Relationship of Two Discrete Dynamical Models: One-dimensional Cellular Automata and Integral Value Transformations
Sreeya Ghosh, Sudhakar Sahoo, Sk. Sarif Hassan, Jayanta Kumar Das, Pabitra Pal Choudhury
aa r X i v : . [ n li n . C G ] J un Relationship of Two Discrete Dynamical Models: One-dimensional CellularAutomata and Integral Value Transformations
Sreeya Ghosh a,b , Sudhakar Sahoo c , Sk. Sarif Hassan d, ∗ , Jayanta Kumar Das e,b , Pabitra Pal Choudhury a a Department of Applied Mathematics, University of Calcutta, Kolkata-700009, India b Applied Statistics Unit, Indian Statistical Institute, Kolkata-108 , India c Institute of Mathematics and Applications, Bhubaneshwar-751029, India d Dept. of Mathematics, Pingla Thana Mahavidyalaya, Paschim Medinipur-721140, India e School of Medicine, Johns Hopkins University, MD-21210, USA
Abstract
Cellular Automaton(CA) and an Integral Value Transformation(IVT) are two well established mathematical modelswhich evolve in discrete time steps. Theoretically, studies on CA suggest that CA is capable of producing a greatvariety 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-dimensionalCA as well as that of a set of IVTs using binary operations. Also DNA sequence evolution has been modeled usingIVTs.
Keywords:
Cellular Automaton, Integral Value Transformations
1. Introduction
Cellular Automaton(pl. cellular automata, abbrev. CA) is a discrete model which has applications in computerscience, mathematics, physics, complexity science, theoretical biology and microstructure modeling. This model wasintroduced by J.von Neumann and S.Ulam in 1940 for designing self replicating systems [1, 2, 3].A CA consists of a finite/countably infinite number of finite-state semi-automata [4, 5] known as ‘cells’ arrangedin an ordered n -dimensional grid. Each cell receives input from the neighbouring cells and changes according tothe transition function. The transitions at each of the cells together induces a change of the grid pattern. Thesimplest CA is a CA where the grid is a one-dimensional line. Stephen Wolfram’s work in the 1980s contributed to asystematic study of one-dimensional CA, providing the first qualitative classification of their behaviour [6]. CA hasbeen studied for solving many interesting problems on utilizing mathematical bases such as polynomial, matrix algebra,Boolean derivative etc.[7, 8]. Further, dynamic behaviour of CA can also be studied using various mathematical tools[9, 10, 11, 12], that help to understand the crucial properties and modeling of various classes of discrete dynamicalsystem [12].An Integral Value Transformation (abbrev. IVT), a class of discrete dynamical system, were first introduced during2009-10 ([13]). A IVT of k -dimension is a function defined using p -adic numbers over N k where N = N ∪ { } , p ∈ N . ∗ Corresponding author
Email addresses: [email protected] (Sreeya Ghosh), [email protected] (Sudhakar Sahoo), [email protected] (Sk.Sarif Hassan), [email protected] (Jayanta Kumar Das), [email protected] (Pabitra Pal Choudhury)
2. Mathematical preliminariesDefinition 1.
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 → Q . The set Q Z isthe set of all global configurations where Q Z = { C | C : Z → Q } . Definition 2.
A mapping τ : Q Z → Q Z is called a global transition function .A CA (denoted by C Qτ )(reported in [16, 17]) 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 3.
The set Q Z = { τ | τ : Q Z → Q Z } is the set of all possible global transition functions of CA having stateset Q .A mapping τ is invertible if ∀ C i , C j ∈ Q Z , ∃ τ − such that τ ( C i ) = C j ⇔ τ − ( C j ) = C i Definition 4.
For i ∈ Z , r ∈ N , let S i = { i − r, ..., i − , i, i + 1 , ..., i + r } ⊆ Z . S i is the neighbourhood of the i th cell, r is the radius of the neighbourhood of a cell. It follows that Z = S i S i .A restriction from Z to S i induces a restriction of C to c i given by c i : S i → Q ; where c i may be called localconfiguration of the i th cell.The mapping µ i : Q S i → Q is known as a local transition function for the i th cell having radius r . Thus ∀ i ∈ Z , µ i ( c i ) ∈ Q and it follows that, τ ( C ) = τ ( ..., c i − , c i , c i +1 , ... ) = .....µ i − ( c i − ) .µ i ( c i ) .µ i +1 ( c i +1 ) ....... Definition 5.
The set M = { µ i | µ i : Q S i → Q, i ∈ Z } is the set of all possible local transition functions of CA havingstate set Q . Definition 6.
If for a particular CA, | Q | = 2 so that we can write Q = { , } , then the CA is said to be a binaryCA or a Boolean CA . A Boolean CA having radius 1 is known as an
Elementary CA(ECA) . Definition 7. Wolfram code is a naming system often used for a one-dimensional CA introduced by StephenWolfram (see[6]). 2or a one-dimensional CA with Q states, the local rule µ i for some i th cell, i ∈ Z of radius r (neighbourhood 2 r + 1)can be specified by an Q r +1 -bit sequence. The decimal equivalent form of this sequence is known as the Wolframcode.Thus, the Wolfram code for a particular rule is a number in the range from 0 to Q Q r +1 −
1, converted from Q -aryto decimal notation. Example 2.1.
Let the local rule of an ECA for some i th cell, i ∈ Z be, µ i ( c i ) = µ i ( c i − , c i , c i +1 ) = ( c i − ∨ c i +1 ) ∧ c i where ′ ∨ ′ stands for OR operation, ′ ∧ ′ stands for AND operation, c j is the j th cell configuration for j = i − , i, i + 1.Then we get the 2 -bit sequence as, µ (111) µ (110) µ (101) µ (100) µ (011) µ (010) µ (001) µ (000) = 1 1 0 0 1 0 0 0The decimal equivalent number for 11001000 is 200 and so the Wolfram code is RULE . Definition 8. A p -adic k − dimensional Integral Value Transformation(IVT) denoted by
IV T p,kj for p ∈ N , k ∈ N is a function of p -base numbers from N k to N defined(in [18]) as IV T p,kj ( n , n , ..., n k ) = ( f j ( a n , ..., a n k ) f j ( a n , ..., a n k ) ...f j ( a n l − , ..., a n k l − )) p = m where N = N ∪ { } , p ∈ N , n s = ( a n s a n s ...a n s l − ) p for s = 1 , , ..., k . f j is a function from { , , ..., p − } k to { , , ..., p − } for j = 0 , , ..., p p k − m is the decimal conversion of the p -base number. Remark 2.1.
In particular(discussed in [19, 20, 21, 22]), if ∀ i = 0 , , ..., l −
1, the function f j be defined as1. f j ( a n i , ..., a n k i ) = ⌊ ( a n i + ... + a n k i ) /p ⌋ , then IV T p,kj is known as a
Modified Carry Value Transforma-tion(MCVT) .Again,
M CV T with a 0 padding at the right end is known as a
Carry Value Transformation(CVT) .2. f j ( a n i , ..., a n k i ) = ( a n i + ... + a n k i ) mod p , then IV T p,kj is known as an
Exclusive OR Transformation(XORT) .3. f j ( a n i , ..., a n k i ) = max ( a n i , ..., a n k i ), then IV T p,kj 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 ∈ { , , , ..., } obtained from the 8-bitsequence 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 a2 base 3-dimensional IVT.The function µ in the above 8-bit sequence can be represented by a function f j : { , } → { , } 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)) IV T , j (11110000 , , ) = IV T , j (240 , , )Hence it follows that for a 3 − neighbourhood Boolean CA, any Wolfram code j ∈ { , , , ... } can be equivalentlyrepresented by IV T , j (240 , , ). Example 3.1.
Wolfram code 200 can be equivalently represented as200 = (11001000) = IV T , (240 , , )However the following example shows that for j ∈ { , , , ... } , any IV T , j may not correspond to a Wolframcode. Example 3.2.
IV T , j (240 , , ) = IV T , j (11110000 , , )= ( f j (111) f j (110) f j (101) f j (100) f j (011) f j (010) f j (001) f j (001)) In this 8-bit sequence, f j (000) is missing and so this cannot correspond to any Wolfram code. We know that transformations such as MCVT, XORT, EVT are particular cases of IVTs. Again, any Wolframcode 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 i th cell, i ∈ Z be, µ i ( c i ) = µ i ( c i − , c i , c i +1 ) = ( c i − ∧ c i ) ∨ (( c i − ∨ c i ) ∧ c i +1 )Then we get the 2 -bit sequence as 11101000 and Wolfram code 232.Here, µ i can be represented by function f j : { , } → { , } given by f j ( c i − , c i , c i +1 ) = ( c i − ∧ c i ) ∨ (( c i − ∨ c i ) ∧ c i +1 ) = ⌊ ( c i − + c i + c i +1 ) / ⌋ Thus, Wolfram code 232 can be represented as( f (111) f (110) f (101) f (100) f (011) f (010) f (001) f (000)) = ( ⌊ / ⌋⌊ / ⌋⌊ / ⌋⌊ / ⌋⌊ / ⌋⌊ / ⌋⌊ / ⌋⌊ / ⌋ ) = M CV T , (11110000 , , ) = M CV T , (240 , , )2. Let the local rule of an ECA for some i th cell, i ∈ Z be, µ i ( c i ) = µ i ( c i − , c i , c i +1 ) = ( c i − ∨ c i ∨ c i +1 )Then we get the 2 -bit sequence as 10010110 and Wolfram code 150.Here, µ i can be represented by function f j : { , } → { , } given by f j ( c i − , c i , c i +1 ) = ( c i − ∨ c i ∨ c i +1 ) = ( c i − + c i + c i +1 ) mod f (111) f (110) f (101) f (100) f (011) f (010) f (001) f (000)) = ((3 mod mod mod mod mod mod mod mod = XORT , (11110000 , , ) = XORT , (240 , , )3. Let the local rule of an ECA for some i th cell, i ∈ Z be, µ i ( c i ) = µ i ( c i − , c i , c i +1 ) = ( c i − ∨ c i ∨ c i +1 )Then we get the 2 -bit sequence as 11111110 and Wolfram code 254.Here, µ i can be represented by function f j : { , } → { , } given by f j ( c i − , c i , c i +1 ) = ( c i − ∨ c i ∨ c i +1 ) = max { c i − , c i , c i +1 } Thus, Wolfram code 254 can be represented as( f (111) f (110) f (101) f (100) f (011) f (010) f (001) f (000)) = (11111110) = EV T , (11110000 , , ) = EV T , (240 , , )
4. Transition Function of a CA and IVTDefinition 9.
The set T p,k = { IV T p,kj | IV T p,kj : N k → N } is the set of all p -base k -dimensional IVTs for p ∈ N , k ∈ N . Definition 10.
The v -fold cartesian product T p,k × ... × T p,k | {z } v times for v ∈ N , denoted by T v is given by T v = { ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) } where IV T p,kj s : N k → N ∈ T p,k , for s = 1 , , ..., v ( ∈ N ) Definition 11.
Let a restriction on
IV T p,kj from N k to Q k = { , , ..., p − } k for p ∈ N be denoted by IV T p,kj . Theset T p,k | Q = { IV T p,kj | IV T p,kj : Q k → Q } is the set of all p -base k -dimensional IVTs when N k is restricted to the subset Q k . Therefore for ( η , η , ..., η k ) ∈ Q k we get, IV T p,kj ( η , η , ..., η k ) = f j ( η , η , ..., η k ) p = m ∈ Q Definition 12.
The set T v | Q = { ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) } is the v -fold cartesian product T p,k | Q × ... × T p,k | Q | {z } v times when N k is restricted to subset Q k where IV T p,kj s : Q k → Q ∈ T p,k | Q for s = 1 , , ..., v ( ∈ N )Now, a local transition function of any i th cell of a one-dimensional CA having p ( ∈ N ) states and radius r ( ∈ N )will be of the form µ i ( c i ) = µ i ( c i − r , ..., c i , ..., c i + r ) = c ∗ i c i − r , ..., c i , ..., c i + r , c ∗ i ∈ { , , ..., p − } .This can be represented by some IV T p,kj i , where p ∈ N , k = 2 r + 1 and j i ∈ { , , , ..., ( p p k − } is the underlyingWolfram code(which is in turn equal to IV T p,kj i (240 , , )).Thus ∀ i ∈ Z , µ i ( c i ) ≡ IV T p,kj i ( c i )Now if τ be the global transition function of any v ( ∈ N ≥ k )-celled CA, then for a global configuration C =( c , c , ..., c v ), we get τ ( C ) = τ ( c , c , ..., c v ) = µ ( c ) .µ ( c ) ....µ v ( c v )Therefore it follows that, τ ( c , c , ..., c v ) ≡ ( IV T p,kj ( c ) , IV T p,kj ( c ) , ..., IV T p,kj v ( c v ))Hence for any v -celled ECA it follows that, τ ( C ) ≡ ( IV T , j ( c ) , IV T , j ( c ) , ..., IV T , j v ( c v ))where j , j , ..., j v ∈ { , , ..., } .Now if τ be the global transition function of a countably infinite celled CA with cells having k neighbourhood, thenfor any global configuration C = ( ...c i − , c i , c i +1 , ... ), we get τ ( C ) = .....µ i − ( c i − ) .µ i ( c i ) .µ i +1 ( c i +1 ) ....... Therefore it follows that, τ ( C ) ≡ ( ...., IV T p,kj i − ( c i − ) , IV T p,kj i ( c i ) , IV T p,kj i +1 ( c i +1 ) , .... )where ...j i − j i , j i +1 , ... ∈ { , , , ..., ( p p k − } . Example 4.1.
Let the initial configuration of a CA having state set { , , } be C = (01201) and the transitionfunction be given by τ (01201) = µ (101) µ (012) µ (120) µ (201) µ (010) = (02001) where µ , µ , µ follow Wolfram code 377 and µ , µ follow Wolfram code 588 for 3-state CA.A local transition function can be equivalently represented by some IV T , j and thus it follows that τ (01201) isequivalent to IV T , (1 , , IV T , (0 , , IV T , (1 , , IV T , (2 , , IV T , (0 , , IV T p,kj if k be odd, i.e. if k = 2 r + 1 for some r < k ( ∈ N ), then IV T p,kj ( η , η , ..., η r +1 ) willbe equivalent to a local transition function of the ( r + 1) th cell in a CA with p ( ∈ N ) states and 2 r + 1 neighbourhoodwhose underlying Wolfram code is j ∈ { , , ..., p p k − } , given by IV T q,lj ( η , η , ..., η r +1 ) = IV T p,kj ( η r +1 ) ≡ µ r +1 ( η r +1 ) Remark 4.1.
For a p ( ∈ N )-state k (= 2 r + 1 ∈ N )-neighbourhood CA, clearly ∀ c i ∈ Q S i ⊆ Q Z µ j ( c i ) ≡ IV T p,kj ( c i )where j ∈ { , , ..., p p k − } is the underlying Wolfram Code. It follows that the set of local transition functions M isequivalent to the set T p,k | Q and vice-versa. 6ence a function φ : M → T p,k | Q defined by φ ( µ j )( c i ) = IV T p,kj ( c i ) is an isomorphism.Moreover, for any v ( ∈ N ≥ k )-celled CA, having a global configuration C = ( c , c , ..., c v ), if Q v = { τ | τ : Q v → Q v } ,then ∀ , c i ∈ Q k ⊆ Q v , i = 1 , , ..., v , it follows that a function φ : Q v → T v | Q defined by φ ( τ )( c , c , ..., c v ) ≡ ( IV T p,kj ( c ) , IV T p,kj ( c ) , ..., IV T p,kj v ( c v ))is an isomorphism.
5. Some Algebraic Results on CA and IVTTheorem 5.1. ( Q Z , ◦ ) 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 ∀ c i ∈ Q , µ e ( c i ) = c i is the local identity and it follows that τ e ∈ Q Z is the globalidentity such that ∀ C ∈ Q Z , i ∈ Z , τ e ( C ) = .....µ e ( c i − ) µ e ( c i ) µ e ( c i +1 ) ..... = C Hence the theorem.
Corollary 5.1. ( Q Z , ◦ ) forms a group w.r.t. composition of global transition functions w Q Z ⊆ Q Z is the set of allinvertible global transition functions of CA having state set Q . Proof.
Since any τ ∈ Q Z ⊆ Q Z is invertible, the corollary holds true. Theorem 5.2. ( T v , ◦ ) forms a monoid w.r.t. composition of IVTs. Proof.
The set T v , for v ∈ N , is closed under composition ′ ◦ ′ since for any( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) , ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) ∈ T v and for any( n , n ..., n v ) ∈ N v ∃ ( m , m , ..., m v ) ∈ N v such that, (cid:16) ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) ◦ ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) (cid:17) ( n , n , ..., n v )= (cid:16) ( IV T p,ki ◦ IV T p,kj ) , ( IV T p,ki ◦ IV T p,kj ) , ..., ( IV T p,ki v ◦ IV T p,kj v ) (cid:17) ( n , n , ...n v )= (cid:16) ( IV T p,ki ◦ IV T p,kj )( n ) , ( IV T p,ki ◦ IV T p,kj )( n ) , ..., ( IV T p,ki v ◦ IV T p,kj v )( n v ) (cid:17) = (cid:16) IV T p,ki ( IV T p,kj ( n )) , IV T p,ki ( IV T p,kj ( n )) , ..., IV T p,ki v ( IV T p,kj v ( n v )) (cid:17) = IV T p,ki ( n ∗ ) IV T p,ki ( n ∗ ) ...IV T p,ki v ( n ∗ v ) = ( m , m , ..., m v ) p where n ∗ s = IV T p,kj s ( n s ) = IV T p,kj s ( η s , η s , ..., η sk ) for s = 1 , , ..., v . T v is associative under ′ ◦ ′ since composition of functions are associative.Again, for ( n , n ..., n v ) ∈ N v ∃ ( IV T p,ke , IV T p,ke , ..., IV T p,ke v ) ∈ T v such that( IV T p,ke , IV T p,ke , ..., IV T p,ke v )( n , n , ..., n v )= IV T p,ke ( n ) IV T p,ke ( n ) ...IV T p,ke v ( n v ) = ( n , n , ..., n v )Since IV T p,ke s ( n s ) = n s ∀ s = 1 , ..., v, it follows that e = e = ... = e v = e ( say )Thus ( IV T p,ke , IV T p,ke , ..., IV T p,ke ) is the identity element of T v Hence the theorem. 7 efinition 13.
Modular addition and multiplication of two local transition functions for a p ( ∈ N )-state CA are defined ∀ i ∈ Z as ( µ i ⊕ p µ i )( c i ) = µ i ( c i ) ⊕ p µ i ( c i ) and ( µ i ⊗ p µ i )( c i ) = µ i ( c i ) ⊗ p µ i ( c i ) Definition 14.
Modular addition and multiplication of two p ( ∈ N )-base k ( ∈ N )-dimensional IVTs(see [18]) are defined ∀ ( n , n , ..., n k ) ∈ N k and j , j ∈ { , , ..., p p k − } , as( IV T p,kj ⊕ p IV T p,kj )( n , n , ..., n k ) = IV T p,kj ( n , n , ..., n k ) ⊕ p IV T p,kj ( n , n , ..., n k )= (cid:0) f j ( a n , ..., a n k ) ⊕ p f j ( a n , ..., a n k ) ...f j ( a n l − , ..., a n k l − ) ⊕ p f j ( a n l − , ..., a n k l − ) (cid:1) p and, ( IV T p,kj ⊗ p IV T p,kj )( n , n , ..., n k ) = IV T p,kj ( n , n , ..., n k ) ⊗ p IV T p,kj ( n , n , ..., n k )= (cid:0) f j ( a n , ..., a n k ) ⊗ p f j ( a n , ..., a n k ) ...f j ( a n l − , ..., a n k l − ) ⊗ p f j ( a n l − , ..., a n k l − ) (cid:1) p where, n s = ( a n s , a n s ..., a n s l − ) p for s = 1 , , ..., k . Theorem 5.3. ( Q Z , ⊕ p , ⊗ p ) forms a commutative ring with identity under the operations ⊕ p and ⊗ p defined as( τ ⊕ p τ )( C ) = ... ( µ i − ⊕ p µ i − )( c i − ) . ( µ i ⊕ p µ i )( c i ) . ( µ i +1 ⊕ p µ i +1 )( c i +1 ) ... and ( τ ⊗ p τ )( C ) = ... ( µ i − ⊗ p µ i − )( c i − ) . ( µ i ⊗ p µ i )( c i ) . ( µ i +1 ⊗ p µ i +1 )( c i +1 ) ... where τ , τ ∈ Q Z , C ∈ Q Z , ⊕ p denotes addition modulo p and ⊗ p denotes multiplication modulo p for | Q | = p ( ∈ N ). Proof.
In a p ( ∈ N )-state CA, for any τ , τ ∈ Q Z , ( τ ⊕ p τ )( C ) ∈ Q Z and ( τ ⊗ p τ )( C ) ∈ Q Z since ∀ i ∈ Z , ( µ i ⊕ p µ i )( c i ) ∈ Q and ( µ i ⊗ p µ i )( c i ) ∈ Q Therefore Q Z is closed w.r.t. ⊕ p and ⊗ p .Associativity follows from associativity of ⊕ p and ⊗ p .The local transition function µ such that ∀ c i ∈ Q , µ ( c i ) = 0 is the additive identity since ∀ i ∈ Z ( µ ⊕ p µ i )( c i ) = ( µ i ⊕ p µ )( c i ) = µ i ( c i )It follows that τ ∈ Q Z is the additive identity such that ∀ C ∈ Q Z , τ ( C ) = ...µ ( c i − ) µ ( c i ) µ ( c i +1 ) ... The local transition function µ id such that ∀ c i ∈ Q , µ id ( c i ) = 1 is the multiplicative identity since ∀ i ∈ Z ( µ id ⊗ p µ i )( c i ) = ( µ i ⊗ p µ id )( c i ) = µ i ( c i )It follows that τ id ∈ Q Z is the multiplicative identity such that ∀ C ∈ Q Z , τ id ( C ) = ...µ id ( c i − ) µ id ( c i ) µ id ( c i +1 ) ... µ i , is a p k -bit sequence b b ...b p k (say), for a p ( ∈ N )-state k ( ∈ N )-neighbourhoodCA where b , ..., b p k ∈ Q = { , , ..., p } . Clearly for any b s ∈ Q , ∃ b − s ∈ Q such that ∀ s = 1 , , ..., p k , b s ⊕ p b − s = 0.Thus for any µ i ( ≡ b ...b p k ), ∃ µ − i ( ≡ b − ...b − p k ) such that ∀ c i ∈ Q ,( µ − i ⊕ p µ i )( c i ) = ( µ i ⊕ p µ − i )( c i ) = µ ( c i ) = 0It follows that for any τ ∈ Q Z , ∃ τ − ∈ Q Z such that ∀ C ∈ Q ,( τ ⊕ p τ − )( C ) = ... ( µ i − ⊕ p µ − i − )( c i − )( µ i ⊕ p µ − i )( c i )( µ i +1 ⊕ p µ − i +1 )( c i +1 ) ... = τ ( C ) , ( τ − ⊕ p τ )( C ) = ... ( µ − i − ⊕ p µ i − )( c i − )( µ − i ⊕ p µ i )( c i )( µ − i +1 ⊕ p µ i +1 )( c i +1 ) ... = τ ( C ) , Thus for any global transition function τ , its additive inverse exists in Q Z .Commutativity follows from commutativity of ⊕ p and ⊗ p .Now for any local transitions µ i , µ i , µ i , ∀ c i ∈ Q, i ∈ Z we get,( µ i ⊗ p ( µ i ⊕ p µ i ))( c i ) = ( µ i ⊗ p µ i )( c i ) ⊕ p ( µ i ⊗ p µ i )( c i ) and, (( µ i ⊕ p µ i ) ⊗ p µ i )( c i ) = ( µ i ⊗ p µ i )( c i ) ⊕ p ( µ i ⊗ p µ i )( c i )Therefore for τ , τ , τ ∈ ( Q ) Z , ∀ C ∈ Q Z we get,( τ ⊗ p ( τ ⊕ p τ ))( C ) = ( τ ⊗ p τ )( C ) ⊕ p ( τ ⊗ p τ )( C ) and, (( τ ⊕ p τ ) ⊗ p τ )( C ) = ( τ ⊗ p τ )( C ) ⊕ p ( τ ⊗ p τ )( C )Hence the theorem. Remark 5.1.
The following example shows that ( Q Z , ⊕ p , ⊗ p ) will not form a field under the operations ⊕ p and ⊗ p even if | Q | = p is prime. Example 5.1.
For initial configuration C = (10001) , let τ (10001) = ( µ (110) µ (100) µ (000) µ (001) µ (011)) Let µ follow Wolfram code 3( ¬ ( c i − ∨ c i )).Then τ (10001) = ( µ (110) µ (100) µ (000) µ (001) µ (011)) = (00110) .Now, τ id (10001) = ( µ id (110) µ id (100) µ id (000) µ id (001) µ id (011)) = (11111) .But ∄ τ − such that ( τ ⊗ τ − )(10001) = (11111) since ∄ µ − such that for all local configurations µ ⊗ µ − = 1. Theorem 5.4. ( T v , ⊕ p , ⊗ p ) forms a commutative ring under the operations ⊕ p and ⊗ p defined as (cid:16) ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) ⊕ p ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) (cid:17) ( n , n , ..., n v )= (cid:16) ( IV T p,ki ⊕ p IV T p,kj ) , ( IV T p,ki ⊕ p IV T p,kj ) , ....., ( IV T p,ki v ⊕ p IV T p,kj v ) (cid:17) ( n , n , ..., n v )= (cid:16) ( IV T p,ki ⊕ p IV T p,kj )( n ) , ( IV T p,ki ⊕ p IV T p,kj )( n ) , ....., ( IV T p,ki v ⊕ p IV T p,kj v )( n v ) (cid:17) and, (cid:16) ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) ⊗ p ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) (cid:17) ( n , n , ..., n v )= (cid:16) ( IV T p,ki ⊗ p IV T p,kj ) , ( IV T p,ki ⊗ p IV T p,kj ) , ...., ( IV T p,ki v ⊗ p IV T p,kj v ) (cid:17) ( n , n , ..., n v )= (cid:16) ( IV T p,ki ⊗ p IV T p,kj )( n ) , ( IV T p,ki ⊗ p IV T p,kj )( n ) , ...., ( IV T p,ki v ⊗ p IV T p,kj v )( n v ) (cid:17) where i s , j s ∈ { , , ..., p p k − } , p, k ∈ N , n s = ( η s , η s , ..., η sk ) for s = 1 , , ..., v roof. For any (
IV T p,ki , IV T p,ki , ..., IV T p,ki v ) , ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) ∈ T v , and ( n , n , ..., n v ) ∈ N v , (cid:16) ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) ⊕ p ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) (cid:17) ( n , n , ..., n v ) ∈ N v and, (cid:16) ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) ⊗ p ( IV T p,kj , IV T p,kj , ..., IV T p,kj v ) (cid:17) ( n , n , ..., n v ) ∈ N v since for every s = 1 , , ..., v, ( IV T p,ki s ⊕ p IV T p,kj s )( n s ) ∈ N and ( IV T p,ki s ⊗ p IV T p,kj s )( n s ) ∈ N where n s = ( η s , η s , ..., η sk )Therefore T v is closed w.r.t. ⊕ p and ⊗ p for p ∈ N .Associativity follows from associativity of ⊕ p and ⊗ p . IV T p,k ∈ T p,k is such that ∀ n s ∈ N k , IV T p,k ( n s ) = 0 is the additive identity of T p,k since ∀ s = 1 , , ..., v ( IV T p,k ⊕ p IV T p,ks )( n s ) = ( IV T p,ks ⊕ p IV T p,k )( n s ) = IV T p,ks ( n s )where, for each ( n s ) = ( η , η , ..., η k ) ∈ N k , IV T p,k ( η , η , ..., η k ) = ( f ( a η , ..., a η k ) , ..., f ( a η l − , ..., a η k l − )) = 0where, η u = ( a η u a η u ...a η u l − ) p for u = 1 , , ..., k. It follows that (
IV T p,k , IV T p,k , ..., IV T p,k ) is the additive identity of T v such that for any ( n , n , ..., n v ) ∈ N v ( IV T p,k , IV T p,k , ..., IV T p,k )( n , n , ..., n v ) = (cid:16) IV T p,k ( n ) , IV T p,k ( n ) , ..., IV T p,k ( n v ) (cid:17) For any
IV T p,ks ∈ T p,k ∃ IV T p,k − s ∈ T p,k such that ∀ n s ∈ N k ,( IV T p,ks ⊕ p IV T p,k − s )( n s ) = ( IV T p,k − s ⊕ p IV T p,ks )( n s ) = IV T p,k ( n s ) = 0It follows that for any ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) ∈ T v , ∃ ( IV T p,k − i , IV T p,k − i , ..., IV T p,k − i v ) ∈ T v such that ∀ ( n , n , ..., n v ) ∈ N v , (cid:16) ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) ⊕ p ( IV T p,k − i , IV T p,k − i , ..., IV T p,k − i v ) (cid:17) ( n , n , ..., n v )= ( IV T p,k , IV T p,k , ..., IV T p,k )( n , n , ..., n v )= (cid:16) ( IV T p,k − i , IV T p,k − i , ..., IV T p,k − i v ) ⊕ p ( IV T p,ki , IV T p,ki , ..., IV T p,ki v ) (cid:17) ( n , n , ..., n v )Thus for any element of T v its additive inverse exists in T v .Commutativity follows from commutativity of ⊕ p and ⊗ p .Now for any IV T p,kh , IV T p,ki , IV T p,kj ∈ T p,k , ∀ n s ∈ N k , we get,( IV T p,kh ⊗ p ( IV T p,ki ⊕ p IV T p,kj ))( n s ) = ( IV T p,kh ⊗ p IV T p,ki )( n s ) ⊕ p ( IV T p,kh ⊕ p IV T p,kj )( n s ) , (( IV T p,ki ⊕ p IV T p,kj ) ⊗ p IV T p,kh )( n s ) = ( IV T p,ki ⊗ p IV T p,kh )( n s ) ⊕ p ( IV T p,kj ⊕ p IV T p,kh )( n s )Thus for any ( IV T p,kh , ..., IV T p,kh v ) , ( IV T p,ki , ..., IV T p,ki v ) , ( IV T p,kj , ..., IV T p,kj v ) ∈ T v , ∀ ( n , ..., n v ) ∈ N v ,10 IV T p,kh , ..., IV T p,kh v ) ⊗ p (cid:16) ( IV T p,ki , ..., IV T p,ki v ) ⊕ p ( IV T p,kj , ..., IV T p,kj v ) (cid:17)(cid:17) ( n , ..., n v )= (cid:16) ( IV T p,kh , ..., IV T p,kh v ) ⊗ p ( IV T p,ki , ..., IV T p,ki v ) (cid:17) ( n , ..., n v ) ⊕ p (cid:16) ( IV T p,kh , ..., IV T p,kh v ) ⊗ p ( IV T p,kj , ..., IV T p,kj v ) (cid:17) ( n , ..., n v ) , and (cid:16)(cid:16) ( IV T p,ki , ..., IV T p,ki v ) ⊕ p ( IV T p,kj , ..., IV T p,kj p ) (cid:17) ⊗ p ( IV T p,kh , ..., IV T p,kh v ) (cid:17) ( n , ..., n v )= (cid:16) ( IV T p,ki , ..., IV T p,ki v ) ⊗ p ( IV T p,kh , ..., IV T p,kh v ) (cid:17) ( n , ..., n v ) ⊕ p (cid:16) ( IV T p,kj , ..., IV T p,kj v ) ⊗ p ( IV T p,kh , ..., IV T p,kh v ) (cid:17) ( n , ..., n v )Hence the theorem. Theorem 5.5. ( T v | Q , ⊕ p , ⊗ p ) forms a commutative ring with identity under the operations ⊕ p and ⊗ p . Proof.
Clearly ( T v | Q , ⊕ p , ⊗ p ) for p, k, v ∈ N is a commutative ring since T v | Q ⊆ T v when N v is restricted to Q v .Now, for any ( η , η , ..., η k ) ∈ Q k , ∃ multiplicative identity IV T p,kid ∈ T p,k | Q such that IV T p,kid ( η , η , ..., η k ) = ( f id ( η , η , ..., η k )) p = 1 since ( IV T p,kj ⊗ p IV T p,kid )( η , η , ..., η k ) = IV T p,kj ( η , η , ..., η k ) = ( f j ( η , η , ..., η k )) p ,and ( IV T p,kid ⊗ p IV T p,kj )( η , η , ..., η k ) = IV T p,kj ( η , η , ..., η k ) = ( f j ( η , η , ..., η k )) p It follows that (
IV T p,kid , IV T p,kid , ..., IV T p,kid ) ∈ T v | Q is the multiplicative identity of T v | Q such that ∀ ( n , n , ..., n v ) ∈ Q v , ( IV T p,kid , IV T p,kid , ..., IV T p,kid )( n , n , ..., n v ) = IV T p,kid ( n ) IV T p,kid ( n ) ...IV T p,kid ( n v )where IV T p,kid ( n s ) = IV T p,kid ( η s , η s , ..., η sk ) = 1 for each s = 1 , , ..., v .Hence the theorem.
6. Modelling of DNA Sequence Evolution
DNA can be modelled as a one-dimensional CA using rule matrix multiplication for linear CA(reported in [23]).The DNA sequence corresponds to the CA lattice and the deoxyribose sugars to the CA cells. The 4 sugar bases A,C, T and G correspond to 4 possible states of the CA cell.Now, 1-neighbourhood CA transition rule is practically absurd since the transition of cell states of a CA is dependenton neighbouring cells. Consequently, computation using 4-state, k ( ∈ N ≥ = 256 different functions of which 4 = 4 are linearfunctions and the others non-linear. The following example depicts DNA sequence evolution in terms of IVTs wherethe sugar bases have been represented with numbers as follows: A → , C → , T → , G → xample 6.1. Let CTCTAGAGGGAA be a particular DNA strand of length 12 at some time. This strand correspondsto the number sequence 121203033300.Evolution of this strand at the next time step can be represented by any (
IV T , j , ..., IV T , j ) ∈ τ where j , ...j ∈{ , , ..., − } .Let the DNA strand be broken into 2 blocks of 6 sugar bases each.If the first block evolves according to non-linear function f (0 → , → , → , →
1) and the second block evolvesaccording to linear function f (0 → , → , → , →
1) then,(
IV T , j , ..., IV T , j )(121203 | {z } block | {z } block ) = ( f (1) f (2) f (1) f (2) f (0) f (3) f (0) f (3) f (3) f (3) f (0) f (0)) = (303021011100) Thus the DNA strand evolved at the next step would be GAGATCACCCAA. Hence the evolution pattern at any latertime step can be obtained recursively.
7. Conclusion