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

06/24/2020
by   Sreeya Ghosh, et al.
0

Cellular Automaton (CA) and an Integral Value Transformation (IVT) are two well established mathematical models which evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a great variety of evolution patterns. However computation of non-linear CA or higher dimensional CA maybe complex, whereas IVTs can be manipulated easily. The main purpose of this paper is to study the link between a transition function of a one-dimensional CA and IVTs. Mathematically, we have also established the algebraic structures of a set of transition functions of a one-dimensional CA as well as that of a set of IVTs using binary operations. Also DNA sequence evolution has been modelled using IVTs.

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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 -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 -dimension is a function defined using -adic numbers over where , . 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 be a finite set of memory elements also called the state set.
A global configuration is a mapping from the group of integers to the set given by . The set is the set of all global configurations where .

Definition 0.

A mapping is called a global transition function.
A CA(denoted by )(reported in ghosh2017some ; kari2005theory ) is a triplet , where,

  • is the finite state set

  • is the set of all configurations

  • is the global transition function

Definition 0.

The set is the set of all possible global transition functions of CA having state set .

A mapping is invertible if , such that

Definition 0.

For , let . is the neighbourhood of the cell, is the radius of the neighbourhood of a cell. It follows that .
A restriction from to induces a restriction of to given by ; where may be called local configuration of the cell.
The mapping is known as a local transition function for the cell having radius . Thus and it follows that,

Definition 0.

The set is the set of all possible local transition functions of CA having state set .

Definition 0.

If for a particular CA, so that we can write , then the CA is said to be a binary CA or a Boolean CA. A Boolean CA having radius 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 states, the local rule for some cell, of radius (neighbourhood ) can be specified by an -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 to , converted from -ary to decimal notation.

Example 2.1.

Let the local rule of an ECA for some cell, be,

where stands for OR operation, stands for AND operation, is the cell configuration for .
Then we get the -bit sequence as,

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

Definition 0.

A -adic dimensional Integral Value Transformation(IVT) denoted by for , is a function of -base numbers from to defined(in hassan2012onedmensional ) as

where , , for .
is a function from to for ,
is the decimal conversion of the -base number.

Remark 2.0.

In particular(discussed in choudhury2009theory ; choudhury2011act ; das2016multi ; pal2012properties ), if , the function be defined as

  1. , then is known as a Modified Carry Value Transformation(MCVT).
    Again, with a padding at the right end is known as a Carry Value Transformation(CVT).

  2. , then is known as an Exclusive OR Transformation(XORT).

  3. , then 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 is the decimal number obtained from the -bit sequence

Therefore, Wolfram code for each local transition function of a -neighbourhood Boolean CA can be represented by a base -dimensional IVT.
The function in the above -bit sequence can be represented by a function which gives

Hence it follows that for a neighbourhood Boolean CA, any Wolfram code can be equivalently represented by .

Example 3.1.

Wolfram code 200 can be equivalently represented as

However the following example shows that for , any may not correspond to a Wolfram code.

Example 3.2.

In this -bit sequence, 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.

  1. Let the local rule of an ECA for some cell, be,

    Then we get the -bit sequence as and Wolfram code .
    Here, can be represented by function given by

    Thus, Wolfram code can be represented as

  2. Let the local rule of an ECA for some cell, be,

    Then we get the -bit sequence as and Wolfram code .
    Here, can be represented by function given by

    Thus, Wolfram code can be represented as

  3. Let the local rule of an ECA for some cell, be,

    Then we get the -bit sequence as and Wolfram code .
    Here, can be represented by function given by

    Thus, Wolfram code can be represented as

4 Transition Function of a CA and IVT

Definition 0.

The set is the set of all -base -dimensional IVTs for .

Definition 0.

The -fold cartesian product for , denoted by is given by

where , for

Definition 0.

Let a restriction on from to for be denoted by . The set is the set of all -base -dimensional IVTs when is restricted to the subset . Therefore for we get,

Definition 0.

The set is the -fold cartesian product when is restricted to subset where for

Now, a local transition function of any cell of a one-dimensional CA having states and radius will be of the form

where .
This can be represented by some , where and is the underlying Wolfram code(which is in turn equal to ).
Thus ,

Now if be the global transition function of any -celled CA, then for a global configuration , we get

Therefore it follows that,

Hence for any -celled ECA it follows that,

where .
Now if be the global transition function of a countably infinite celled CA with cells having neighbourhood, then for any global configuration
, we get

Therefore it follows that,

where .

Example 4.1.

Let the initial configuration of a CA having state set be and the transition function be given by

where follow Wolfram code and follow Wolfram code for -state CA.
A local transition function can be equivalently represented by some and thus it follows that is equivalent to

Conversely, for any if

be odd, i.e. if

for some , then will be equivalent to a local transition function of the cell in a CA with states and neighbourhood whose underlying Wolfram code is , given by

Remark 4.0.

For a -state -neighbourhood CA, clearly where is the underlying Wolfram Code. It follows that the set of local transition functions is equivalent to the set and vice-versa.
Hence a function defined by is an isomorphism.
Moreover, for any -celled CA, having a global configuration , if , then , it follows that a function defined by

is an isomorphism.

5 Some Algebraic Results on CA and IVT

Theorem 5.15.

forms a monoid w.r.t. composition of global transition functions.

Proof.

Clearly is closed and associative under composition of global transition functions.
The transition function such that , is the local identity and it follows that is the global identity such that ,

Hence the theorem. ∎

Corollary 5.0.

forms a group w.r.t. composition of global transition functions w is the set of all invertible global transition functions of CA having state set .

Proof.

Since any is invertible, the corollary holds true. ∎

Theorem 5.17.

forms a monoid w.r.t. composition of IVTs.

Proof.

The set , for , is closed under composition since for any
and for any
such that,

where for .
is associative under since composition of functions are associative.
Again, for such that