Some Algebraic Properties Of Linear Synchronous Cellular Automata
aa r X i v : . [ n li n . C G ] A ug Some Algebraic Properties Of Linear Synchronous CellularAutomata
Sreeya Ghosh and Sumita BasuOctober 1, 2018
Abstract
Relation between global transition function and local transition function of a homoge-neous one dimensional cellular automaton (CA) is investigated for some standard transitionfunctions. It could be shown that left shift and right shift CA are invertible. The final resultof this paper states that the set of all left and right shift CA together with the identity CAon the same set forms an abelian group.
Key words :
Cellular Automaton, Transition function, Invertible Cellular Automaton.
Cellular Automata is the computational model of a dynamical system.This model was introducedby J.von Neumann and S.Ulam in 1940 for designing self replicating systems which later sawapplications in Physics, Biology and Computer Science.Neumann conceived a cellular automaton as a two-dimensional mesh of finite state machinescalled cells which are locally interconnected with each other. Each of the cells change theirstates synchronously depending on the states of some neighbouring cells (for details see [8, 13]and references therein). The local changes of each of the cells together induces a change of theentire mesh. Later one dimensional CA, i.e a CA where the elementary cells are distributed ona straight line was studied. Stephen Wolfram’s work in the 1980s contributed to a systematicstudy of one-dimensional CA, providing the first qualitative classification of their behaviour (reported in [14, 15, 7] ).In this paper we consider only synchronous one dimensional Cellular Automaton (CA) wherethe underlying topology is a one dimensional grid line. A finite automaton with finite memorymodels a simple computation. A CA is a computation model of a dynamical system wherefinite/countably infinite number of automaton are arranged in an ordered linear grid.... ... A i − A i A i +1 ... ... Fig.1A typical grid of a linear CA
Each of the automaton work synchronously leading to evolution of the entire grid through anumber of discrete time steps. If the set of memory elements of each automaton is { } then atypical pattern evolved over time t may be as follows :1ime ↓ Grid Position ( i ) → ... -3 -2 -1 0 1 2 3 ... t = 0 Configuration C → ... 0 0 0 1 0 0 0 ... t = 1 C → ... 0 0 1 0 1 0 0 ... t = 2 C → ... 0 1 0 1 0 1 0 ... t = 3 C → ... 1 0 1 0 1 0 1 ...... ... ... ... ... ... ... ... ... ... ... Fig.2The grid line at time t gives the configuration of the CA at time t Algebraic properties of a CA and its relation to group theory and topology is gaining interestin recent years (see[6, 9, 11, 2, 3]). All the CA’s referred above are deterministic in nature. Nondeterministic CA have also been studied in [1].In this paper we study one dimensional homogeneous infinite CA and study the relationshipof the macro and micro level changes of the configuration in specific cases. In Section 2 basicconcepts are introduced and some fundamental results are reported. We compare the behaviorof local and global transition function for some standard CA in Section 3. Section 4 is devoted tofinding inverse of some standard CA. We could show that inverse of an m place left shift functionis an m place right shift function and vice versa. Finally, in Section 5 a binary operation onthe set of all CA having the same state set is defined and its properties are studied. It could beshown that the set of CA containing identity function and all shift (both left and right) functionsform an abelian group under the binary function.
We give a formal definition of a Cellular Automaton.
Definition 2.1.
Let us consider a finite set Q called the state set . The memory elements ofthe automata placed on the grid line belong to this state set Q .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 is the set of all global configurations where Q Z = { C | C : Z → Q } .A mapping τ : Q Z → Q Z is called a global transition function .A CA (denoted by C Qτ ) is a triplet ( Q, Q Z , τ ) where Q is the finite state set, Q Z is the set of allconfigurations, τ is the global transition function. Remark 2.1.
For a particular state set Q and a particular global transition function τ a triple ( Q, Q Z , τ ) denoted by C Q τ defines the set of all possible cellular automata on ( Q, τ ). However,the evolution of a CA at times is dependent on the initial configuration (starting configuration)of the CA. A particular CA C Q τ ( C ) ∈ C Q τ is defined as the quadruple ( Q, Q Z , τ, C ) such that C ∈ Q Z is the initial configuration of the particular CA C Q τ ( C ) . The configuration at time t is denoted by C t such that C t ∈ Q Z for all time t .Also, τ ( C t ) = C t +1 C = ... ... ; τ ( C ) = τ ( ... ... ) = ... ... = C ; τ ( C ) = τ ( ... ... ) = ... ... = C ,etc.Evolution of a Cellular Automaton is mathematically expressed by the global transition func-tion. However, this global transition is induced by transitions of automaton at each grid point ofthe CA. The transition of the state of the automaton at the ith grid point of a CA at a particulartime, depends on the state of the automaton at the i th grid point and its adjacent cells. Theseadjacent cells constitute the neighbourhood of that cell. The transition of the automaton ateach grid point is called local transition. Definition 2.2.
For i ∈ Z , r ∈ N , let S i = { i − r, ..., i − , i, i + 1 , ..., i + r } ⊆ Z . S i is theneighbourhood 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 the following:1. Restriction of C to c i is given by c i : S i → Q ; and c i may be called local configuration of the i th cell.2. Restriction of Q Z to Q S i is given by Q S i = { c i | c i : S i → Q } ;and Q S i may be called the setof all local configurations of the i th cell.The mapping µ i : Q S i → Q is known as a local transition function for the i th automatonhaving radius r . So, ∀ i ∈ Z , µ i ( c i ) ∈ Q . So,if the local configuration of the ith cell at time t isdenoted by c ti then µ i ( c ti ) = c ( t +1) i ( i ) . Remark 2.2.
Since S i has (2 r + 1) elements and Q is finite, Q S i is finite having | Q | (2 r +1) elements. Remark 2.3.
From Definition 2.2 ,1. ∀ i ∈ Z , C ( i ) = c i ( i ) ∀ j ∈ S i ⊆ Z , c i ( j ) = C ( j ) . Remark 2.4. If τ ( C ) = C ∗ then C ∗ ( i ) = τ ( C )( i ) = µ i ( c i ) . So we have,1. C ( t +1) ( i ) = τ ( C t )( i ) = µ i ( c ti ) = c ( t +1) i ( i ) τ ( C ) = .....µ i − ( c i − ) .µ i ( c i ) .µ i +1 ( c i +1 ) ....... Definition 2.3.
If all µ ′ i s are identical then the CA is homogeneous.A homogeneous CellularAutomaton may also be defined as a triplet ( Q, r, µ ) where Q is the finite state set, r is theradius of the neighbourhood of a cell, µ is the local transition function. Remark 2.5.
The set Q Z , where Q Z = ( Q Z ) Q Z is the set of all global transition functions of aCA defined as Q Z = ( Q Z ) Q Z = { τ | τ : Q Z → Q Z } . Remark 2.6.
The set M, where M = { ( Q S i ) Q | i ∈ Z } is the set of all local transition functionsof a CA defined as M = { ( Q S i ) Q | i ∈ Z } = { µ | µ : Q S i → Q, i ∈ Z } . If there is no ambiguity regarding
Q, C a CA is often denoted by τ where τ ∈ Q Z . Definition 2.4.
If for a particular CA, | Q | = 2 so that we can write Q = { , } , then the CAis said to be a binary CA.For a binary CA ( Q, Q Z , τ ) if C , C ∈ Q Z such that τ ( C ) = C where ∀ i ∈ Z , C ( i ) = 0 ↔ C ( i ) = 1 and C ( i ) = 1 ↔ C ( i ) = 0 then C is the complement of C and vice versa. τ is said to be the complementary transitionfunction and is denoted by τ c . Definition 2.5.
A local transition function µ is an identity function denoted by µ e if, ∀ i ∈ Z , µ ( c i ) = c i ( i ) A global transition function τ e is an identity function provided for all C ∈ Q Z , τ e ( C ) = C .A CA is said to be an identity Cellular Automaton if the global transition function is anidentity function. Definition 2.6.
A local transition function µ is a constant function denoted by µ q if, ∀ i ∈ Z , µ ( c i ) = q for a particular state q ∈ Q .A global transition function τ is a constant function provided for all C ∈ Q Z , τ ( C ) = C ∗ fora particular constant configuration C ∗ ∈ Q Z .A CA is said to be a constant Cellular Automaton if the global transition function is aconstant function. Definition 2.7.
A local transition function µ is an m-place left shift function denoted by µ Lm where m ∈ N is finite, if the state of the i th automaton c i ( i ) shifts m − place leftwards. So, ∀ i ∈ Z , µ Lm ( c i ) = c i ( i + m ) where c i is the restriction of C .A global transition function τ is an m-place left shift function denoted by τ Lm where m ∈ N is finite, if ∀ i ∈ Z , τ Lm ( C )( i ) = C ( i + m ) .A CA is said to be a m-place left shift Cellular Automaton if the global transition functionis a m-place left shift function. Definition 2.8.
A local transition function µ is an m-place right shift function denoted by µ Rm where m ∈ N is finite, if the state of the i th automaton c i ( i ) shifts m − place rightwards.So, ∀ i ∈ Z , µ Rm ( c i ) = c i ( i − m ) where c i is the restriction of C .A global transition function τ is an m-place right shift function denoted by τ Rm where m ∈ N is finite, if ∀ i ∈ Z , τ Rm ( C )( i ) = C ( i − m ) .A CA is said to be a m-place right shift Cellular Automaton if the global transition functionis a m-place right shift function. heorem 2.1. For a homogeneous CA, if the global transition function τ is a constant functionthen ∀ i ∈ Z , τ ( C )( i ) is identical.Proof. Let the global transition function τ of a homogneous CA be a constant function.Then for all C ∈ Q Z , τ ( C ) = τ q ( C ) = C ∗ where C ∗ ∈ Q Z is a particular configuration.Thus if ∀ i ∈ Z , µ = µ i be the corresponding local transition function then τ ( C )( i ) = C ∗ ( i ) = µ i ( c i ) = q i ∈ Q Now, for j, k ∈ Z , τ ( C )( j ) = τ ( C )( k ) = ⇒ C ∗ j = C ∗ k = ⇒ µ j ( c j ) = µ k ( c k ) = ⇒ µ ( c j ) = µ ( c k )This is a contradiction to the fact that the CA is homogeneous.Hence the theorem. The global transition function τ describes the evolution of the dynamical system at the macrolevel. The corresponding local transition function( µ ) having radius of the neighbourhood r denoted by µ r describes the same at micro level. Theorem 3.1.
For a homogeneous CA, having the global transition function τ the corresponding µ r may not be unique.Proof. The result is proved by example.Let, x n = 2 n . Then the squence { x n } in decimal system is { , , ..... } The binary representation of this sequence is { , , , , ..... } . This sequencecan be generated by the CA with state set Q = { , } and τ = τ L . Let µ r : Q S i → Q be thelocal transition function of the CA having radius r ∈ N .Depending on the value of r we have thefollowing cases. • Case 1 : r < r = 2Since c i ( i + 3) is not defined, µ ( c i ) = c i ( i + 3). So, the local transition cannot berepresented by µ . Similarly µ cannot be the local transition function. • Case 2 : r ≥ c i ( i + 3) is defined, and C ( i + 3) = c i ( i + 3). So, ∀ r ≥
3, we may define, µ r ( c i ) = c i ( i + 3) and µ r may act as local transition function of the CACase 2 shows that for a given global transition function the local transition function is notunique. 5 emma 3.1. Given a local transition function µ r the corresponding global transition function τ is unique. Theorem 3.2.
For a homogeneous CA, the global transition function of the CA is an identityfunction τ e if and only if the local transition function is an identity function µ e .Proof. Let µ be the local transition function and τ be the global transition function of a homo-geneous CA.Suppose µ is an identity function. Then for any i ∈ Z , we have, µ ( c i ) = µ e ( c i ) = c i ( i )If τ ( C ) = C ∗ then ∀ i ∈ Z we have, τ ( C )( i ) = C ∗ ( i ) = µ ( c i ) = c i ( i )Again, ∀ i ∈ Z we know that, c i ( i ) = C ( i ) . So, ∀ i ∈ Z ,C ∗ ( i ) = c i ( i ) = C ( i ) ⇔ τ ( C )( i ) = C ( i )Since the CA is homogeneous, we have, τ ( C ) = C .It follows that the global transition function is an identity function and is denoted by τ e .Conversely, let the global transition function be an identity function.Then, τ ( C ) = τ e ( C ) = C Since, i ∈ Z , C ( i ) = c i ( i ) ⇔ τ e ( C )( i ) = µ ( c i ) = c i ( i )Therefore the local transition function is an identity function.Hence the theorem. Theorem 3.3.
For a homogeneous CA, the global transition function of the CA is a constantfunction τ q if and only if the local transition function is a constant function µ q .Proof. Let µ be the local transition function and τ be the global transition function of a homo-geneous CA.Let the local transition function µ be a constant function µ q . Then for any i ∈ Z there exists apartcular q ∈ Q such that µ ( c i ) = µ q ( c i ) = q Now, if τ ( C ) = C ∗ then ∀ i ∈ Z we have, τ ( C )( i ) = C ∗ ( i ) = µ ( c i ) = q So ∀ i ∈ Z , C ∗ ( i ) = q . Thus C ∗ is a constant configuration.Therefore, for the constant configuration C ∗ ∈ Q Z , and forall C ∈ Q Z , τ ( C ) = C ∗ .It follows that the global transition function is a constant function.Conversely, let the global transition function be a constant function τ q .Then for all C ∈ Q Z , τ ( C ) = τ q ( C ) = C ∗ for a particular configuration C ∗ ∈ Q Z such that ∀ i ∈ Z C ∗ ( i ) = q. Again, ∀ i ∈ Z , τ ( C )( i ) = µ ( c i ) = q . 6herefore it follows that the local transition function is a constant function and is denoted by µ q .Hence the theorem. Theorem 3.4.
For a homogeneous CA, the global transition function of the CA is an m − place left shift function τ Lm if and only if the local transition function is an m − place left shift function µ Lm where m ∈ N is finite.Proof. Let µ r be the local transition function and τ be the global transition function of a homo-geneous CA.Let the local transition function be an m − place left shift function where m ≤ r .Then for any i ∈ Z ,if τ ( C ) = C ∗ µ r ( c i ) = ( µ r ) Lm ( c i ) = c i ( i + m ) = C ( i + m ) τ ( C )( i ) = C ∗ ( i ) = µ r ( c i ) = C ( i + m )Hence it follows that the global transition function is an m − place left shift function where m ≤ r such that r ∈ N is finite.Conversely, let the global transition function be an m − place left shift function.Then, ∀ C ∈ Q Z , ∀ i ∈ Z , τ ( C )( i ) = τ Lm ( C )( i ) = C ( i + m )However, ∀ i ∈ Z , m ≤ r µ r ( c i ) = τ ( C )( i ) = C ( i + m ) = c i ( i + m )Thus it follows that µ the local transition function of the CA is an m − place left shift functionwhere m ∈ N is finite.Hence the theorem.Similarly we have the following result which we state without proof. Theorem 3.5.
For a homogeneous CA, the global transition function of the CA is an m − place right shift function τ Rm if and only if the local transition function is an m − place right shiftfunction µ Rm where m ∈ N is finite. Definition 4.1.
Let a class of CA be given by ( Q, Q Z , τ ) . Any global transition function τ − ∈ ( Q Z ) Q Z such that for all C i , C j ∈ Q Z , τ ( C i ) = C j ⇔ τ − ( C j ) = C i is called the inverse of the global transition function τ .Consequently, a CA given by ( Q, Q Z , τ − ) is the inverse of the CA ( Q, Q Z , τ ) . Theorem 4.1.
The inverse of an identity function is an identity function. roof. Let us consider an identity global transition function τ e .Therefore , ∀ C ∈ Q Z , τ e ( C ) = C If τ − e denotes the inverse the of τ e , then we have, τ e ( C ) = C ⇔ τ − e ( C ) = C Thus, for all C ∈ Q Z , τ e ( C ) = τ − e ( C ) ⇔ τ e = τ − e Hence the theorem.
Theorem 4.2.
For a binary homogeneous CA the inverse of the complementary transitionfunction is the function itself.Proof.
By definition, τ c ( C ) = C ↔ τ c ( C ) = C Hence ( τ c ) − = τ c Theorem 4.3.
The inverse of an m − place left shift global transition function τ Lm where m ∈ N is finite, is an m − place right shift global transition function τ Rm and vice-versa.Proof. Let us consider an m-place left shift global transition function τ Lm .Then, ∀ i ∈ Z , τ Lm ( C )( i ) = C ( i + m )So, τ Lm ( C )( i − m ) = C ( i − m + m ) = C ( i )Again, for an m-place right shift global transition function τ Rm ,we have, ∀ i ∈ Z , τ Rm ( C )( i ) = C ( i − m )So, τ Rm ( C )( i + m ) = C ( i + m − m ) = C ( i )Thus , ∀ i ∈ Z , τ Lm ( C )( i ) = C ( i + m ) ⇔ τ Rm ( C )( i + m ) = C ( i )Therefore, we conclude that the inverse of τ Lm is τ Rm .Hence, the theorem. 8 Binary Operations on One Dimensional Cellular Automatawith the Same State Set
In this section we discuss the properties of binary operations on the set of CA with the samestate set Q . Let, Q Z = ( Q Z ) Q Z . Thus any element of the set Q Z is a global transition functionor often called a CA with the state set Q . Definition 5.1.
A binary operation ∗ on Q Z is defined as follows:If τ and τ ∈ Q Z , then ∀ C ∈ Q Z , ( τ ∗ τ )( C ) = τ ( τ ( C )) Lemma 5.1.
For a particular state set
Q, τ e , the global identity transition function belongs to Q Z . Lemma 5.2.
For a particular state set Q, ∀ m, n ∈ N , τ Lm , τ Rn , the global m-place left shifttransition function and global n-place right shift function belongs to Q Z . Lemma 5.3.
For a particular state set Q, Q Z is closed under the binary operation ∗ .Proof. As τ and τ ∈ Q Z , ∀ C i ∈ Q Z , ∃ C j ∈ Q Z so that τ ( C i ) = C j . Hence,( τ ∗ τ )( C i ) = τ ( τ ( C i )) = τ ( C j ) ∈ Q Z So, ( τ ∗ τ ) ∈ Q Z Lemma 5.4.
For a particular state set Q, ∗ , the binary operation on Q Z is associative.Proof. Let τ , τ and τ ∈ Q Z , then ∀ C ∈ Q Z , τ ∗ (( τ ∗ τ )( C )) = τ ∗ (( τ ( τ ( C ))) = τ (( τ ( τ ( C ))(( τ ∗ τ ) ∗ τ )( C ) = ( τ ∗ τ ) τ ( C ) = τ (( τ ( τ ( C )) Lemma 5.5.
For a binary CA if τ c is the complementary transition function then τ c ∈ Q Z . Lemma 5.6.
For a particular state set Q, ∀ m, n ∈ N , τ Lm , τ Rn , the global m-place left shifttransition function and global n-place right shift function then1. τ Lm ∗ τ Rn = τ Rn ∗ τ Lm .2. If, m > n, τ Lm ∗ τ Rn is a left shift function.3. If n > m, τ Lm ∗ τ Rn is a right shift function. roof. ∀ C ∈ Q Z , we have( τ Lm ∗ τ Rn ) C ( i ) = τ Lm ( C ( i − n )) = C ( i − n + m ) and ( τ Rn ∗ τ Lm ) C ( i ) = τ Rn ( C ( i + m )) = C ( i + m − n )Hence we get result 1. If m − n = p, τ Lm ∗ τ Rn is a left shift function when p > p < Theorem 5.1. h Q Z , ∗i forms a monoid. From Lemma 5.5, Theorem 4.2 and Lemma 5.3 we have the following,
Theorem 5.2.
For a binary CA if τ c is the complementary transition function then h{ τ e , τ c } , ∗i forms a group and is the smallest such nontrivial group. Lemma 5.2, Lemma 5.6, Theorem 4.3 and Theorem 5.1 together gives the final result givenbelow.
Theorem 5.3.
Let us consider the set G = { τ e } S { τ Lm | m ∈ N } S { τ Rm | m ∈ N } where τ Lm isa m-place left-shift function and τ Rm is a m-place right-shift function. Then ( G, ∗ ) forms anabelian group with respect to binary operation ∗ . The results obtained in this paper is for homogeneous and linear Cellular Automaton. An inves-tigation /extension of results for other types of Cellular Automaton may be worth attempting.
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Sreeya Ghosh(Corresponding author),
Dept. of Applied Mathematics; Calcutta University, Rash-behari Shiksha Prangan,92,A.P.C. Road; Kolkata-700009.
Email: [email protected]
Sumita Basu,
Bethune College; 181 Bidhan Sarani; Kolkata700006
Email: sumi [email protected] [email protected]