An example of computation of the density of ones in probabilistic cellular automata by direct recursion
aa r X i v : . [ n li n . C G ] J un An example of computation of the density ofones in probabilistic cellular automata by directrecursion
Henryk Fuk´s
Abstract
We present a method for computing probability of occurence of 1s ina configuration obtained by iteration of a probabilistic cellular automata (PCA),starting from a random initial configuration. If the PCA is sufficiently simple, onecan construct a set of words (or blocks of symbols) which is complete, meaningthat probabilities of occurence of words from this set can be expressed as linearcombinations of probabilities of occurence of these words at the previous time step.One can then setup and solve a recursion for block probabilities. We demonstratean example of such PCA, which can be viewed as a simple model of diffusion ofinformation or spread of rumors. Expressions for the density of ones are obtainedfor this rule using the proposed method.
Binary probabilistic cellular automata (PCA) in one dimension are one of the mostfrequently studied types of cellular automata, and one of the most natural and mostfrequently encountered problems in PCA is what the author proposes to call thedensity response problem : If the proportion of ones in the initial configuration drawnfrom a Bernoulli distribution is r , what is the expected proportion of ones after t iterations of the PCA rule?Of course, one could ask a similar question about the probability of occurence oflonger blocks of symbols after t iterations of the PCA rule. Due to the complexity ofPCA dynamics, it is clear that questions of this type are rather hopeless if one wantsto know the answer for an arbitrary rule. In spite of this, it may still be possible toprovide the answer if the rule is sufficiently simple. Henryk Fuk´sDepartment of Mathematics and Statistics, Brock University, St. Catharines, Canada. e-mail: [email protected]
One of the methods which can be used to do this is studying the structure ofpreimages of short blocks and detecting patterns present in them. This approach hasbeen successfully used by the author for a number of deterministic CA rules, such aselementary rules 172, 142, 130 (references [4], [3], and [6] respectively), and severalothers. It has also been used for a special class of PCA known as single-transition a -asynchronous rules [7].In this chapter, however, we would like to describe yet another method of com-puting probabilities of blocks of symbols, by setting up a system of recursive equa-tions which can then be explicitly solved. Such a recursive system can be easilyconstructed for any rule for probabilities of all blocks, but it is normally too big andto complex to be solved. In certain cases, however, one can find a smaller set ofblocks for which the recursion is solvable. We will present one such example, usinga PCA which can be viewed as a simple model of diffusion of innovations or spreadof rumors.To give the reader a flavour of what to expect, let us informally define the afore-mentioned PCA rule. Suppose we have an infinite one-dimensional lattice whereeach site is occupied by an individual who has already adopted the innovation (1) orwho has not adopted it yet (0). Initially the proportion of adopters is r . Once theindividual adopts the innovation, he remains in state 1 forever. Individuals in state0 change can their states to 1 (adopt the innovation) with probabilities dependingon the state of nearest neighbours: if only the right (resp., left) neighbour alreadyadopted, the probability is p (resp., q ), and if both of them already adopted, theprobability is r . What is the proportion of adopters r t of after t iterations of the rule,assuming that the initial configuration is drawn from a Bernoulli distribution? Wewill show that the explicit formula for r t can derived, r t = ( − E (( r q − )( r p − )) t − F ( − r ) t if pq r − ( p + q ) r + r = , − ( G + Ht )( − r ) t − if pq r − ( p + q ) r + r = , where E , F , G , H are constants depending on parameters p , q , r and r .In order to accomplish this, we will start from some general theoretical remarks,considering PCA as maps in the space of shift-invariant probability measures, sim-ilarly as done in [10], [11], [9], and other works. More precisely, we will look atorbits of uniform Bernoulli measures under the action of PCA. Probabilistic CA are often defined as stochastic dynamical systems. In this article,we will concentrate on Boolean CA in one dimension. Let s i ( t ) denote the state ofthe lattice site i at time t , where i ∈ Z , t ∈ N . We will further assume that s i ( t ) ∈{ , } and we will say that the site i is occupied (empty) at time t if s i ( t ) = s i ( t ) = itle Suppressed Due to Excessive Length 3 In a probabilistic cellular automaton, lattice sites simultaneously change statesfrom 0 to 1 or from 1 to 0 with probabilities depending on states of local neigh-bours. A common method for defining PCA is to specify a set of local transitionprobabilities. For example, in order to define a nearest-neighbour PCA one has tospecify the probability w ( s i ( t + )) | s i − ( t ) , s i ( t ) , s i + ( t )) that the site s i ( t ) with near-est neighbors s i − ( t ) , s i + ( t ) changes its state to s i ( t + ) in a single time step.A more formal definition of nearest-neighbour PCA can be constructed as fol-lows. Let r be a positive integer, called radius of PCA , and let us consider a set ofindependent Boolean random variables X i , b , where i ∈ Z and b ∈ { , } r + . Proba-bility that the random variable X i , b takes the value a ∈ { , } will be assumed to beindependent of i and denoted by w ( a | b ) , Pr ( X i , b = a ) = w ( a | b ) . (1)Obviously, w ( | b ) + w ( | b ) = b ∈ { , } r + . The update rule for the PCAis then defined by s i ( t + ) = X i , { s i − r ( t ) ,..., s i ( t ) ,..., s i + r ( t ) } . (2)Note that new random variables X are used at each time step t , that is, randomvariables X used at the time step t are independent of those used at previous timesteps.Having the above definition in mind, we note that in order to fully define anearest-neighbour PCA rule (i.e., rule with r = w ( | x x x ) for all x , x , x ∈ { , } . Remaining eight prob-abilities, w ( | x x x ) , can be obtained by w ( | x x x ) = − w ( | x x x ) .In any dynamical system, the main object of interest is the orbit of the systemstarting from a given initial point, and properties of this orbit. In the case of PCA,we often assume that the initial condition is “random” or “disordered”, typicallymeaning that each s i ( ) is set to 1 with a given probability r and to 0 with prob-ability 1 − r o , independently of each other. We then want to answer question ofthe type “After t iterations, what is the proportion of sites in state 1?” or “After t iterations, what is the probability of finding a pair of adjacent zeros”? In order topose and answer questions of this kind rigorously, we will present an alternativedefinition of PCA, as maps in the space of probability measures. Let A = { , } and X = A Z . A finite sequence of elements of A , b = b b . . . , b n will be called a block (or word ) of length n . Set of all blocks of elements of A ofall possible lengths will be denoted by A ⋆ .A cylinder set generated by the block b = b b . . . , b n and anchored at i is definedas [ b ] i = { x ∈ A Z : x [ i , i + n ) = b } . (3) Henryk Fuk´s
The set of probability measures on the s -algebra generated by cylinder sets of X will be denoted by M ( X ) . Details of construction of such measures, using Hahn-Kolmogorov theorem, can be found in [5]. These details, however, are not essentialfor our subsequent considerations. Given a probability measure m ∈ M ( X ) , measureof a cylinder set [ b ] i , denoted by m ([ b ] i ) , is often informally called a “probability ofoccurence of block b at site i ”.Let the function w : A × A r + → [ , ] , whose values are denoted by w ( a | b ) for a ∈ A , b ∈ A r + , satisfying (cid:229) a ∈ A w ( a | b ) =
1, be called local transition function of radius r , and let its values be called local transition probabilities . A probabilisticcellular automaton with local transition function w is a map F : M ( X ) → M ( X ) defined as ( F m )([ a ] i ) = (cid:229) b ∈ A | a | + r w ( a | b ) m ([ b ] i − r ) for all i ∈ Z , a ∈ A ⋆ , (4)where we define w ( a | b ) = | a | (cid:213) j = w ( a j | b j b j + . . . b j + r ) . (5)When the function w takes values in the set { , } , the corresponding cellular au-tomaton is called a deterministic CA .In this paper, we will exclusively deal with shift-invariant probability measuresfor which m ([ b ] i ) is independent of i . We will, therefore, drop the index i and simplywrite m ([ b ]) . Moreover, we will be interested in orbits of Bernoulli measures n l defined for l ∈ [ , ] by n l ([ b ]) = l ( b ) ( − l ) ( b ) for any b ∈ A ⋆ , (6)where ( b ) and ( b ) denote the number of zeros and ones in b . In order to sim-plify the notation, we define P t ( b ) = ( F t n l )([ b ]) , (7)which will be informally referred to as “probability of occurence of block b after t iterations of PCA rule F ”. With this notation, eq. (4) can be written as P t + ( a ) = (cid:229) b ∈ A | a | + r w ( a | b ) P t ( b ) , (8)for any a ∈ A ⋆ and t ∈ N . We will furthermore define P ( a ) = n l ([ a ]) = l ( a ) ( − l ) ( a ) (9)for any a ∈ A ⋆ .Elements of A ⋆ can be enumerated in lexicographical order, and correspondingprobabilities arranged in an infinite column vector itle Suppressed Due to Excessive Length 5 P t = ( P t ( ) , P t ( ) , P t ( ) , P t ( ) , P t ( ) , P t ( ) , P t ( ) . . . ) T . (10)Before we continue, note that not all these probabilities are independent. Due toadditivity of measure, the following relationships, know as consistency conditions,are valid for any a ∈ A ⋆ , P t ( a ) = P t ( a ) + P t ( a ) = P t ( a ) + P t ( a ) . (11)These conditions will be frequently used in our subsequent considerations.Since each P t + ( a ) , by the virtue of eq. (8), is a linear combination of a finitenumber of P t ( b ) values, we can write P t + = MP t , (12)where the infinite matrix M is defined by eq. (8). This yields the following expres-sion for probabilities of all finite words, P t = M t P , (13)where components of P are defined in eq. (9). In theory, the above equation givesus a complete solution of the problem of determining the orbit of Bernoulli measureunder iterations of a PCA rule. In practice, however, computing powers of an infinitematrix is a daunting, if not impossible, task.In practical applications, however, we rarely need all probabilities P t ( a ) , that is,all components of the vector P t . Sometimes we are interested only in one specificprobability, for example, P t ( ) . For a binary PCA, the expected value of a givenlattice site after t iterations of the rule is equal to 1 · P t ( ) + · P t ( ) = P t ( ) , andfor that reason, P t ( ) is sometimes referred to as an expected density of ones , to bedenoted by r t , r t = P t ( ) . (14)Note that for Bernoulli measure n l we have r = l . Given r , could one find anexplicit expression for r t for a given PCA using eq. (12)? This problem will becalled a density response problem . Although it cannot be solved in a general case,we will demonstrate that for a sufficiently simple PCA it is a doable task.The idea is to setup a recursion similar to ( ) , but using a “smaller” set ofblock probabilities, for which the matrix M has somewhat simpler structure, lendingitself to direct computation of M t . If we could then express r t in terms of blockprobabilities from this “smaller” set, we would solve the density response problem.Let us define the concept of the “smaller” set first. A set of words A ⋆ ⊃ C = { a , a , a , . . . } will be called complete with respect to a PCA rule F if forevery a ∈ C and t ∈ N , P t + ( a ) can be expressed as a linear combination of P t ( a ) , P t ( a ) , P t ( a ) , . . . . We will show a concrete example of a complete set in thenext section. Henryk Fuk´s
As an example, we will consider a PCA rule which generalizes some of the CArules investigated in [1]. This PCA can be viewed as a simple model for diffusionof innovations, spread of rumors, or a similar process involving transport of infor-mation between neighbours. We consider an infinite one-dimensional lattice whereeach site is occupied by an individual who has already adopted the innovation (1)or who has not adopted it yet (0). Once the individual adopts the innovation, heremains in state 1 forever. Individuals in state 0 change can their states to 1 (adoptthe innovation) with probabilities depending on the state of nearest neighbours. Allchanges of states take place simultaneously. This process can be formally describedas a radius 1 binary PCA with the following transition probabilities, w ( | ) = , w ( | ) = p , w ( | ) = , w ( | ) = , (15) w ( | ) = q , w ( | ) = r , w ( | ) = , w ( | ) = , where p , q , r are fixed parameters of the model, p , q , r ∈ [ , ] . In order to illustratethe difficulty of computing block probabilities for this rule, let us write eq. (8) forblocks a of length 1 and 2, P t + ( ) = P t ( ) + ( − p ) P t ( ) + ( − q ) P t ( ) + ( − r ) P t ( ) , P t + ( ) = pP t ( ) + P t ( ) + P t ( ) + qP t ( ) + rP t ( ) + P t ( ) + P t ( ) , P t + ( ) = P t ( ) + ( − p ) P t ( ) + ( − q ) P t ( ) + ( − p )( − q ) P t ( ) , P t + ( ) = pP t ( ) + ( − p ) P t ( ) + ( − p ) P t ( ) + p ( − q ) P t ( )+ ( − r ) P t ( ) + ( − r ) P t ( ) , P t + ( ) = ( − q ) P t ( ) + ( − r ) P t ( ) + qP t ( ) + ( − p ) qP t ( )+ ( − q ) P t ( ) + ( − r ) P t ( ) , P t + ( ) = pP t ( ) + pP t ( ) + qP t ( ) + rP t ( ) + P t ( ) + P t ( )+ pqP t ( ) + rP t ( ) + rP t ( ) + qP t ( )+ rP t ( ) + P t ( ) + P t ( ) . As we can see, in order to know P t + ( ) , we need to know probabilities of blocks oflength 3 at time step t , and in order to compute these, we would need probabilitiesof blocks of length 5 at time step t −
2, etc.We will now show, however, that for the PCA rule defined in eq. (15) a completesubset of A ⋆ can be constructed. This subset consists of clusters of zeros boundedby 1 on each side, that is, of blocks of the type 10 n
1, where n ∈ N and 0 n denotes n consecutive zeros. Proposition 1.
The set C = { , , , . . . } is complete with respect to thePCA rule defined in in eq. (15). itle Suppressed Due to Excessive Length 7 In order to prove this, we need to show that every P t + ( n ) can be expressed asa linear combination of probabilities of the type P t ( k ) . Let us write eq. (8) for a = n
1. Two cases must be distinguished, n = n >
1. For n =
1, we have P t + ( ) = p ( − q ) P t ( ) + ( − r ) P t ( ) + ( − r ) P t ( )+ pqP t ( ) + ( − p ) qP t ( ) + ( − p ) qP t ( ) + p ( − q ) P t ( )+ ( − r ) P t ( ) + ( − r ) P t ( ) . (16)By consistency conditions, P t ( ) + P t ( ) = P t ( ) and P t ( ) + P t ( ) = P t ( ) , as well as P t ( ) + P t ( ) = P t ( ) . This yields P t + ( ) = ( − r ) P t ( ) + ( − r ) P t ( ) + pqP t ( )+ ( − p ) qP t ( ) + p ( − q ) P t ( ) + ( − r ) P t ( ) , (17)and further reduction is possible using P t ( ) + P t ( ) = P t ( ) and P t ( ) + P t ( ) = P t ( ) . The final result is P t + ( ) = ( − r ) P t ( ) + ( p − pq + q ) P t ( ) + pqP t ( ) . (18)For n >
1, using a similar procedure (omitted here), we obtain P t + ( n ) = ( − p )( − q ) P t ( n ) + ( p − pq + q ) P t ( n + ) + pqP t ( n + ) . (19)Equations (18) and (19) clearly show that the set C is complete. (cid:3) P t ( n ) Having a complete set of block probabilities, we can now write eq. (18) and (19) inmatrix form, P t + ( ) P t + ( ) ... P t + ( n ) ... = ˜ a b c · · · a b c a b c
00 0 0 a b c ... . . . P t ( ) P t ( ) ... P t ( n ) ... , (20)where a = ( − p )( − q ) , ˜ a = − r , b = p − pq + q , and c = pq .Let us define Henryk Fuk´s M = ˜ a b c · · · a b c a b c
00 0 0 a b c ... . . . , P t = P t ( ) P t ( ) ... P t ( n ) ... . (21)We will use diag ( x , x , x , . . . ) to denote an infinite matrix with x , x , x , . . . onthe diagonal and zeros elsewhere. Similarly, sdiag ( x , x , x , . . . ) will denote shifteddiagonal matrix having x , x , x , . . . on the line above the diagonal and zeros else-where, and sdiag ( x , x , x , . . . ) will denote doubly-shifted diagonal matrix, with x , x , x , . . . on the second line above the diagonal and zeros elsewhere. With thisnotation, we have M = A + B + C , (22)where A = diag ( ˜ a , a , a , . . . ) , (23) B = sdiag ( b , b , b , . . . ) , (24) C = sdiag ( c , c , c , . . . ) . (25)Now, P t = M t P , (26)and we need to compute M t . We will do it by considering a special case first. ˜ a = a When ˜ a = a , matrices A , B , and C pairwise commute, thus we can use the trinomialexpansion formula, M t = ( A + B + C ) t = (cid:229) i + j + k = t (cid:18) ti , j , k (cid:19) A i B j C k , (27)where (cid:18) ti , j , k (cid:19) = t ! i ! j ! k ! . (28)Generalizing the previously introduced notation, let n sdiag ( x , x , x , . . . ) (29)denote n -times shifted diagonal matrix, which has x , x , x , . . . entries on the n -thline above the diagonal and zeros elsewhere. It is straightforward to prove that itle Suppressed Due to Excessive Length 9 A i = diag ( a i , a i , a i , . . . ) , (30) B j = j sdiag ( b j , b j , b j , . . . ) , (31) C k = k sdiag ( c k , c k , c k , . . . ) , (32)and, consequently, A i B j C k = j + k sdiag ( a i b j c k , a i b j c k , a i b j c k , . . . ) . (33)In the first row of the above matrix, the only non-zero element ( a i b j c k ) is in thecolumn 1 + j + k . In the second row, the only non-zero element ( a i b j c k ) is in thecolumn 2 + j + k , and so on. This means that A i B j C k P = a i b j c k P ( + j + k ) a i b j c k P ( + j + k ) a i b j c k P ( + j + k ) ... . (34)Using the above and the fact that P ( n ) = r ( − r ) n , we can now write P t = M t P = (cid:229) i + j + k = t (cid:18) ti , j , k (cid:19) ˜ a i b j c k r ( − r ) + j + k a i b j c k r ( − r ) + j + k a i b j c k r ( − r ) + j + k ... . (35)We finally obtain P t ( l ) = (cid:229) i + j + k = t (cid:18) ti , j , k (cid:19) a i b j c k r ( − r ) l + j + k = r ( − r ) l (cid:229) i + j + k = t (cid:18) ti , j , k (cid:19) a i [ b ( − r )] j [ c ( − r ) ] k = r ( − r ) l (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t . (36) We are now ready to handle the general case, without the ˜ a = a assumption. Let usfirst note that t -th powers of matrices ˜ a b c · · · a b c a b c
00 0 0 a b c ... . . . t , a b c · · · a b c a b c
00 0 0 a b c ... . . . t (37)differ only in their first row. This implies that the expression for P t ( l ) given in eq.(36) remains valid for l > a = a . We only need to consider l = P t ( ) . This can be done by writing eq. (18) and replacing P t ( ) and P t ( ) by appropriate expressions obtained from eq. (36), P t + ( ) = ˜ aP t ( ) + b r ( − r ) (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t + c r ( − r ) (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t . (38)This can be written as P t + ( ) = ˜ aP t ( ) + K q t , (39)where K = b r ( − r ) + c r ( − r ) , (40) q = a + b ( − r ) + c ( − r ) . (41)Eq. (39) is a first-order non-homogeneous difference equation for P t ( ) , and, assuch, it can be easily solved by standard methods [2]. The solution is P t ( ) = P ( ) ˜ a t + K t (cid:229) i = ˜ a t − i q i − . (42)The sum on the right hand side is a partial sum of geometric series if ˜ a = q , or ofan arithmetic series when ˜ a = q . Using appropriate formulae for partial sums ofgeometric and arithmetic series one obtains P t ( ) = ( P ( ) ˜ a t + K ( ˜ a t − q t ) / ( ˜ a − q ) if ˜ a = q , P ( ) ˜ a t + K ˜ a t − t if ˜ a = q . (43)Taking P ( ) = r ( − r ) and replacing K and q by their definitions we obtain,for a − ˜ a + b ( − r ) + c ( − r ) = a = q ), P t ( ) = r ( − r ) ( b + c − c r ) a − ˜ a + b ( − r ) + c ( − r ) (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t + r ( − r ) ( a − ˜ a ) a − ˜ a + b ( − r ) + c ( − r ) ˜ a t . (44) itle Suppressed Due to Excessive Length 11 For a − ˜ a + b ( − r ) + c ( − r ) = a = q ), the solution is slightlysimpler, P t ( ) = r ( − r ) (cid:0)(cid:0) c r − ( b + c ) r + b + c (cid:1) t + ˜ a (cid:1) ˜ a t − . (45)We now have expressions for P t ( l ) for l = l > We are finally ready to compute r t . To do this, we will use the formula P t ( ) = ¥ (cid:229) k = kP t ( k ) , (46)which we will refer to as “cluster expansion”. Various proofs of this formula can begiven (see, for example, [12]), but we will show here that it is a direct consequenceof additivity of measure.Consider a cylinder set of a single zero anchored at i , [ ] i . A single zero mustbelong to a cluster of zeros of size k with possible values of k varying from 1 toinfinity. If it belongs to a cluster of k zeros, than it must be the j -th zero of thecluster, with possible values of j varying from 1 to k . Therefore, [ ] i = ¥ [ k = k [ j = [ k ] i − j . (47)Since all the cylinder sets on the right hand side are mutually disjoint, their measuresadd up, thus ( F t n l )([ ] i ) = ¥ (cid:229) k = k (cid:229) j = ( F t n l )([ k ] i − j ) . (48)The measure is shift-invariant, thus ( F t n l )([ k ] i − j ) = P t ( k ) , and we obtain P t ( ) = ¥ (cid:229) k = k (cid:229) j = P t ( k ) , (49)which yields eq. (46), as desired.We can now compute P t ( ) using the cluster expansion formula and eq. (44), (45)and (36). We will first consider the case of a − ˜ a + b ( − r ) + c ( − r ) =
0, thatis, using eq. (44) for P t ( ) . P t ( ) = ¥ (cid:229) l = lP t ( l ) = r ( − r ) ( b + c − c r ) a − ˜ a + b ( − r ) + c ( − r ) (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t + r ( − r ) ( a − ˜ a ) a − ˜ a + b ( − r ) + c ( − r ) ˜ a t + ¥ (cid:229) l = l r ( − r ) l (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t . (50)Since ¥ (cid:229) l = l ( − r ) l = ( + r )( − r ) r , (51)we obtain P t ( ) = r ( − r ) ( b + c − c r ) a − ˜ a + b ( − r ) + c ( − r ) (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t + r ( − r ) ( a − ˜ a ) a − ˜ a + b ( − r ) + c ( − r ) ˜ a t +( + r )( − r ) (cid:0) a + b ( − r ) + c ( − r ) (cid:1) t . (52)After substitution of ˜ a , a , b , c and simplification, as well as taking r t = − P t ( ) , thefollowing expression for r t is obtained, r t = − ( − r ) ( r − ( p − r + q ) r ) pq r − ( p + q ) r + r (( r q − )( r p − )) t − r ( − r ) (( q − ) p − q + r ) pq r − ( p + q ) r + r ( − r ) t . (53)When a − ˜ a + b ( − r ) + c ( − r ) =
0, similar calculations can be performed,but this time using using eq. (45) for P t ( ) . After simplification, this yields r t = − ( − r ) (cid:0) r ( r − )( pq r − p + pq − q ) t + − r (cid:1) ( − r ) t − . (54)Let us summarize this in a more readable form, noticing that after substitution of˜ a , a , c , b by their definitions the condition a − ˜ a + b ( − r )+ c ( − r ) = pq r − ( p + q ) r + r =
0. Our final expression for the density of ones can be writtenas r t = ( − E (( r q − )( r p − )) t − F ( − r ) t if pq r − ( p + q ) r + r = , − ( G + Ht )( − r ) t − if pq r − ( p + q ) r + r = , (55)where definitions of E , F , G , H can be figured out by comparing the above to eq. (52)and (53).We can see that in the non-degenerate case (when pq r − ( p + q ) r + r = r ¥ = lim t → ¥ r t always exists, and that r t approaches r ¥ exponentiallyfast, excluding special cases when r t = const (such as r = p = q = r = itle Suppressed Due to Excessive Length 13 In the degenerate case, r ¥ always exists as well, but the approach of r t to r ¥ islinearly-exponential.It is worth noting that the existence of the degenerate case is a fairly subtle phe-nomenon, and that it would be very difficult to discover the linearly-exponentialconvergence by computer simulations alone. This illustrates the point that havinga formula for r t brings some advantages, and that the search for such formulae isworthwhile.As a separate remark, let us note that deterministic CA are nothing else but spe-cial cases of PCA, thus we can choose integer values of p , q , r and obtain relevantexpression for r t for a number of elementary CA rules (ECA), as follows.ECA rule 206 ( p = , q = , r =
0) or rule 220 ( p = , q = , r = r t = − r ( − r ) − ( − r ) t + , (56)ECA rule 222 ( p = q = , r = r t = + ( − r ) t + + r ( − r ) r − , (57)ECA rule 236 ( p = , q = , r = r t = − ( r + )( − r ) , (58)ECA rule 238 ( p = , q = , r =
1) or rule 252 ( p = , q = , r =
1) , r t = − ( − r ) t + , (59)ECA rule 254 ( p = q = r = r t = − ( − r ) t + . (60)The above formulae agree with those derived informally in [1]. We presented a method for computing the density of ones in the orbit of theBernoulli measure under the action of a probabilistic cellular automaton, using asimple PCA rule as an example. For this rule, we were able to construct a completeset of block probabilities, and then solve the resulting recurrence relations. By usingthe cluster expansion, we then obtained the required density of ones. Although thismethod is obviously applicable only to PCA rules with rather simple dynamics, itmay be possible to find other PCA rules with complete sets, thus making the methoduseful for them. Generalization of the rule used in this paper to larger neighbour-hood sizes comes to mind as a first possibility, and sufficiently simple deterministic rules, such as asymptotic emulators of identity investigated in [8], are also possiblecandidates.
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