### 数学代考|计算复杂性理论代写computational complexity theory代考|Additive Cellular Automata

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## 数学代考|计算复杂性理论代写computational complexity theory代考|Notation and Formal Definitions

Let $S(\mathcal{L})=\left{s_{i}\right}$ be the set of lattice sites of a $d$-dimensional lattice $\mathcal{L}$ with $n_{r}$ equal to the number of lattice sites on dimension $r$. Denote by $\mathcal{A}$ a finite symbols set with $|\mathcal{A}|=p$ (usually prime). An $\mathcal{A}$-configuration on $\mathcal{L}$ is a surjective map $v: \mathcal{A} \mapsto S(\mathcal{L})$ that assigns a symbol from $\mathcal{A}$ to each site in $S(\mathcal{L})$. In this way, every $\mathcal{A}$-configuration defines a size $n_{1} \times \cdots \times n_{d}, d$-dimensional matrix $\mu$ of symbols drawn from $\mathcal{A}$. Denote the set of all $\mathcal{A}$-configurations on $\mathcal{L}$ by $\mathcal{E}(\mathcal{A}, \mathcal{L})$.

Each $s_{i} \in S(\mathcal{L})$ is labeled by an integer vector $\vec{i}=$ $\left(i_{1}, \ldots, i_{d}\right)$ where $i_{r}$ is the number of sites along the $r$ th dimension separating $s_{i}$ from the assigned origin in $\mathcal{L}$. The shift operator on the $r$ th dimension of $\mathcal{L}$ is the map $\sigma_{r}: \mathcal{L} \mapsto \mathcal{L}$ defined by
$$\sigma_{r}\left(s_{i}\right)=s_{j}, \quad \vec{j}=\left(i_{1}, \ldots, i_{r}-1, \ldots, i_{d}\right)$$
Equivalently, the shift maps the value at site $\vec{i}$ to the value at site $\vec{j}$.

Let $\mu\left(s_{i} ; t\right)=\mu\left(i_{1}, \ldots, i_{d} ; t\right) \in \mathcal{A}$ be the entry of $\mu$ corresponding to site $s_{i}$ at iteration $t$ for any discrete dynamical system having $\mathcal{E}(\mathcal{A}, \mathcal{L})$ as state space. Given a finite set of integer $d$-tuples $\mathcal{N}=\left{\left(k_{1}, \ldots, k_{d}\right)\right}$, define the

$$N\left(s_{i}\right)=\left{s_{j} \mid \vec{j}=\vec{i}+\vec{k}, \vec{k} \in \mathcal{N}\right}$$
A neighborhood configuration is a surjective map $y$ : $\mathcal{A} \mapsto N\left(s_{0}\right)$. Denote the set of all neighborhood configurations by $\mathcal{E}_{\mathcal{N}}(\mathcal{A})$.

The rule table for a cellular automata acting on the state space $\mathcal{E}(\mathcal{A}, \mathcal{L})$ with standard neighborhood $N\left(s_{0}\right)$ is defined by a map $x: \mathcal{E}{\mathcal{N}}(\mathcal{A}) \mapsto \mathcal{A}$ (note that this map need not be surjective or injective). The value of $x$ for a given neighborhood configuration is called the (value of the) rule component of that configuration. The map $x: \mathcal{E}{\mathcal{N}}(\mathcal{A}) \mapsto \mathcal{A}$ induces a global map $\mathcal{X}: \mathcal{E}(\mathcal{A}, \mathcal{L}) \mapsto$ $\mathcal{E}(\mathcal{A}, \mathcal{L})$ as follows: For any given element $\mu(t) \in$ $\mathcal{E}(\mathcal{A}, \mathcal{L})$, the set $C\left(s_{i}\right)=\left{\mu\left(s_{j} ; t\right) \mid s_{j} \in N\left(s_{i}\right)\right}$ is a neighborhood configuration for the site $s_{i}$, hence the map $\mu\left(s_{i} ; t\right) \mapsto x\left(C\left(s_{i}\right)\right)$ for all $s_{i}$ produces a new symbol $\mu\left(s_{i} ; t+1\right)$. The site $s_{i}$ is called the mapping site. When taken over all mapping sites, this produces a matrix $\mu(t+1)$ that is the representation of $\mathcal{X}(\mu(t))$. A cellular automaton is indicated by reference to its rule table or to the global map defined by this rule table.

A cellular automaton with global map $\chi$ is additive if and only if, for all pairs of states $\mu$ and $\beta$,
$$\chi(\mu+\beta)=\chi(\mu)+\chi(\beta)$$
Addition of states is carried out site-wise $\bmod (p)$ on the matrix representations of $\mu$ and $\beta$; for example, for a onedimensional six-site lattice with $p=3$ the sum of 120112 and 021212 is 111021 .

The definition for additivity given in [52] differs slightly from this standard definition. There, a binary valued cellular automaton is called “linear” if its local rule only involves the XOR operation and “additive” if it involves XOR and/or XNOR. A rule involving XNOR can be written as the binary complement of a rule involving only XOR. In terms of the global operator of the rule, this means that it has the form $1+X$ where $\mathcal{X}$ satisfies Eq. (3) and 1 represents the rule that maps every site to 1 . Thus, $(1+X)(\mu+\beta)$ equals $1 \ldots 1+\mathcal{X}(\mu+\beta)$ while
\begin{aligned} (1&+X)(\mu)+(1+X)(\beta) \ &=1 \ldots 1+1 \ldots 1+X(\mu)+\chi(\beta) \ &=X(\mu)+\chi(\beta) \bmod (2) \end{aligned}
In what follows, an additive rule is defined strictly as one obeying Eq. (3), corresponding to rules that are “linear” in [52].

Much of the formal study of cellular automata has focused on the properties and forms of representation of the map $\mathcal{X}: \mathbb{E}(\mathcal{A}, \mathcal{L}) \mapsto \mathcal{E}(\mathcal{A}, \mathcal{L})$. The structure of the state transition diagram $(\operatorname{STD}(\mathcal{X}))$ of this map is of particular interest.

## 数学代考|计算复杂性理论代写computational complexity theory代考|Boundary Conditions and Additivity

In the case of one-dimensional cellular automata, the lattice $\mathcal{L}$ can be isomorphic to the integers; to the nonnegative integers; to the finite set ${0, \ldots, n-1} \in Z$; or to the integers modulo an integer $n$. In the first case, there are no boundary conditions; in the remaining three cases, different boundary conditions apply. If $\mathcal{L}$ is isomorphic to $Z_{n}$, the integers $\bmod (n)$, the boundary conditions are periodic and the lattice is circular (it is a $p$-adic necklace). This is called a cylindrical cellular automata [77] because evolution of the rule can be represented as taking place on a cylinder. If the lattice is isomorphic to ${0, \ldots, n-1}$, null, or Dirchlet boundary conditions are set $[78,79,80]$. That is, the symbol assigned to all sites in $\mathcal{L}$ outside of this set is the null symbol. When the lattice is isomorphic to the non-negative integers $Z^{+}$, null boundary conditions are set at the left boundary. In these latter two cases, the neighborhood structure assumed may influence the need for null conditions.

Example 4 (Elementary Rule 90) Let $\delta$ represent the global map for the elementary cellular automata rule 90 , with rule table
$\begin{array}{cccccccc}000 & 001 & 010 & 011 & 100 & 101 & 110 & 111 \ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 0\end{array}$
For a binary string $\mu$ in $Z$ or $Z_{n}$ the action of rule 90 is defined by $[\delta(\mu)]{i}=\mu{i-1}+\mu_{i+1} \bmod (2)$, where all indices are taken $\bmod (n)$ in the case of $Z_{n}$. In the remaining cases,
\begin{aligned} {[\delta(\mu)]{i} } &= \begin{cases}\mu{1} & i=0 \ \mu_{i-1}+\mu_{i+1} & 0<i< \ \mu_{n-2} & i=n\end{cases} \ {[\delta(\mu)]{i} } &= \begin{cases}\mu{1} & i=0 \ \mu_{i-1}+\mu_{i+1} & 0<i\end{cases} \end{aligned}
half-infinite conditions
Note that $\mathcal{L}$ and $Z_{n}$ are representations of the intervals $[-1,1]$ and $[0,1]$ respectively. Cellular automata rules are not quite functions on these intervals, however, since they are generally double valued on rational points having distinct representations as binary strings [81]. For example, both $0 1$ and $1 0$ in $Z^{+}$, where underlining indicates infinite repetition, are numerically $1 / 2$ but $\delta(0 1)=1 0=3 / 4$ while $\delta(10)=01 0=1 / 4$.

The state space $\mathcal{E}({0,1}, \mathcal{L})$ for binary valued one-dimensional cellular automata is just the set of binary sequences over the specified one-dimensional lattice. For the

cases of $Z$ and $Z_{n}$ all such rules commute with the shift operator $\sigma$. When null boundary conditions are involved, however, commutativity fails at the boundary sites. For example, let $\mathcal{X}$ be the global operator for an elementary cellular automata operating on strings $\mu=\mu_{0} \ldots \mu_{n-1}$ of length $n$ with null boundary conditions. Noting that $-1=1 \bmod (2)$, the commutator $[\mathcal{X}, \sigma]$ has components
\begin{aligned} &{[\mathcal{X}, \sigma]{i}=[\mathcal{X} \sigma(\mu)+\sigma \mathcal{X}(\mu)]{i}} \ &= \begin{cases}\chi\left(0 \mu_{1} \mu_{2}\right)+X\left(\mu_{0} \mu_{1} \mu_{2}\right) \bmod (2) & i=0 \ 0 & 0<i<n-1 \ \mathcal{X}\left(\mu_{n-1} 00\right) & i=n-1\end{cases} \end{aligned}

## 数学代考|计算复杂性理论代写computational complexity theory代考|Additive Cellular Automata and Fractals

There is a direct connection between the space-time output patterns of additive cellular automata and self-similar fractal patterns $[82,83,84,85,86,87,88]$. The simplest examples are elementary rules 102 and 90 . When acting on a doubly infinite sequence with the initial state $0 1 0$, iteration of these rules yields the space-time output indicated in Fig. 1. In the case of rule 60, this output is the $\bmod (2)$ Pascal triangle while for rule 90 it consists of every other row of this triangle [89].

The pattern generated by rule 60 (or, equivalently, by rule 102) rescales to yield the fractal known as the Sirpinski gasket $[90,91]$. Direct connections between cellular automata outputs and the fractal generation schemes of matrix substitution systems and hierarchical iterated function systems are shown in $[92,93,94,95,96,97,98,99]$. In $[100,101]$ the dimension spectrum associated to the space-time output of additive cellular automata is shown to be equal to the singularity spectrum of an associated multifractal.

## 数学代考|计算复杂性理论代写computational complexity theory代考|Notation and Formal Definitions

σr(s一世)=sj,j→=(一世1,…,一世r−1,…,一世d)

χ(μ+b)=χ(μ)+χ(b)

[52] 中给出的可加性定义与这个标准定义略有不同。在那里，如果局部规则仅涉及 XOR 操作，则二进制值元胞自动机称为“线性”，如果涉及 XOR 和/或 XNOR，则称为“加法”。涉及 XNOR 的规则可以写成只涉及 XOR 的规则的二进制补码。就规则的全局运算符而言，这意味着它具有以下形式1+X在哪里X满足方程。(3) 和 1 表示将每个站点映射到 1 的规则。因此，(1+X)(μ+b)等于1…1+X(μ+b)尽管
(1+X)(μ)+(1+X)(b) =1…1+1…1+X(μ)+χ(b) =X(μ)+χ(b)反对(2)

## 数学代考|计算复杂性理论代写computational complexity theory代考|Boundary Conditions and Additivity

000001010011100101110111 01011010

[d(μ)]一世={μ1一世=0 μ一世−1+μ一世+10<一世< μn−2一世=n [d(μ)]一世={μ1一世=0 μ一世−1+μ一世+10<一世

\begin{aligned} &{[ \mathcal{X}, \sigma ]{i}=[\mathcal{X} \sigma(\mu)+\sigma \mathcal{X}(\mu)] {i}} \ &={χ(0μ1μ2)+X(μ0μ1μ2)反对(2)一世=0 00<一世<n−1 X(μn−100)一世=n−1 \end{对齐}

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