Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations

1 Introduction

Cellular Automaton(pl. cellular automata, abbrev. CA) is a discrete model which has applications in computer science, mathematics, physics, complexity science, theoretical biology and microstructure modeling. This model was introduced by J.von Neumann and S.Ulam in 1940 for designing self replicating systems von1966theory ; schiff2011cellular ; ulam1962some .

A CA consists of a finite/countably infinite number of finite-state semi-automata ghosh2016evo ; ghosh2016finite known as ‘cells’ arranged in an ordered $n$ -dimensional grid. Each cell receives input from the neighbouring cells and changes according to the transition function. The transitions at each of the cells together induces a change of the grid pattern. The simplest CA is a CA where the grid is a one-dimensional line. Stephen Wolfram’s work in the 1980s contributed to a systematic study of one-dimensional CA, providing the first qualitative classification of their behaviour wolfram2002new . CA has been studied for solving many interesting problems on utilizing mathematical bases such as polynomial, matrix algebra, Boolean derivative etc.choudhury2009investigation ; das1992characterization . Further, dynamic behaviour of CA can also be studied using various mathematical tools wuensche1997attractor ; wuensche2003discrete ; xu2009dynamical ; edwards2019class , that help to understand the crucial properties and modeling of various classes of discrete dynamical system edwards2019class .

An Integral Value Transformation (abbrev. IVT), a class of discrete dynamical system, were first introduced during 2009-10 (hassan2010collatz ). A IVT of $k$ -dimension is a function defined using $p$ -adic numbers over $N_{0}^{k}$ where $N_{0} = N \cup {0}$ , $p \in N$ . The IVTs have been studied in different viewpoints including the understanding of the evolution of integer sequences and behavioral patterns of integers in discrete time points hassan2011integral ; das2017two .

In this paper, Wolfram code of an elementary CA and global transition function of a one-dimensional CA has been represented through IVTs. The algebraic structure of a set of transition functions of a one-dimensional CA as well as the algebraic structure of a set of IVTs under some binary operations have been studied. In the last section, some specific applications have been discussed in which we have seen certain novelty lies in IVTs over one-dimensional CA.

2 Mathematical preliminaries

Definition 0.

Let $Q$ be a finite set of memory elements also called the state set.
A global configuration is a mapping from the group of integers $Z$ to the set $Q$ given by $C : Z \to Q$ . The set $Q^{Z}$ is the set of all global configurations where $Q^{Z} = {C | C : Z \to Q}$ .

Definition 0.

A mapping $τ : Q^{Z} \to Q^{Z}$ is called a global transition function.
A CA(denoted by $C_{τ}^{Q}$ )(reported in ghosh2017some ; kari2005theory ) is a triplet $(Q, Q^{Z}, τ)$ , where,

$Q$ is the finite state set
$Q^{Z}$ is the set of all configurations
$τ$ is the global transition function

Definition 0.

The set $Q^{Z} = {τ | τ : Q^{Z} \to Q^{Z}}$ is the set of all possible global transition functions of CA having state set $Q$ .

A mapping $τ$ is invertible if $\forall C_{i}, C_{j} \in Q^{Z}$ , $\exists τ^{- 1}$ such that

τ (C_{i}) = C_{j} \Leftrightarrow τ^{- 1} (C_{j}) = C_{i}

Definition 0.

For $i \in Z, r \in N$ , let $S_{i} = {i - r, . . ., i - 1, i, i + 1, . . ., i + r} \subseteq Z$ . $S_{i}$ is the neighbourhood of the $i^{t h}$ cell, $r$ is the radius of the neighbourhood of a cell. It follows that $Z = ⋃_{i} S_{i}$ .
A restriction from $Z$ to $S_{i}$ induces a restriction of $C$ to $_{i}$ given by $_{i} : S_{i} \to Q$ ; where $_{i}$ may be called local configuration of the $i^{t h}$ cell.
The mapping $μ_{i} : Q^{S_{i}} \to Q$ is known as a local transition function for the $i^{t h}$ cell having radius $r$ . Thus $\forall i \in Z, μ_{i} (_{i}) \in Q$ and it follows that,

τ (C) = τ (. . ., c_{i - 1}, c_{i}, c_{i + 1}, . . .) = . . . . . μ_{i - 1} (_{i - 1}) . μ_{i} (_{i}) . μ_{i + 1} (_{i + 1}) . . . . . . .

Definition 0.

The set $M = {μ_{i} | μ_{i} : Q^{S_{i}} \to Q, i \in Z}$ is the set of all possible local transition functions of CA having state set $Q$ .

Definition 0.

If for a particular CA, $| Q | = 2$ so that we can write $Q = {0, 1}$ , then the CA is said to be a binary CA or a Boolean CA. A Boolean CA having radius $1$ is known as an Elementary CA(ECA).

Definition 0.

Wolfram code is a naming system often used for a one-dimensional CA introduced by Stephen Wolfram (seewolfram2002new ).
For a one-dimensional CA with $Q$ states, the local rule $μ_{i}$ for some $i^{t h}$ cell, $i \in Z$ of radius $r$ (neighbourhood $2 r + 1$ ) can be specified by an $Q^{2 r + 1}$ -bit sequence. The decimal equivalent form of this sequence is known as the Wolfram code.

Thus, the Wolfram code for a particular rule is a number in the range from $0$ to $Q^{Q^{2 r + 1}} - 1$ , converted from $Q$ -ary to decimal notation.

Example 2.1.

Let the local rule of an ECA for some $i^{t h}$ cell, $i \in Z$ be,

μ_{i} (_{i}) = μ_{i} (c_{i - 1}, c_{i}, c_{i + 1}) = (c_{i - 1} \lor c_{i + 1}) \land c_{i}

where $^{'} \lor^{'}$ stands for OR operation, $^{'} \land^{'}$ stands for AND operation, $c_{j}$ is the $j^{t h}$ cell configuration for $j = i - 1, i, i + 1$ .
Then we get the $2^{3}$ -bit sequence as,

μ (111) μ (110) μ (101) μ (100) μ (011) μ (010) μ (001) μ (000) = 11001000

The decimal equivalent number for $11001000$ is $200$ and so the Wolfram code is RULE 200.

Definition 0.

A $p$ -adic $k -$ dimensional Integral Value Transformation(IVT) denoted by $I V T_{j}^{p, k}$ for $p \in N$ , $k \in N$ is a function of $p$ -base numbers from $N_{0}^{k}$ to $N_{0}$ defined(in hassan2012onedmensional ) as

I V T_{j}^{p, k} (n_{1}, n_{2}, . . ., n_{k}) = (f_{j} (a_{0}^{n_{1}}, . . ., a_{0}^{n_{k}}) f_{j} (a_{1}^{n_{1}}, . . ., a_{1}^{n_{k}}) . . . f_{j} (a_{l - 1}^{n_{1}}, . . ., a_{l - 1}^{n_{k}}))_{p} = m

where $N_{0} = N \cup {0}$ , $p \in N$ , $n_{s} = (a_{0}^{n_{s}} a_{1}^{n_{s}} . . . a_{l - 1}^{n_{s}})_{p}$ for $s = 1, 2, . . ., k$ .
$f_{j}$ is a function from ${0, 1, . . ., p - 1}^{k}$ to ${0, 1, . . ., p - 1}$ for $j = 0, 1, . . ., p^{p^{k}} - 1$ ,
$m$ is the decimal conversion of the $p$ -base number.

Remark 2.0.

In particular(discussed in choudhury2009theory ; choudhury2011act ; das2016multi ; pal2012properties ), if $\forall i = 0, 1, . . ., l - 1$ , the function $f_{j}$ be defined as

$f_{j} (a_{i}^{n_{1}}, . . ., a_{i}^{n_{k}}) = ⌊ (a_{i}^{n_{1}} + . . . + a_{i}^{n_{k}}) / p ⌋$ , then $I V T_{j}^{p, k}$ is known as a Modified Carry Value Transformation(MCVT).
Again, $M C V T$ with a $0$ padding at the right end is known as a Carry Value Transformation(CVT).
$f_{j} (a_{i}^{n_{1}}, . . ., a_{i}^{n_{k}}) = (a_{i}^{n_{1}} + . . . + a_{i}^{n_{k}}) m o d p$ , then $I V T_{j}^{p, k}$ is known as an Exclusive OR Transformation(XORT).
$f_{j} (a_{i}^{n_{1}}, . . ., a_{i}^{n_{k}}) = m a x (a_{i}^{n_{1}}, . . ., a_{i}^{n_{k}})$ , then $I V T_{j}^{p, k}$ is known as an Extreme Value Transformation(EVT).

3 Wolfram code of an ECA and IVT

For an ECA, Wolfram code for a local rule $μ$ is the decimal number $j \in {0, 1, 2, . . ., 255}$ obtained from the $8$ -bit sequence

μ (111) μ (110) μ (101) μ (100) μ (011) μ (010) μ (001) μ (000) = j

Therefore, Wolfram code for each local transition function of a $3$ -neighbourhood Boolean CA can be represented by a $2$ base $3$ -dimensional IVT.
The function $μ$ in the above $8$ -bit sequence can be represented by a function $f_{j} : {0, 1}^{3} \to {0, 1}$ which gives

(f_{j} (111) f_{j} (110) f_{j} (101) f_{j} (100) f_{j} (011) f_{j} (010) f_{j} (001) f_{j} (000))_{2}

= I V T_{j}^{2, 3} (11110000_{2}, 11001100_{2}, 10101010_{2}) = I V T_{j}^{2, 3} (240_{10}, 204_{10}, 170_{10})

Hence it follows that for a $3 -$ neighbourhood Boolean CA, any Wolfram code $j \in {0, 1, 2, . . .255}$ can be equivalently represented by $I V T_{j}^{2, 3} (240_{10}, 204_{10}, 170_{10})$ .

Example 3.1.

Wolfram code 200 can be equivalently represented as

200_{10} = (11001000)_{2} = I V T_{200}^{2, 3} (240_{10}, 204_{10}, 170_{10})

However the following example shows that for $j \in {0, 1, 2, . . .255}$ , any $I V T_{j}^{2, 3}$ may not correspond to a Wolfram code.

Example 3.2.

I V T_{j}^{2, 3} (240_{10}, 204_{10}, 171_{10}) = I V T_{j}^{2, 3} (11110000_{2}, 11001100_{2}, 10101011_{2})

= (f_{j} (111) f_{j} (110) f_{j} (101) f_{j} (100) f_{j} (011) f_{j} (010) f_{j} (001) f_{j} (001))_{2}

In this $8$ -bit sequence, $f_{j} (000)$ is missing and so this cannot correspond to any Wolfram code.

3.1 Some Particular Wolfram codes and Particular IVTs

We know that transformations such as MCVT, XORT, EVT are particular cases of IVTs. Again, any Wolfram code can be represented by an IVT. Therefore some particular Wolfram codes which can be represented by an MCVT, XORT or EVT are as follows.

Let the local rule of an ECA for some $i^{t h}$ cell, $i \in Z$ be,

$μ_{i} (_{i}) = μ_{i} (c_{i - 1}, c_{i}, c_{i + 1}) = (c_{i - 1} \land c_{i}) \lor ((c_{i - 1} \lor c_{i}) \land c_{i + 1})$

Then we get the $2^{3}$ -bit sequence as $11101000$ and Wolfram code $232$ .
Here, $μ_{i}$ can be represented by function $f_{j} : {0, 1}^{3} \to {0, 1}$ given by

$f_{j} (c_{i - 1}, c_{i}, c_{i + 1}) = (c_{i - 1} \land c_{i}) \lor ((c_{i - 1} \lor c_{i}) \land c_{i + 1}) = ⌊ (c_{i - 1} + c_{i} + c_{i + 1}) / 2 ⌋$

Thus, Wolfram code $232$ can be represented as

$(f_{232} (111) f_{232} (110) f_{232} (101) f_{232} (100) f_{232} (011) f_{232} (010) f_{232} (001) f_{232} (000))_{2}$

$= (⌊ 3 / 2 ⌋ ⌊ 2 / 2 ⌋ ⌊ 2 / 2 ⌋ ⌊ 1 / 2 ⌋ ⌊ 2 / 2 ⌋ ⌊ 1 / 2 ⌋ ⌊ 1 / 2 ⌋ ⌊ 0 / 2 ⌋)_{2}$

$= M C V T_{232}^{2, 3} (11110000_{2}, 11001100_{2}, 10101010_{2}) = M C V T_{232}^{2, 3} (240_{10}, 204_{10}, 170_{10})$
Let the local rule of an ECA for some $i^{t h}$ cell, $i \in Z$ be,

$μ_{i} (_{i}) = μ_{i} (c_{i - 1}, c_{i}, c_{i + 1}) = (c_{i - 1} \lor - - c_{i} \lor - - c_{i + 1})$

Then we get the $2^{3}$ -bit sequence as $10010110$ and Wolfram code $150$ .
Here, $μ_{i}$ can be represented by function $f_{j} : {0, 1}^{3} \to {0, 1}$ given by

$f_{j} (c_{i - 1}, c_{i}, c_{i + 1}) = (c_{i - 1} \lor - - c_{i} \lor - - c_{i + 1}) = (c_{i - 1} + c_{i} + c_{i + 1}) m o d 2$

Thus, Wolfram code $150$ can be represented as

$(f_{150} (111) f_{150} (110) f_{150} (101) f_{150} (100) f_{150} (011) f_{150} (010) f_{150} (001) f_{150} (000))_{2}$

$= ((3 m o d 2) (2 m o d 2) (2 m o d 2) (1 m o d 2) (2 m o d 2) (1 m o d 2) (1 m o d 2) (0 m o d 2))_{2}$

$= X O R T_{150}^{2, 3} (11110000_{2}, 11001100_{2}, 10101010_{2}) = X O R T_{150}^{2, 3} (240_{10}, 204_{10}, 170_{10})$
Let the local rule of an ECA for some $i^{t h}$ cell, $i \in Z$ be,

$μ_{i} (_{i}) = μ_{i} (c_{i - 1}, c_{i}, c_{i + 1}) = (c_{i - 1} \lor c_{i} \lor c_{i + 1})$

Then we get the $2^{3}$ -bit sequence as $11111110$ and Wolfram code $254$ .
Here, $μ_{i}$ can be represented by function $f_{j} : {0, 1}^{3} \to {0, 1}$ given by

$f_{j} (c_{i - 1}, c_{i}, c_{i + 1}) = (c_{i - 1} \lor c_{i} \lor c_{i + 1}) = m a x {c_{i - 1}, c_{i}, c_{i + 1}}$

Thus, Wolfram code $254$ can be represented as

$(f_{254} (111) f_{254} (110) f_{254} (101) f_{254} (100) f_{254} (011) f_{254} (010) f_{254} (001) f_{254} (000))_{2}$

$= (11111110)_{2}$

$= E V T_{254}^{2, 3} (11110000_{2}, 11001100_{2}, 10101010_{2}) = E V T_{254}^{2, 3} (240_{10}, 204_{10}, 170_{10})$

4 Transition Function of a CA and IVT

Definition 0.

The set $T^{p, k} = {I V T_{j}^{p, k} | I V T_{j}^{p, k} : N_{0}^{k} \to N_{0}}$ is the set of all $p$ -base $k$ -dimensional IVTs for $p \in N, k \in N$ .

Definition 0.

The $v$ -fold cartesian product $T^{p, k} \times . . . \times T^{p, k}      v t i m e s$ for $v \in N$ , denoted by $T^{v}$ is given by

T^{v} = {(I V T_{j_{1}}^{p, k}, I V T_{j_{2}}^{p, k}, . . ., I V T_{j_{v}}^{p, k})}

where $I V T_{j_{s}}^{p, k} : N_{0}^{k} \to N_{0} \in T^{p, k}$ , for $s = 1, 2, . . ., v (\in N)$

Definition 0.

Let a restriction on $I V T_{j}^{p, k}$ from $N_{0}^{k}$ to $Q^{k} = {0, 1, . . ., p - 1}^{k}$ for $p \in N$ be denoted by ${I V T - --- -}_{j}^{p, k}$ . The set $T_{| Q}^{p, k} = {{I V T - --- -}_{j}^{p, k} | {I V T - --- -}_{j}^{p, k} : Q^{k} \to Q}$ is the set of all $p$ -base $k$ -dimensional IVTs when $N_{0}^{k}$ is restricted to the subset $Q^{k}$ . Therefore for $(η_{1}, η_{2}, . . ., η_{k}) \in Q^{k}$ we get,

{I V T - --- -}_{j}^{p, k} (η_{1}, η_{2}, . . ., η_{k}) = f_{j} (η_{1}, η_{2}, . . ., η_{k})_{p} = m \in Q

Definition 0.

The set $T_{| Q}^{v} = {({I V T - --- -}_{j_{1}}^{p, k}, {I V T - --- -}_{j_{2}}^{p, k}, . . ., {I V T - --- -}_{j_{v}}^{p, k})}$ is the $v$ -fold cartesian product $T_{| Q}^{p, k} \times . . . \times T_{| Q}^{p, k}      v t i m e s$ when $N_{0}^{k}$ is restricted to subset $Q^{k}$ where ${I V T - --- -}_{j_{s}}^{p, k} : Q^{k} \to Q \in T_{| Q}^{p, k}$ for $s = 1, 2, . . ., v (\in N)$

Now, a local transition function of any $i^{t h}$ cell of a one-dimensional CA having $p (\in N)$ states and radius $r (\in N)$ will be of the form

μ_{i} (_{i}) = μ_{i} (c_{i - r}, . . ., c_{i}, . . ., c_{i + r}) = c_{i}^{*}

where $c_{i - r}, . . ., c_{i}, . . ., c_{i + r}, c_{i}^{*} \in {0, 1, . . ., p - 1}$ .
This can be represented by some ${I V T - --- -}_{j_{i}}^{p, k}$ , where $p \in N, k = 2 r + 1$ and $j_{i} \in {0, 1, 2, . . ., (p^{p^{k}} - 1)}$ is the underlying Wolfram code(which is in turn equal to $I V T_{j_{i}}^{p, k} (240_{10}, 204_{10}, 170_{10})$ ).
Thus $\forall i \in Z$ ,

μ_{i} (_{i}) \equiv {I V T - --- -}_{j_{i}}^{p, k} (_{i})

Now if $τ$ be the global transition function of any $v (\in N \geq k)$ -celled CA, then for a global configuration $C = (c_{1}, c_{2}, . . ., c_{v})$ , we get

τ (C) = τ (c_{1}, c_{2}, . . ., c_{v}) = μ_{1} (_{1}) . μ_{2} (_{2}) . . . . μ_{v} (_{v})

Therefore it follows that,

τ (c_{1}, c_{2}, . . ., c_{v}) \equiv ({I V T - --- -}_{j_{1}}^{p, k} (_{1}), {I V T - --- -}_{j_{2}}^{p, k} (_{2}), . . ., {I V T - --- -}_{j_{v}}^{p, k} (_{v}))

Hence for any $v$ -celled ECA it follows that,

τ (C) \equiv ({I V T - --- -}_{j_{1}}^{2, 3} (_{1}), {I V T - --- -}_{j_{2}}^{2, 3} (_{2}), . . ., {I V T - --- -}_{j_{v}}^{2, 3} (_{v}))

where $j_{1}, j_{2}, . . ., j_{v} \in {0, 1, . . ., 255}$ .
Now if $τ$ be the global transition function of a countably infinite celled CA with cells having $k$ neighbourhood, then for any global configuration
$C = (. . . c_{i - 1}, c_{i}, c_{i + 1}, . . .)$ , we get

τ (C) = . . . . . μ_{i - 1} (_{i - 1}) . μ_{i} (_{i}) . μ_{i + 1} (_{i + 1}) . . . . . . .

Therefore it follows that,

τ (C) \equiv (. . . ., {I V T - --- -}_{j_{i - 1}}^{p, k} (_{i - 1}), {I V T - --- -}_{j_{i}}^{p, k} (_{i}), {I V T - --- -}_{j_{i + 1}}^{p, k} (_{i + 1}), . . . .)

where $. . . j_{i - 1} j_{i}, j_{i + 1}, . . . \in {0, 1, 2, . . ., (p^{p^{k}} - 1)}$ .

Example 4.1.

Let the initial configuration of a CA having state set ${0, 1, 2}$ be $C_{0} = (01201)_{3}$ and the transition function be given by

τ (01201)_{3} = μ_{1} (101) μ_{2} (012) μ_{3} (120) μ_{4} (201) μ_{5} (010) = (02001)_{3}

where $μ_{1}, μ_{3}, μ_{5}$ follow Wolfram code $377$ and $μ_{2}, μ_{4}$ follow Wolfram code $588$ for $3$ -state CA.
A local transition function can be equivalently represented by some ${I V T - --- -}_{j}^{3, 3}$ and thus it follows that $τ (01201)_{3}$ is equivalent to

{I V T - --- -}_{377}^{3, 3} (1, 0, 1) {I V T - --- -}_{588}^{3, 3} (0, 1, 2) - {I V T}_{377}^{3, 3} (1, 2, 0) {I V T - --- -}_{588}^{3, 3} (2, 0, 1) {I V T - --- -}_{377}^{3, 3} (0, 1, 0)

Conversely, for any ${I V T - --- -}_{j}^{p, k}$ if $k$

be odd, i.e. if

k = 2 r + 1

for some

r < k (\in N_{0})

, then

{I V T - --- -}_{j}^{p, k} (η_{1}, η_{2}, . . ., η_{2 r + 1})

will be equivalent to a local transition function of the

(r + 1)^{t h}

cell in a CA with

p (\in N)

states and

2 r + 1

neighbourhood whose underlying Wolfram code is

j \in {0, 1, . . ., p^{p^{k}} - 1}

, given by

{I V T - --- -}_{j}^{q, l} (η_{1}, η_{2}, . . ., η_{2 r + 1}) = {I V T - --- -}_{j}^{p, k} ({¯ ¯ ¯ η}_{r + 1}) \equiv μ_{r + 1} ({¯ ¯ ¯ η}_{r + 1})

Remark 4.0.

For a $p (\in N)$ -state $k (= 2 r + 1 \in N)$ -neighbourhood CA, clearly $\forall_{i} \in Q^{S_{i}} \subseteq Q^{Z}$ $μ_{j} (_{i}) \equiv {I V T - --- -}_{j}^{p, k} (_{i})$ where $j \in {0, 1, . . ., p^{p^{k}} - 1}$ is the underlying Wolfram Code. It follows that the set of local transition functions $M$ is equivalent to the set $T_{| Q}^{p, k}$ and vice-versa.
Hence a function $ϕ : M \to T_{| Q}^{p, k}$ defined by $ϕ (μ_{j}) (_{i}) = {I V T - --- -}_{j}^{p, k} (_{i})$ is an isomorphism.
Moreover, for any $v (\in N \geq k)$ -celled CA, having a global configuration $C = (c_{1}, c_{2}, . . ., c_{v})$ , if $Q^{v} = {τ | τ : Q^{v} \to Q^{v}}$ , then $\forall,_{i} \in Q^{k} \subseteq Q^{v}, i = 1, 2, . . ., v$ , it follows that a function $ϕ : Q^{v} \to T_{| Q}^{v}$ defined by

ϕ (τ) (c_{1}, c_{2}, . . ., c_{v}) \equiv ({I V T - --- -}_{j_{1}}^{p, k} (_{1}), {I V T - --- -}_{j_{2}}^{p, k} (_{2}), . . ., {I V T - --- -}_{j_{v}}^{p, k} (_{v}))

is an isomorphism.

5 Some Algebraic Results on CA and IVT

Theorem 5.15.

$(Q^{Z}, \circ)$ forms a monoid w.r.t. composition of global transition functions.

Proof.

Clearly $Q^{Z}$ is closed and associative under composition of global transition functions.
The transition function $μ_{e}$ such that $\forall c_{i} \in Q$ , $μ_{e} (_{i}) = c_{i}$ is the local identity and it follows that $τ_{e} \in Q^{Z}$ is the global identity such that $\forall C \in Q^{Z}, i \in Z$ ,

τ_{e} (C) = . . . . . μ_{e} (_{i - 1}) μ_{e} (_{i}) μ_{e} (_{i + 1}) . . . . . = C

Hence the theorem. ∎

Corollary 5.0.

$(^{Z}, \circ)$ forms a group w.r.t. composition of global transition functions w $^{Z} \subseteq Q^{Z}$ is the set of all invertible global transition functions of CA having state set $Q$ .

Proof.

Since any $τ \in^{Z} \subseteq Q^{Z}$ is invertible, the corollary holds true. ∎

Theorem 5.17.

$(T^{v}, \circ)$ forms a monoid w.r.t. composition of IVTs.

Proof.

The set $T^{v}$ , for $v \in N$ , is closed under composition $^{'} \circ^{'}$ since for any
$(I V T_{i_{1}}^{p, k}, I V T_{i_{2}}^{p, k}, . . ., I V T_{i_{v}}^{p, k}), (I V T_{j_{1}}^{p, k}, I V T_{j_{2}}^{p, k}, . . ., I V T_{j_{v}}^{p, k}) \in T^{v}$ and for any
$(n_{1}, n_{2} . . ., n_{v}) \in N_{0}^{v}$ $\exists (m_{1}, m_{2}, . . ., m_{v}) \in N_{0}^{v}$ such that,

	$((I V T_{i_{1}}^{p, k}, I V T_{i_{2}}^{p, k}, . . ., I V T_{i_{v}}^{p, k}) \circ (I V T_{j_{1}}^{p, k}, I V T_{j_{2}}^{p, k}, . . ., I V T_{j_{v}}^{p, k})) (n_{1}, n_{2}, . . ., n_{v})$
	$= ((I V T_{i_{1}}^{p, k} \circ I V T_{j_{1}}^{p, k}), (I V T_{i_{2}}^{p, k} \circ I V T_{j_{2}}^{p, k}), . . ., (I V T_{i_{v}}^{p, k} \circ I V T_{j_{v}}^{p, k})) (n_{1}, n_{2}, . . . n_{v})$
	$= ((I V T_{i_{1}}^{p, k} \circ I V T_{j_{1}}^{p, k}) (_{1}), (I V T_{i_{2}}^{p, k} \circ I V T_{j_{2}}^{p, k}) (_{2}), . . ., (I V T_{i_{v}}^{p, k} \circ I V T_{j_{v}}^{p, k}) (_{v}))$
	$= (I V T_{i_{1}}^{p, k} (I V T_{j_{1}}^{p, k} (_{1})), I V T_{i_{2}}^{p, k} (I V T_{j_{2}}^{p, k} (_{2})), . . ., I V T_{i_{v}}^{p, k} (I V T_{j_{v}}^{p, k} (_{v})))$
	$= I V T_{i_{1}}^{p, k} (¯ ¯¯¯ ¯ n_{1}^{}) I V T_{i_{2}}^{p, k} (¯ ¯¯¯ ¯ n_{2}^{}) . . . I V T_{i_{v}}^{p, k} (¯ ¯¯¯ ¯ n_{v}^{*}) = (m_{1}, m_{2}, . . ., m_{v})_{p}$

where $n_{s}^{*} = I V T_{j_{s}}^{p, k} (_{s}) = I V T_{j_{s}}^{p, k} (η_{1}^{s}, η_{2}^{s}, . . ., η_{k}^{s})$ for $s = 1, 2, . . ., v$ .
$T^{v}$ is associative under $^{'} \circ^{'}$ since composition of functions are associative.
Again, for $(n_{1}, n_{2} . . ., n_{v}) \in N_{0}^{v}$ $\exists (I V T_{e_{1}}^{p, k}, I V T_{e_{2}}^{p, k}, . . ., I V T_{e_{v}}^{p, k}) \in T^{v}$ such that

(I V T_{e_{1}}^{p, k}, I V T_{e_{2}}^{p, k}, . . ., I V T_{e_{v}}^{p, k}) (n_{1}, n_{2}, . . ., n_{v})

= I V T_{e_{1}}^{p, k} (_{1}) I V T_{e_{2}}^{p, k} (¯ ¯¯¯

Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations

Authors

An overview of Quantum Cellular Automata

Multi-Number CVT-XOR Arithmetic Operations in any Base System and its Significant Properties

Classification of Complex Systems Based on Transients

Integrality of matrices, finiteness of matrix semigroups, and dynamics of linear cellular automata

Recovering decimation-based cryptographic sequences by means of linear CAs

On Lebesgue Integral Quadrature

Causal dynamics of discrete manifolds

1 Introduction

2 Mathematical preliminaries

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Example 2.1.

Definition 0.

Remark 2.0.

3 Wolfram code of an ECA and IVT

Example 3.1.

Example 3.2.

3.1 Some Particular Wolfram codes and Particular IVTs

4 Transition Function of a CA and IVT

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Example 4.1.

Remark 4.0.

5 Some Algebraic Results on CA and IVT

Theorem 5.15.

Proof.

Corollary 5.0.

Proof.

Theorem 5.17.

Proof.

Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations

Authors

Related Research

An overview of Quantum Cellular Automata

Multi-Number CVT-XOR Arithmetic Operations in any Base System and its Significant Properties

Classification of Complex Systems Based on Transients

Integrality of matrices, finiteness of matrix semigroups, and dynamics of linear cellular automata

Recovering decimation-based cryptographic sequences by means of linear CAs

On Lebesgue Integral Quadrature

Causal dynamics of discrete manifolds

This week in AI

1 Introduction

2 Mathematical preliminaries

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Example 2.1.

Definition 0.

Remark 2.0.

3 Wolfram code of an ECA and IVT

Example 3.1.

Example 3.2.

3.1 Some Particular Wolfram codes and Particular IVTs

4 Transition Function of a CA and IVT

Definition 0.

Definition 0.

Definition 0.

Definition 0.

Example 4.1.

Remark 4.0.

5 Some Algebraic Results on CA and IVT

Theorem 5.15.

Proof.

Corollary 5.0.

Proof.

Theorem 5.17.

Proof.