$. Each state $\mu$ maps to a unique state $\mathcal{X}(\mu)$ so $\operatorname{STD}(\mathcal{X})$ consists of a set of trees rooted on fixed points or cycles. States at the top of trees are Garden-of-Eden states.

If $h(\mathcal{X}, n)$ is the maximum tree height, states at heights $h \leq h(\mathcal{X}, n)$ cannot appear after $h(\mathcal{X}, n)-h+1$ iterations and after $h(\mathcal{X}, n)$ iterations only fixed points and states on cycles remain. Thus, iteration of a non-injective rule on $\mathcal{E}\left(\mathcal{A}, \mathcal{Z}{n}\right)$ decreases the number of available states with a corresponding reduction in entropy. On the other hand, non-injective additive rules acting on $\mathcal{E}\left(\mathcal{A}, Z^{+}\right)$ do not reduce entropy [117] even though the do so on $\mathcal{E}\left(\mathcal{A}, Z{n}\right)$ for all $n$. The explanation for this apparent paradox is that the Garden-of-Eden states that appear in $\mathcal{E}\left(\mathcal{A}, Z_{n}\right)$ are artifacts of the finite length of states in this space. When embedded in $Z^{+}$, states in $Z_{n}$ correspond to periodic configurations, hence to rational numbers in $[0,1]$ and the set of all rationals has measure 0 in the reals.

Parameters of interest for characterizing state transition diagrams of rules acting on $\mathcal{E}\left(\mathcal{A}, Z_{n}\right)$ are the maximum tree height $h(\mathcal{X}, n)$ and the cycle periods $c_{s}(\mathcal{X}, n)$

## 数学代考|计算复杂性理论代写computational complexity theory代考|Computing Predecessor States

A problem of general interest for cellular automata is computation of predecessor states. For a rule $\mathcal{X}: \mathbb{E}(\mathcal{A}, \mathcal{L}) \mapsto \mathcal{E}(\mathcal{A}, \mathcal{L})$ with a state $\beta$ given this requires solution of the equation $\mathcal{X}(\mu)=\beta$. It is always possible to construct solutions for this equation, or to show that none exist by a method of backward reconstruction based on the rule table.

Example 7 (Rule 60 Acting on $Z_{4}$ With Periodic Boundary Conditions) Rule 60 is a 2 -site rule, defined by $(00,11) \mapsto 0,(01,10) \mapsto 1$. Given the state 0110 the predecessors of this state can be computed as follows:

1. The initial 0 in 0110 can arise from either 00 or 11 .
2. Starting with a 00, the next symbol in 0110 is a 1 and this can arise from a 01 or a 10 , but this must also connect to the original 00 so only 01 is allowed, giving 001 . Starting from a 11, on the other hand, the same reasoning requires 110 .
3. The third symbol in 0110 is also a 1 . To be consistent with 001 requires that 10 be selected, and to be consistent with 110 requires that 01 be selected, thus giving the two partially constructed possibilities as 0010 and 1101 .
4. Finally, the fourth symbol must be a 0 . This requires that the predecessor string conclude with either 00 or 11. Since the strings are in $Z_{4}$ with periodic boundary conditions, the final symbol in the predecessor string must also be the first symbol in that string. Thus, both 0010 and 1101 are seen to be predecessors of 0110 .
Other ways of computing predecessor states for finite strings is through the construction of a rule matrix [81] or the use of de Bruijn diagrams $[81,113]$. Backward reconstruction, the rule matrix, and use of a de Bruijn diagram are valid methods for computing predecessor states for all one-dimensional rules. For additive rules, how ever, there is an analytic means for computing predecessor states, starting from left justified neighborhoods defined on $\mathcal{E}\left(\mathcal{A}, \mathcal{Z}{n}\right)$ or $\mathcal{E}\left(\mathcal{A}, Z^{+}\right)[81,128]$. This can be illustrated for rules defined on $\mathcal{E}\left({0,1}, Z^{+}\right)$. This method also works for rules defined on $\mathcal{E}\left({0,1}, Z{n}\right)$ if it is embedded in $\mathcal{E}\left({0,1}, Z^{+}\right)$as the subset of halfinfinite periodic sequences with periods that divide $n$. Define operators $\mathcal{B}: \mathcal{E}\left({0,1}, Z^{+}\right) \mapsto \mathcal{E}\left({0,1}, Z^{+}\right)$and $\sigma^{-1}: \mathcal{E}\left({0,1}, Z^{+}\right) \mapsto \mathcal{E}\left({0,1}, Z^{+}\right)$by
$$\begin{gathered} {[B(\mu)]{s}=\sum{i=0}^{s} \mu_{i} \bmod (2)} \ {\left[\sigma^{-1}(\mu)\right]{s}= \begin{cases}0 & s=0 \ \mu{s-1} & s>0\end{cases} } \end{gathered}$$

## 数学代考|计算复杂性理论代写computational complexity theory代考|$d$-Dimensional Rules

Both $[102,105]$ discuss the extension from one-dimensional to $d$-dimensional rules defined on tori. In [102] this discussion uses a formalism of multinomials defined over finite fields. In [105], the one-dimensional analysis based on circulant matrices is generalized. The matrix formulism of state transitions is retained by defining a $d$-fold “circulant of circulants,” which is not, of itself, necessarily a circulant. Computation of the the non-zero eigenvalues of this matrix yields results on transient lengths and cycle periods.

More recently, an extensive analysis of additive rules defined on multi-dimensional tori has appeared [129]. A $d$-dimensional integer vector $\vec{n}=\left(n_{1}, \ldots, n_{d}\right)$ defines a discrete toridal lattice $\mathcal{L}(\vec{n})$. Every $d$-dimensional matrix of size $\vec{n}$ with entries in $\mathcal{A},|\mathcal{A}|=p$ (prime), defines an additive rule acting on $\mathcal{E}(\mathcal{A}, \mathcal{L}(\vec{n}))$ as follows: Let $\mathcal{T}$ and $\mu(t)$ be elements of $\mathcal{E}(\mathcal{A}, \mathcal{L}(\vec{n}))$ with $X$ the rule defined by $\mathcal{T}$ and $\mu(t)$ a state at time $t$. The state transition defined by $\mathcal{X}$ is $\mu(t+1)=\mathcal{X}(\mu(t))$ and this is given by
$$\begin{array}{r} {[\mu(t+1)]{i{1} \ldots i_{d}}=\sum_{k_{1}, \ldots, k_{d}}[C(\mathcal{T})]{i{1} \ldots i_{d}}^{k_{1} \ldots k_{d}}[\mu(t)]{k{1} \ldots k_{d}}} \ {[C(\mathcal{T})]{i{1} \ldots i_{d}}^{k_{1} \ldots k_{d}}=\mathcal{T}{j{1} \ldots j_{d}} \quad j_{s}=k_{s}-i_{s} \bmod \left(n_{s}\right)} \end{array}$$

The matrix $C(T)$ is the $d$-dimensional generalization of a circulant matrix with $T$ as the equivalent of its first row. For example, if $d=1$ and $p=2$ with $\mathcal{T}=(0,1,0,0,0,1)$ this defines the additive rule $\sigma+\sigma^{5}$ (rule 90 ) and the matrix $C(T)$ is given in Eq. (10a).

Let $S$ and $\mathcal{T}$ be elements of $\mathcal{E}(\mathcal{A}, \mathcal{L}(\vec{n}))$ and define the binary operation $\psi: \mathcal{E}(\mathcal{A}, \mathcal{L}(\vec{n})) \times \mathcal{E}(\mathcal{A}, \mathcal{L}(\vec{n}))$ $\mapsto \mathcal{E}(\mathcal{A}, \mathcal{C}(\vec{n}))$ by
\begin{aligned} {[\psi(S, \mathcal{T})]{i{1} \ldots i_{d}}=} & \sum_{k_{1}, \ldots, k_{d}} S_{k_{1} \ldots k_{d}} \mathcal{T}{i{1}-k_{1} \ldots i_{d}-k_{d}} \ & 0 \leq k_{s}<n_{s} \end{aligned}
with all sums taken $\bmod (p)$.

## 数学代考|计算复杂性理论代写computational complexity theory代考|Computing Predecessor States

1. 0110 中的初始 0 可以来自 00 或 11 。
2. 从 00 开始，0110 中的下一个符号是 1，这可以从 01 或 10 产生，但这也必须连接到原始 00，因此只允许 01，给出 001。另一方面，从 11 开始，同样的推理需要 110 。
3. 0110 中的第三个符号也是 1 。与 001 一致需要选择 10，而与 110 一致则需要选择 01，因此给出了 0010 和 1101 两种部分构造的可能性。
4. 最后，第四个符号必须是 0 。这要求前导字符串以 00 或 11 结尾。由于字符串在Z4在周期性边界条件下，前导字符串中的最后一个符号也必须是该字符串中的第一个符号。因此， 0010 和 1101 都被视为 0110 的前身。
计算有限字符串的前驱状态的其他方法是通过构建规则矩阵 [81] 或使用 de Bruijn 图[81,113]. 后向重构、规则矩阵和使用 de Bruijn 图是计算所有一维规则的先行状态的有效方法。然而，对于加法规则，有一种用于计算先行状态的分析方法，从定义的左对齐邻域开始E(A,Zn)或者E(A,Z+)[81,128]. 这可以用定义的规则来说明E(0,1,Z+). 此方法也适用于定义的规则E(0,1,Zn)如果它嵌入E(0,1,Z+)作为半无限周期序列的子集，其周期为n. 定义运算符B:E(0,1,Z+)↦E(0,1,Z+)和σ−1:E(0,1,Z+)↦E(0,1,Z+)经过
[B(μ)]s=∑i=0sμimod(2) [σ−1(μ)]s={0s=0 μs−1s>0

## 数学代考|计算复杂性理论代写computational complexity theory代考|d-Dimensional Rules

[ 102,105都讨论了从一维到维规则的扩展，定义在 tori 上。在[102]中，这个讨论使用了在有限域上定义的多项式的形式。在[105]中，基于循环矩阵的一维分析得到了推广。状态转换的矩阵公式通过定义一个倍的“循环的循环”来保留，它本身不一定是循环的。计算该矩阵的非零特征值会产生瞬态长度和循环周期的结果。[102,105]dd

[μ(t+1)]i1…id=∑k1,…,kd[C(T)]i1…idk1…kd[μ(t)]k1…kd [C(T)]i1…idk1…kd=Tj1…jdjs=ks−ismod(ns)

mod(p)

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