Orbits of Bernoulli Measures in Cellular Automata
OOrbits of Bernoulli Measures in CellularAutomata
Henryk Fuk´s
Glossary
Configuration space
Set of all bisequences of symbols from the alphabet A of N symbols, A = { , , . . . , N − } , denoted by A Z . Elements of A Z are calledconfigurations and denoted by bold lowercase letters: x , y , etc. Block or word A finite sequence of symbols of the alphabet A . Set of all blocksof length n is denoted by A n , while the sent of all possible blocks of all lengthsby A (cid:63) . Blocks are denoted by bold lowercase letters a , b , c , etc. Individualsymbols of the block b are denoted by indexed italic form of the same let-ter, b = b , b , . . . , b n . To make formulae more compact, commas are sometimesdropped (if no confusion arises), and we simply write b = b b . . . b n . Cylinder set
For a block b of length n , the cylinder set generated by b and an-chored at i is the subset of configurations such that symbols at positions from i to i + n − b , while the remainingsymbols are arbitrary. Denoted by [ b ] i = { x ∈ A Z : x [ i , i + n ) = b } . Cellular automaton
In this article, cellular automaton is understood as a map F in the space of shift-invariant probability measures over the configuration space A Z . To define F , two ingredients are needed, a positive integer r called radiusand a function w : A × A r + → [ , ] , whose values are called transition proba-bilities. The image of a measure µ under the action of F is then defined by prob-abilities of cylinder sets, ( F µ )([ a ] i ) = ∑ b ∈ A | a | + r w ( a | b ) µ ([ b ] i − r ) where i ∈ Z , a ∈ A (cid:63) , and where w ( a | b ) is defined as w ( a | b ) = ∏ | a | j = w ( a j | b j b j + . . . b j + r ) .Cellular automaton is called deterministic if transition probabilities take valuesexclusively in the set { , } , otherwise it is called probabilistic. Orbit of a measure
For a given cellular automaton F and a given shift invariantprobability measure µ , the orbit of µ under F is a sequence µ , F µ , F µ , F µ , . . . . Henryk Fuk´sDepartment of Mathematics and Statistics, Brock University, St. Catharines, ON, Canada e-mail: [email protected] a r X i v : . [ n li n . C G ] F e b Henryk Fuk´s
The main subject of this article are orbits of Bernoulli measures on { , } Z ,that is, measures parametrized by p ∈ [ , ] and defined by µ p ([ b ]) = p ( b ) ( − p ) ( b ) , where k ( b ) denotes number of symbols k in b . Block probability
Probability of occurence of a given block b (or word) of sym-bols. Formally defined as a measure of the cylinder set generated by the block b and anchored at i , and denoted by P ( b ) = µ ([ b ] i ) . In this article we are ex-clusively dealing with shift-invariant probability measures, thus µ ([ b ] i ) is inde-pendent of i . Probability of occurence of a block b after n iterations of cellularautomaton F starting from initial measure µ is denoted by P n ( b ) and defined as P n ( b ) = ( F n µ )([ b ] i ) . Here again we assume shift invariance, thus ( F n µ )([ b ] i ) isindependent of i . Short/long block representation
Shift invariant probability measures on A Z areunambiguously determined by block probabilities P ( b ) , b ∈ A (cid:63) . For a given k ,probabilities of blocks of length 1 , , . . . , k are not all independent, as they haveto satisfy measure additivity conditions, known as Kolmogorov consistency con-ditions. One can show that only ( N − ) N k − of them are linearly independent.If one declares as independent the set of ( N − ) N k − blocks chosen so that theyare as short blocks as possible, one can express the remaining blocks probabilitiesin terms of these. An algorithm for selection of shortest possible blocks is calledshort block representation. If, on the other hand, one chooses the longest possibleblocks to be declared independent, this is called long block representation. Local structure approximation
Approximation of points of the orbit of a mea-sure µ under a given cellular automaton F by Markov measures, that is, mea-sures maximizing entropy and completely determined by probabilities of blocksof length k . The number k is called the order or level of local structure approxi-mation. Block evolution operator
When the cellular automaton rule of radius r is deter-ministic, its transition probabilities take values in the set { , } . For such rulesand for A = { , } , define the local function f : A r + → A by f ( x , x , . . . x r + )= w ( | x , x , . . . x r + ) for all x , x , . . . x r + ∈ A . A block evolution operatorcorresponding to f is a mapping f : A (cid:63) (cid:55)→ A (cid:63) defined for a = a a . . . a n − ∈ A n by f ( a ) = { f ( a i , a i + , . . . , a i + r ) } n − r − i = . For a deterministic cellular automaton F its local function is denoted by the corresponding lowercase italic form of thesame letter, f , while the block evolution operator is the bold form of the sameletter, f . The set of preimages of the block a under f is called block preimage set,denoted by f − ( a ) . Complete set set
A set of words C = { a , a , a , . . . } is called complete with re-spect to a CA rule F if for every a ∈ C and n ∈ N , P n + ( a ) can be expressed as alinear combination of P n ( a ) , P n ( a ) , P n ( a ) , . . . . ontents Orbits of Bernoulli Measures in Cellular Automata . . . . . . . . . . . . . . . . . . .
In both theory and applications of cellular automata (CA), one of the most natu-ral and most frequently encountered problems is what one could call the densityresponse problem : If the proportion of ones (or other symbols) in the initial configu-ration drawn from a Bernoulli distribution is known, what is the expected proportion of ones (or other symbols) after n iterations of the CA rule? One could rephrase itin a slightly different form: if the probability of occurence of a given symbol in aninitial configuration is known, what is the probability of occurrence of this symbolafter n iterations of this rule?A similar question could be asked about the probability of occurence of longerblocks of symbols after n iterations of the rule. Due to complexity of CA dynamics,there is no hope to answer questions like this in a general form, for an arbitrary rule.The situation is somewhat similar to hat we encounter in the theory of differentialequations: there is no general algorithm for solving initial value problem for anarbitrary rule, but one can either solve it approximately (by numerical method), or,for some differential equations, one can construct the solution formula in terms ofelementary functions.In cellular automata, there are also two ways to make progress. One is to usesome approximation techniques, and compute approximate values of the desiredprobabilities. Another is to focus on narrower classes of CA rules, with sufficientlysimple dynamics, and attempt to compute these probabilities in a rigorous ways.Both these approaches are discussed in this article.We will treat cellular automata as maps in the space of Borel shift-invariant prob-ability measures, equipped with the so-called weak (cid:63) topology (K˚urka and Maass,2000; K˚urka, 2005; Pivato, 2009; Formenti and K˚urka, 2009). In this setting, theaforementioned problem of computing block probabilities can be posed as the prob-lem of determining the orbit of given initial measure µ (usually a Bernoulli mea-sure) under the action of a given cellular automaton. Since computing the completeorbit of a measure is, in general, very difficult, approximate methods have beendeveloped. The simplest of these methods is called the mean-field theory, and hasits origins in statistical physics (Wolfram, 1983). The main idea behind the mean-field theory is to approximate the consecutive iterations of the initial measure byBernoulli measures, ignoring correlations between sites. While this approximationis obviously very crude, it is sometimes quite useful in applications.In 1987, H. A. Gutowitz, J. D. Victor, and B. W. Knight proposed a generalizationof the mean-field theory for cellular automata which, unlike the mean-field theory,takes into account correlations between sites, although only in an approximate way(Gutowitz et al, 1987). The basic idea is to approximate the consecutive iterationsof the initial measure by Markov measures, also called finite block measures. Fi-nite block measures of order k are completely determined by probabilities of blocksof length k . For this reason, one can construct a map on these block probabilities,which, when iterated, approximates probabilities of occurrence of the same blocksin the actual orbit of a given cellular automaton. The construction of Markov mea-sures is based on the idea of “Bayesian extension”, introduced in 1970s and 80s inthe context of lattice gases (Brascamp, 1971; Fannes and Verbeure, 1984). The localstructure theory produces remarkably good approximations of probabilities of smallblocks, provided that one uses sufficiently high order of the Markov measure.For deterministic CA, if one wants to compute probabilities of small block ex-actly, without using any approximations, one has to study combinatorial structureof preimages of these block under the action of the rule. In many cases, this reveals ontents 5 some regularities which can be exploited in computation of block probabilities. Fora number of elementary CA rules, this approach has been used to construct probabil-ities of short blocks, typically block of up to three symbols. For probabilistic cellularautomata, one can try to compute n -step transition probabilities, and in some casesthese transition probabilities are expressible in terms of elementary functions. Thisallows to construct formulae for block probabilities.In the rest of this article we will discuss how to construct shift-invariant prob-ability measures over the space of bisequences of symbols, and how to describesuch measures in terms of block probabilities. We will then define cellular automataas maps in the space of measures and discuss orbits of shift-invariant probabilitymeasures under these maps. Subsequently, the local structure approximation will bediscussed as a method to approximate orbits of Bernoulli measures under the actionof cellular automata. The final sections presents some known examples of cellularautomata, both deterministic and probabilistic, for which elements of the orbit ofthe Bernoulli measure (probabilities of short blocks) can be determined exactly. Let A = { , , . . . , N − } be called an alphabet , or a symbol set , and let X = A Z becalled the configurations space . The Cantor metric on X is defined as d ( x , y ) = − k ,where k = min {| i | : x i (cid:54) = y i } . X with the metric d is a Cantor space, that is, compact,totally disconnected and perfect metric space. 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 ele-ments of A of all possible lengths will be denoted by A (cid:63) . A cylinder set generatedby the block b = b b . . . , b n and anchored at i is defined as [ b ] i = { x ∈ A Z : x [ i , i + n ) = b } . (1)When one of the indices i , i + , . . . , i + n − elementary . The collection (class) of all elementary cylinder sets of X together with the empty set and the whole space X will be denoted by Cyl ( X ) . Onecan show that Cyl ( X ) constitutes a semi-algebra over X . Moreover, one can showthat any finitely additive map µ : Cyl ( X ) → [ , ∞ ] for which µ ( ∅ ) = Cyl ( X ) .The semi-algebra of elementary cylinder sets, equipped with a measure is still“too small” a class of subsets of X to support all requirements of probability theory.For this we need a σ -algebra, that is, a class of subsets of X that is closed under thecomplement and under the countable unions of its members. Such σ -algebra canbe defined as an “extension” of Cyl ( X ) . The smallest σ -algebra containing Cyl ( X ) will be called σ -algebra generated by Cyl ( X ) . As it turns out, it is possible to ex-tend a measure on semi-algebra to the σ -algebra generated by it, by using Hahn-Kolmogorov theorem. Contents
In what follows, we will only consider measures for which µ ( X ) = b ∈ A (cid:63) , µ ([ b ] i ) is independent of i . To simplify notation, we then define P : A (cid:63) → [ , ] as P ( b ) : = µ ([ b ] i ) . (2)Values P ( b ) will be called block probabilities . Application of Hahn-Kolmogorovtheorem to the case of shift-invariant probability measure µ yields the followingresult. Theorem 1
Let P : A (cid:63) → [ , ] satisfy the conditionsP ( b ) = ∑ a ∈ A P ( b a ) = ∑ a ∈ G P ( a b ) ∀ b ∈ A (cid:63) , (3)1 = ∑ a ∈ A P ( a ) . (4) Then P uniquely determines shift-invariant probability measure on the σ -algebragenerated by elementary cylinder sets of X. The set of shift-invariant probability measures on the σ -algebra generated by ele-mentary cylinder sets of X will be denoted by M ( X ) .Conditions (3) and (4) are often called consistency conditions , although they areessentially equivalent to measure additivity conditions. Some consequences of con-sistency conditions in the context of cellular automata have been studied in detailby McIntosh (2009). Since the probabilities P ( b ) uniquely determine the probability measure, we candefine a shift-invariant probability measure by specifying P ( b ) for all b ∈ A (cid:63) . Ob-viously, because of consistency conditions, block probabilities are not independent,thus in order to define the probability measure, we actually need to specify only some of them, but not necessarily all - the missing ones can be computed by usingconsistency conditions.Define P ( k ) to be the column vector of all probabilities of blocks of length k arranged in lexical order. For example, for A = { , } , these are P ( ) = [ P ( ) , P ( )] T , P ( ) = [ P ( ) , P ( ) , P ( ) , P ( )] T , P ( ) = [ P ( ) , P ( ) , P ( ) , P ( ) , P ( ) , P ( ) , P ( ) , P ( )] T , · · · . ontents 7 The following result (Fuk´s, 2013) is a direct consequence of consistency conditionsof eq. (3) and (4).
Proposition 1
Among all block probabilities constituting components of P ( ) , P ( ) , . . . , P ( k ) only ( N − ) N k − are linearly independent. For example, for N = k =
3, among P ( ) , P ( ) , P ( ) (which have, in total, 8 + + =
14 components), there are only 4 independent blocks. These four block canbe selected somewhat arbitrarily (but not completely arbitrarily). Two methods oralgorithms for selection of independent blocks are especially convenient.The first one is called long block representation . It is based on the followingproperty (cf. ibid.).
Proposition 2
Let P ( k ) be partitioned into two sub-vectors, P ( k ) = ( P ( k ) Top , P ( k ) Bot ) ,where P ( k ) Top contains first N k − N k − entries of P ( k ) , and P ( k ) Bot the remaining N k − entries. Then P ( k ) Bot = ... − (cid:16) B ( k ) (cid:17) − A ( k ) P ( k ) Top . (5)In the above, matrix B ( k ) is constructed from zero N k − × N k − matrix by placing − B ( k ) = − · · · − . (6)The matrix A ( k ) is a bit more complicated, A ( k ) = [ J J . . . J N − ] + [ B ( k ) B ( k ) . . . B ( k ) (cid:124) (cid:123)(cid:122) (cid:125) N − ] , (7)where J m is an N k − × N k − matrix in which m -th row consist of all 1’s, and allother entries are equal to 0.The above proposition means that among block probabilities constituting com-ponents of P ( ) , P ( ) , . . . , P ( k ) , we can treat first N k − N k − entries of P ( k ) as inde-pendent variables. Remaining components of P ( k ) can be obtained by using eq. (5),while P ( ) , P ( ) , . . . , P ( k − ) can be obtained by eq. (3).When applied to the N = k = P ( ) , P ( ) , P ( ) and P ( ) . The remaining 10 probabilitiescan then be expressed as follows, Contents P ( ) P ( ) P ( ) P ( ) = P ( ) − P ( ) + P ( ) + P ( ) P ( ) − P ( ) − P ( ) − P ( ) − P ( ) , P ( ) P ( ) P ( ) P ( ) = P ( ) + P ( ) P ( ) + P ( ) P ( ) + P ( ) − P ( ) − P ( ) − P ( ) − P ( ) , (cid:20) P ( ) P ( ) (cid:21) = (cid:20) P ( ) + P ( ) + P ( ) + P ( ) − P ( ) − P ( ) − P ( ) − P ( ) (cid:21) . (8)Of course, this is not the only choice. Alternatively, we can choose as indepen-dent blocks the shortest possible blocks. The algorithm resulting in such a choicewill be called short block representation . In order to describe it in a formal way,let us define a vector of admissible entries for short block representation, P ( k ) adm , asfollows. Let us take vector P ( k ) in which block probabilities are arranged in lex-icographical order, indexed by an index i which runs from 1 to N k . Vector P ( k ) adm consists of all entries of P ( k ) for which the index i is not divisible by N and forwhich i < N k − N k − . For example, for N = k = P ( ) = [ P ( ) , P ( ) , P ( ) , P ( ) , P ( ) , P ( ) , P ( ) , P ( ) , P ( )] T , and we need to select entries with i not divisible by 3 and i <
6, which leaves i = , , ,
5, hence P ( ) adm = [ P ( ) , P ( ) , P ( ) , P ( )] T . Vector of independent block probabilities in short block representation is nowdefined as P ( k ) short = P ( ) adm P ( ) adm ... P ( k ) adm . (9)The following result can be established. Proposition 3
Among block probabilities constituting components of P ( ) , P ( ) , . . . P ( k ) , we can treat entries of P ( k ) short as independent variables. One can express allother components of P ( ) , P ( ) , . . . , P ( k ) in terms of components P ( k ) short . The exact formulae expressing components of P ( ) , P ( ) , . . . , P ( k ) in terms of compo-nents P ( k ) short are rather complicated, and can be found in (Fuk´s, 2013). As an exam-ple, for N = k =
3, this algorithm yields P ( ) , P ( ) , P ( ) and P ( ) to bethe independent block probabilities, that is, the components of P ( ) short . The remaining10 dependent blocks probabilities can be expressed in terms of P ( ) , P ( ) , P ( ) and P ( ) . ontents 9 P ( ) P ( ) P ( ) P ( ) P ( ) P ( ) = P ( ) − P ( ) P ( ) − P ( ) − P ( ) P ( ) − P ( ) P ( ) − P ( ) + P ( ) P ( ) − P ( ) − P ( ) − P ( ) + P ( ) + P ( ) . P ( ) P ( ) P ( ) = P ( ) − P ( ) P ( ) − P ( ) − P ( ) + P ( ) , P ( ) = − P ( ) . (10) Let w : A × A r + → [ , ] , whose values are denoted by w ( a | b ) for a ∈ A , b ∈ A r + , satisfying ∑ a ∈ A w ( a | b ) =
1, be called local transition function of radius r ,and its values called local transition probabilities . Probabilistic cellular automaton with local transition function w is a map F : M ( X ) → M ( X ) defined as ( F µ )([ a ] i ) = ∑ b ∈ A | a | + r w ( a | b ) µ ([ b ] i − r ) for all i ∈ Z , a ∈ A (cid:63) , (11)where we define w ( a | b ) = | a | ∏ j = w ( a j | b j b j + . . . b j + r ) . (12)When the function w takes values in the set { , } , the corresponding cellular au-tomaton is called deterministic cellular automaton .For any shift-invariant probability measure µ ∈ M ( X ) , we define the orbit of µ under F as { F n µ } ∞ n = , (13)where F µ = µ . Points of the orbit of µ under F are uniquely determined by proba-bilities of cylinder sets. Thus, if we define, for n ≥ P n ( a ) = ( F n µ )([ a ] i ) , then, for a ∈ A k , eq. (11) becomes P n + ( a ) = ∑ b ∈ A | a | + r w ( a | b ) P n ( b ) . (14)In the above we assume that P ( a ) = µ ([ a ] i ) .Given the measure µ , equations (14) define a system of recurrence relationshipfor block probabilities. Solving this recurrence system, that is, finding P n ( a ) forall n ∈ N and all a ∈ A (cid:63) , would be equivalent to determining the orbit of µ un-der F . However, it is very difficult to solve these equations explicitly, and no gen-eral method for doing this is known. To see the source of the difficulty, let us take A = { , } and let us consider the example of rule 14, for which local transitionsprobabilities are given by w ( | ) = , w ( | ) = , w ( | ) = , w ( | ) = , w ( | ) = , w ( | ) = , w ( | ) = , w ( | ) = , (15)and w ( | x x x ) = − w ( | x x x ) for all x , x , x ∈ { , } . For k =
2, eq. (14)becomes P n + ( ) = P n ( ) + P n ( ) + P n ( ) + P n ( ) + P n ( )+ P n ( ) , P n + ( ) = P n ( ) + P n ( ) + P n ( ) + P n ( ) , P n + ( ) = P n ( ) + P n ( ) + P n ( ) + P n ( ) , P n + ( ) = P n ( ) + P n ( ) . (16)It is obvious that this system of equations cannot be iterated over n , because on theleft hand side we have probabilities of blocks of length 2, and on the right handside – probabilities of blocks of length 4. Of course, not all these probabilities areindependent, thus it will be better to rewrite the above using short form represen-tation. Since among block probabilities of length 2 only 2 are independent, we cantake only two of the above equations, and express all block probabilities occurringin them by their short form representation, using eq. (10). This reduces eq. (16) to P n + ( ) = − P n ( ) + P n ( ) , P n + ( ) = − P n ( ) + P n ( ) + P n ( ) . (17)Although much simpler, the above system of equations still cannot be iterated, be-cause on the right hand side we have an extra variable P n ( ) . To put it differently,one cannot reduce iterations of F to iterations of a finite-dimensional map (in thiscase, two-dimensional map). For this reason, a special method has been developedto approximate orbits of F by orbits of finite-dimensional maps. For a given measure µ , it is clear that the knowledge of P ( k ) is enough to determineall P ( i ) with i < k , by using consistency conditions. What about i > k ? Obviously,since the number of independent components in P ( i ) is greater than in P ( k ) for i > k , there is no hope to determine P ( i ) using only P ( k ) . It is possible, however, toapproximate longer block probabilities by shorter block probabilities using the ideaof Bayesian extension.Suppose now that we want to approximate P ( a a . . . a k + ) by P ( a a . . . a k ) .One can say that by knowing P ( a a . . . a k ) we know how values of individual sym- ontents 11 bols in a block are correlated providing that symbols are not farther apart than k − k is extended by adding anothersymbol to it on the right, then the the conditional probability of finding a particularvalue of that symbol does not significantly depend on the left-most symbol, i.e., P ( a a . . . a k + ) P ( a . . . a k ) ≈ P ( a . . . a k + ) P ( a . . . a k ) . (18)This produces the desired approximation of k + k -block and k − P ( a a . . . a k + ) ≈ P ( a . . . a k ) P ( a . . . a k + ) P ( a . . . a k ) , (19)where we assume that the denominator is positive. If the denominator is zero, thenwe take P ( a a . . . a k + ) =
0. In order to avoid writing separate cases for denomina-tor equal to zero, we define “thick bar” fraction as ab : = ab if b (cid:54) =
00 if b = . (20)Now, let µ ∈ M ( X ) be a measure with associated block probabilities P : A (cid:63) → [ , ] , P ( b ) = µ ([ b ] i ) for all i ∈ Z and b ∈ A (cid:63) . For k >
0, define (cid:101) P : A (cid:63) → [ , ] such that (cid:101) P ( a a . . . a p ) = P ( a a . . . a p ) if p ≤ k , ∏ p − k + i = P ( a i . . . a i + k − ) ∏ p − ki = P ( a i + . . . a i + k − ) otherwise . (21)Then (cid:101) P determines a shift-invariant probability measure (cid:101) µ ( k ) ∈ M ( X ) , to be called Bayesian approximation of µ of order k .When there exists k such that Bayesian approximation of µ of order k is equal to µ , we call µ a Markov measure or a finite block measure of order k . The space ofshift-invariant probability Markov measures of order k will be denoted by M ( k ) ( X ) , M ( k ) ( X ) = { µ ∈ M ( X ) : µ = (cid:101) µ ( k ) } . (22)It is often said that the Bayesian approximation “maximizes entropy”. Le us de-fine entropy density of shift-invariant measure µ ∈ M ( X ) as h ( µ ) = lim n → ∞ − n ∑ b ∈ A n P ( b ) log P ( b ) , (23) where, as usual, P ( b ) = µ ([ b ] i ) for all i ∈ Z and b ∈ A (cid:63) . It has been established byFannes and Verbeure (1984) that for any µ ∈ M ( X ) , the entropy density of the k -thorder Bayesian approximation of µ is given by h ( (cid:101) µ ( k ) ) = ∑ a ∈ A k − P ( a ) log P ( a ) − ∑ a ∈ A k P ( a ) log P ( a ) , (24)and that for any µ ∈ M ( X ) and any k >
0, the entropy density of µ does not exceedthe entropy density of its k -th order Bayesian approximation, h ( µ ) ≤ h ( (cid:101) µ ( k ) ) . (25)Moreover, one can show that the sequence of k -th order Bayesian approximationsof µ ∈ M ( X ) weakly converges to µ as k → ∞ (Gutowitz et al, 1987).Using the notion of Bayesian extension, H. Gutowitz et. al. developed a methodof approximating orbits of F , known as the local structure theory (Gutowitz et al,1987; Gutowitz and Victor, 1987). Following these authors, let us define the scram-ble operator of order k , denoted by Ξ ( k ) , and defined as Ξ ( k ) µ = (cid:101) µ ( k ) . (26)The scramble operator, when applied to a shift invariant measure µ , produces aMarkov measure of order k which agrees with µ on all blocks of length up to k . Theidea of local structure approximation is that each time step, instead of just applying F , we apply scramble operator, then F , and then the scramble operator again. Thisyields a sequence of Markov measures ν ( k ) n defined recursively as ν ( k ) n + = Ξ ( k ) F Ξ ( k ) ν ( k ) n , ν ( k ) = µ . (27)The sequence defined as (cid:110)(cid:16) Ξ ( k ) F Ξ ( k ) (cid:17) n µ (cid:111) ∞ n = (28)will be called the local structure approximation of level k of the exact orbit { F n µ } ∞ n = . Note that all terms of this sequence are Markov measures, thus the entirelocal structure approximation of the orbit lies in M ( k ) ( X ) . The following theoremdescribes the local structure approximation in a formal way. Theorem 2
For any positive integer n, and for any shift invariant probability mea-sure µ , ν ( k ) n weakly converges to F n µ as k → ∞ . A nice feature of Markov measures is that they can be entirely described by speci-fying probabilities of a finite number of blocks. This makes construction of finite-dimensional maps generating approximate orbits of measures in CA possible. ontents 13
Define Q n ( b ) = ν ( k ) n ([ b ]) . Using definitions of F and Ξ , eq. (27) yields, for any a ∈ A k , Q n + ( a ) = ∑ a ∈ A | b | + r w ( a | b ) ∏ r + i = Q n ( b [ i , i + k − ] ) ∏ ri = ∑ c ∈ A Q n ( c b [ i + , i + k − ] ) . (29)If we arrange Q n ( a ) for all a ∈ A k in lexicographical order to form a vector Q n ,we will obtain Q n + = L ( k ) ( Q n ) , (30)where L ( k ) : [ , ] | A | k → [ , ] | A | k has components defined by eq. (29). L ( k ) will becalled local structure map of level k .Of course, not all components of Q are independent, due to consistency con-ditions. We can, therefore, further reduce dimensionality of local structure map to ( N − ) N k − dimensions. This will be illustrated for rule 14 considered earlier.Recall that for rule 14, if we start with an initial measure µ and define P n ( b ) =( F n µ )[ b ] , then P n + ( ) = − P n ( ) + P n ( ) , P n + ( ) = − P n ( ) + P n ( ) + P n ( ) . (31)The corresponding local structure map can be obtained from the above by simplyreplacing P by Q and using the fact that block probabilities Q represent Markovmeasure of order k , thus Q n ( ) = Q n ( ) Q n ( ) Q n ( ) . (32)Equations (17) would then become Q n + ( ) = − Q n ( ) + Q n ( ) Q n ( ) , Q n + ( ) = − Q n ( ) + Q n ( ) + Q n ( ) Q n ( ) , (33)where Q ( ) = P ( ) , Q ( ) = P ( ) . The above is a formula for recursive it-eration of a two-dimensional map, thus one could compute Q n ( ) and Q n ( ) forconsecutive n = , . . . without referring to any other block probabilities, in starkcontrast with eq. (17). Block probabilities Q approximate exact block probabilities P , and the quality of this approximation varies depending on the rule. Nevertheless,as the order of approximation k increases, values of Q become closer and closer to P , due to the weak convergence of ν ( k ) n to F n µ .As an illustration of this convergence, let us consider a probabilistic rule definedby w ( | ) = , w ( | ) = α , w ( | ) = − α , w ( | ) = − α , w ( | ) = α , w ( | ) = , w ( | ) = − α , w ( | ) = − α , (34)and w ( | x x x ) = − w ( | x x x ) for all x , x , x ∈ { , } , where α ∈ [ , ] is aparameter. This rule is known as α -asynchronous elementary rule 18 Fat`es (2009),because for α = µ / , then lim n → ∞ P n ( ) = α ≤ α c , and lim n → ∞ P n ( ) > α > α c , where α c ≈ .
7. This phenomenon canbe observed in simulations, if one iterates the rule for large number of time steps T and records P T ( ) . The graph of P T ( ) as a function of α for T = , obtained bysuch direct simulations of the rule, is shown in Figure 1. To approximate P T ( ) bylocal structure theory, one can construct local structure map of order k for this rule,iterate it T times, and obtain Q T ( ) , which should approximate P T ( ) . The graphsof Q T ( ) vs. α , obtained this way, are shown in Figure 1 as dashed lines. One canclearly see that as k increases, the dashed curves approximate the graph of P T ( ) better and better. P T ( ) α simulation LS 2 LS 3 LS 4 LS 5 LS 6 LS 9 Fig. 1
Graph of P T ( ) for T = as a function α for probabilistic CA rule defined in eq. (34).Continuous line represents values of P T ( ) obtained by Monte Carlo simulations, and dashed linesvalues of Q T ( ) obtained by iterating local structure maps of level k = , , , For some simple CA rules, the local structure approximation is exact. Such isthe case of idempotent rules, that is, CA rules for which F = F . Gutowitz et al(1987) found that this is also the case for what he calls linear rules, toggle rules, andasymptotically trivial rules. ontents 15 If approximations provided by the local structure theory are not enough, one canattempt to compute orbits of Bernoulli measures exactly. Typically, it is not possibleto obtain expressions for all block probabilities P n ( a ) along the orbit, yet one canoften compute P n ( a ) if a is short, for example, containing just one, two, or threesymbols.For elementary CA rules, the behaviour of P n ( ) as a function of n has been stud-ied extensively by many authors, starting from Wolfram (1983), who determinednumerical values of P ∞ ( ) for a wide class of CA rules and postulated exact valuesfor some of them. Later one exact values of P n ( ) have been established for someelementary rules, and in some cases, P n ( a ) has been computed for all | a | ≤
3. Wewill discuss these results in what follows.When the rule is deterministic, transition probabilities in eq. (11) take values inthe set { , } . Let us consider elementary cellular automata , that is, binary rulesfor which N = A = { , } and the radius r =
1. For such rules, define the localfunction f by f ( x , x , x ) = w ( | x x x ) for all x , x , x ∈ { , } . Elementary CAwith the local local function f are usually identified by their Wolfram number W ( f ) ,defined as (Wolfram, 1983) W ( f ) = ∑ x , x , x = f ( x , x , x ) ( x + x + x ) . A block evolution operator corresponding to f is a mapping f : A (cid:63) (cid:55)→ A (cid:63) definedas follows. Let a = a a . . . a n − ∈ A n where n ≥
3. Then f ( a ) = { f ( a i , a i + , a i + ) } n − i = . (35)For elementary CA eq. (11) reduces to ( F µ )([ a ]) = ∑ b ∈ f − ( a ) µ ([ b ]) , (36)where we dropped indices indicating where the cylinder set is anchored (we assumeshift-invariance of measure µ ), and where f − ( a ) is the set of preimages of a underthe block evolution operator f . This can be generalized to the n -th iterate of F , P n ( a ) = ( F n µ )([ a ]) = ∑ b ∈ f − n ( a ) µ ([ b ]) , (37)where, again, f − n ( a ) is the set of preimages of a under f n , the n -th iterate of f . Thus,if we know the elements of the set of n -step preimages of the block a under theblock evolution operator f , then we can easily compute the probability P n ( a ) .Now, let us suppose that the initial measure is a Bernoulli measure µ p , definedby µ p ([ a ]) = p ( a ) ( − p ) ( a ) , where s ( a ) denotes number of symbols s in a and where p ∈ [ , ] is a parameter. In such a case eq. (37) reduces to P n ( a ) = ∑ b ∈ f − n ( a ) p ( b ) ( − p ) ( b ) . (38)Furthermore, if p = /
2, then the above reduces to even simpler form, P n ( a ) = ∑ b ∈ f − n ( a ) | b | = card f − n ( a ) | a | + n . (39)For many elementary CA rules and for short blocks a , the sets f − n ( a ) exhibitsimple enough structure to be described and enumerated by combinatorial methods,so that the formula for card f − n can be constructed and/or the sum in eq. (38) can becomputed. Although there is no precise definition of “simple enough structure”, theknown cases can be informally classified into five groups:1. rules with preimage sets that are “balanced” (have the same number of preimagesfor each block),2. rules with preimage sets mostly composed of long blocks of identical symbols(having long runs ) or long blocks of arbitrary symbols,3. rules with preimage sets that can be described as sets of strings in which somelocal property holds everywhere,4. rules with preimage sets that can be described as strings in which some global(non-local) property holds,5. rules for which preimage sets are related to preimage sets of some known solv-able rule.Selection of the most interesting examples in each category is given below. It is well known that the symmetric Bernoulli measure µ / is invariant under theaction of a surjective rule (see Pivato, 2009, and references therein). In one dimen-sion, surjectivity is a decidable property, and the relevant algorithm is known, dueto Amoroso and Patt (1972). Among elementary CA rules, surjective rules have thefollowing Wolfram numbers: 15, 30, 45, 51, 60, 90, 105, 106, 150, 154, 170 and204. For all of them, for the initial measure µ = µ / and for any block a , P n ( a ) = −| a | . (40)The above result is a direct consequence of the Balance Theorem, first proved byHedlund (1969), which states that for a surjective rule, card f − ( a ) is the same forall blocks a of a given length. For elementary rules this implies that card f − ( a ) = f − n ( a ) = n . From eq. (39) one then obtains eq. (40). ontents 17 Consider the elementary CA with the local function f (cid:0) x , x , x (cid:1) = (cid:26) ( x x x ) = ( ) or ( ) , W ( f ) = W ( f ) will be referred to as “rule W ( f ) ”.For rule 130 and for µ p , the the probabilities P n ( a ) are known for | a | ≤ P n ( ) . P n ( ) = − p n + − p (cid:16) − p (cid:100) ( n − ) / (cid:101) + + p (cid:100) ( n − ) / (cid:101) + − p (cid:98) n / (cid:99) + + p − p + (cid:17) p + p + p + . (42)The above result is based on the fact that for rule 130, the set f − n ( ) has onlyone element, namely the block 11 . . .
1, hence card f − n ( ) =
1. Moreover, the set f − n ( ) consists of all blocks of the form (cid:63) . . . (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n − i . . . (cid:124) (cid:123)(cid:122) (cid:125) i (if i is odd) or (cid:63) . . . (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n − i . . . (cid:124) (cid:123)(cid:122) (cid:125) i (if i is even),where i ∈ { . . . n } and (cid:63) denotes an arbitrary value in A . Probabilities of occurenceof blocks 111 and 001 can thus be easily computed. Using the fact that for this rule P n ( ) = − P n − ( ) − P n − ( ) , one then obtains eq. (42). The floor and ceilingoperators appear in that formula because different expressions are needed for oddand even n , as it is evident from the structure of preimages of 001 described above.Rule 130 is an example of a rule where convergence of P n ( ) to its limiting value isessentially exponential (like in rule 172 discussed below, except that there are somesmall variations between values corresponding to even and odd n . The local function of rule 172 is defied as f ( x , x , x ) = (cid:26) x if x = x if x =
1. (43)The combinatorial structure of f − n ( a ) for this rule can be described, for some blocks a , as binary strings with forbidden sub-blocks. More precisely, one can prove thefollowing proposition (Fuk´s, 2010). Proposition 4
Block b of length n + belongs to f − n ( ) for rule 172 if and only ifit has the structure b = (cid:63) (cid:63) . . . (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n − (cid:63) (cid:63) . . . (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n , or b = (cid:63) (cid:63) . . . (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n − a a . . . a n + c c , wherea a . . . a n is a binary string which does not contain any pair of adjacent zeros, andc c = (cid:40) (cid:63), if a n + = ,(cid:63) , otherwise . (44)Since the number of binary strings of length n without any pair of consecutive zerosis know to be F n + , where F n is the n -th Fibonacci number, it is not surprising thatFibonacci numbers appear in expressions for block probabilities of rule 172. Forthis rule and µ = µ / , probabilities P n ( a ) are known for | a | ≤
3, as shown below. P n ( ) = − F n + n + , (45) P n ( ) = / − − n − F n + − − n − F n + , P n ( ) = / − − n − F n + − − n − F n + , P n ( ) = / − − n − F n + , Note that the above are probabilities in short block representation, thus all remain-ing probabilities of blocks of length up to 3 can be obtained using eq. (10). Morerecently, P n ( ) has been computed for arbitrary µ p (Fuk´s, 2016b), P n ( ) = − ( − p ) p − p λ − λ (cid:0) α λ n − + α λ n − (cid:1) , (46)where λ , = p ± (cid:112) p ( − p ) , (47) α , = (cid:16) p − (cid:17) (cid:112) p ( − p ) ± (cid:18) p − (cid:19) . (48) While in rule 172 the ombinatorial description of sets f − n ( a ) involved some localconditions (e.g., two consecutive zeros are forbidden), in rule 184, with the localfunction f ( x , x , x ) = x + x x − x x , the conditions are more of a global nature,that is, involving properties of longer substrings. In particular, one can show thefollowing. Proposition 5
The block b b . . . b n + belongs to f − n ( ) under rule if andonly if b = , b = and + ∑ ki = ξ ( b i ) > for every ≤ k ≤ n + , where ξ ( ) = , ξ ( ) = − . ontents 19 Proof of this property relies on the fact that rule 184 is known to be equivalent toa ballistic annihilation process (Krug and Spohn, 1988; Fuk´s, 1999; Belitsky andFerrari, 2005). Another crucial property of rule 184 is that it is number-conserving,that is, conserves the number of zeros and ones. Using this fact and the above propo-sition, probabilities P n ( a ) can be computed for for µ p and | a | ≤ P n ( ) = − p , P n ( ) = n + ∑ j = jn + (cid:18) n + n + − j (cid:19) p n + − j ( − p ) n + + j . (49)The main idea which is used in deriving the above expression is the fact that preim-age sets f − ( ) have a similar structure to trajectories of one-dimensional randomwalk starting from the origin and staying on the positive semi-axis. Enumeration ofsuch trajectories is a well known combinatorial problem, and the binomial coeffi-cient appearing in the expression for P n ( ) indeed comes from this enumerationprocedure. In the limit of large n one can demonstrate thatlim n → ∞ P n ( ) = (cid:26) − p if p < / , . (50)All the above results can be extended to generalizations of rule 184 with largerradius (Fuk´s, 1999).A special case of µ / is especially interesting, as in this case probabilities ofblocks up to length 3 can be obtained, P n ( ) = , (51) P n ( ) = − − n (cid:18) n + n + (cid:19) , (52) P n ( ) = − n − (cid:18) n + n + (cid:19) , (53) P n ( ) = − · − − n (cid:18) n + n + (cid:19) . (54)Using Stirling’s approximation for factorials for large n , one obtains P n ( ) ∼ n − / ,thus P n ( ) converges to 0 as a power law with exponent 1 / The local function of rule 14 is defied by f ( , , ) = f ( , , ) = f ( , , ) =
1, and f ( x , x , x ) = ( x , x , x ) ∈ { , } . For rule 14 and µ = µ / ,the probabilities P n ( a ) are known for | a | ≤
3, and are given by P n ( ) = (cid:18) + n − n C n − (cid:19) , (55) P n ( ) = − − n ( n + ) C n + , (56) P n ( ) = − n − ( n + ) C n , (57) P n ( ) = − − n ( n + ) C n , (58)where C n is the n -th Catalan number (Fuk´s and Haroutunian, 2009). These formulaewere obtained using the fact that this rule conserves the number of blocks 10 andthat the combinatorial structure of preimage sets of some short blocks resembles thestructure of related preimage sets under the rule 184. More precisely, computationof the above block probabilities relies on the following property (see ibid. for proof). Proposition 6
For any n ∈ N , the number of n-step preimages of 101 under the rule14 is the same as the number of n-step preimages of 000 under the rule 184, that is, card f − n ( ) = card f − n ( ) , (59) where subscripts 184 and 14 indicate block evolution operators for, respectively,CA rules 184 and 14. Moreover, the bijection M n from the set f − n ( ) to the set f − n ( ) is defined byM n ( x x . . . x m ) = (cid:40) n + j + + j ∑ i = x i mod 2 (cid:41) mj = (60) for m ∈ N and for x x . . . x m ∈ { , } m . As in the case of rule 184, one can show that for rule 14 and large n , P n ( ) ≈ + √ π n − . (61)The power law which appears here exhibits the same exponent as in the case of rule184 for P n ( ) . The examples shown in the previous sections indicate that in all cases for which P n ( a ) can be computed exactly, as n → ∞ , P n ( a ) remains either constant, or con-verges to its limiting value exponentially or as a power law. The exponential conver-gence is the most prevalent. Indeed, for many other elementary CA rules for whichformulae for P n ( ) are either known or conjectured, the exponential convergence to P ∞ ( ) can be observed most frequently. This includes 15 elementary rules which areknown as asymptotic emulators of identity (Rogers and Want, 1994; Fuk´s and Soto, ontents 21 P n ( ) for the initial measure µ / for these rules are shownbelow. Starred rules are those for which a formal proof has been published in theliterature (see Fuk´s and Soto, 2014, and references therein). • Rule 13: P n ( ) = / − ( − ) − n − • Rule 32 (cid:63) : P n ( ) = − − n • Rule 40: P n ( ) = − n − • Rule 44: P n ( ) = / + − n • Rule 77 (cid:63) : P n ( ) = / • Rule 78: P n ( ) = / • Rule 128 (cid:63) : P n ( ) = − − n • Rule 132 (cid:63) : P n ( ) = / + − n • Rule 136 (cid:63) : P n ( ) = − n − • Rule 140 (cid:63) : P n ( ) = / + − n − • Rule 160 (cid:63) : P n ( ) = − n − • Rule 164: P n ( ) = / − − n + − n • Rule 168 (cid:63) : P n ( ) = n − n − • Rule 172 (cid:63) : P n ( ) = + ( − √ )( −√ ) n +( + √ )( + √ ) n · n • Rule 232: P n ( ) = / P n ( ) ∼ n − / . The above power law can be explained by the fact that in rule 18 one can viewsequences of 0’s of even length as “defects” which perform a random walk andannihilate upon collision, as discovered numerically by Grassberger (1984) and laterformally demonstrated by Eloranta and Nummelin (1992). A very general treatmentof particle kinematics in CA confirming this result can be found in the work ofPivato (2007).Another example of an interesting power law appears in rule 54, for which Boc-cara et al (1991) numerically verified that P n ( ) ∼ n − γ , where γ ≈ .
15. Particle kinematics of rule 54 is now very well understood (Pivato,2007), but the above power law has not been formally demonstrated, and the exactvalue of the exponent γ remains unknown. For probabilistic rules, one cannot use eq. (38) because the block evolution operator f − n does not have any obvious non-deterministic version. Once thus has to workdirectly with eq. (11).Equation (11) can be written for the n -th iterate of F , ( F n µ )([ a ] i ) = ∑ b ∈ A | a | + nr w n ( a | b ) µ ([ b ] i − nr ) for all i ∈ Z , a ∈ A (cid:63) , (62)where we define the n-step block transition probability w n recursively, so that, when n ≥ c a ∈ A (cid:63) and b ∈ A | a | + rn , w n ( a | b ) = ∑ b (cid:48) ∈ A | a | + r ( n − ) w n − ( a | b (cid:48) ) w ( b (cid:48) | b ) . (63)The n -step block transition probability w n ( a | b ) can be intuitively understood as theconditional probability of seeing the block a after n iterations of F , conditioned onthe fact that the original configuration contained the block b .Using definition of w given in eq. (12), one can produce an explicit formula for w n , w n ( a | b ) = ∑ b n − ∈ A | a | + r ( n − ) ... b ∈ A | a | + r w ( a | b ) (cid:32) n − ∏ i = w ( b i | b i + ) (cid:33) w ( b n − | b ) . (64)For a shift-invariant initial probability measure µ , equation (62) becomes P n ( a ) = ∑ b ∈ A | a | + nr w n ( a | b ) P ( b ) . (65)Since some of the transition probabilities may be zero, we define, for any block b ∈ A (cid:63) , supp w n ( a |· ) = { b ∈ A | a + nr : w n ( a | b ) > } , (66)and then we have P n ( a ) = ∑ b ∈ supp w n ( a |· ) w n ( a | b ) P ( b ) . (67)In some cases, supp w n ( a |· ) is small and has a simple structure, and the needed w n ( a | b ) can be computed directly from eq. (64). This approach has been success-fully used for a class of probabilistic CA rules known as α -asynchronous rules withsingle transitions (Fuk´s and Skelton, 2011a). We show two examples of such rulesbelow. ontents 23 Rule 200A, known as α -asynchronous rule 200, is defined by transition probabilities w ( | b ) = b ∈ { , , , } , b ∈ { , , } , − α if b = , (68)and w ( | b ) = − w ( | b ) for all b ∈ { , } , where α ∈ [ , ] is a parameter. The setsupp w n ( |· ) for this rule consists of all blocks of the form (cid:63) · · · (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n (cid:63) · · · (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n . (69)Moreover, one can show that for any block b ∈ supp w n ( |· ) where the central 1 has0s as both neighbours, w n ( | b ) = ( − α ) n , while w n ( | b ) = P n ( ) and P n ( ) from eq. (67) directly. By the same methodprobabilities of other blocks of length up to 3 can be computed for rule 200A and µ p , and results are shown below. P n ( ) = − p ( − p ) − ( − p ) n p ( − p ) , (70) P n ( ) = − ( − + p ) ( − p + ( − α ) n p − ) , P n ( ) = − p ( − + p ) ( − α ) n + p (cid:0) p − (cid:1) ( − + p ) ( − α ) n − (cid:0) p + p − p − (cid:1) ( − + p ) , P n ( ) = ( − α ) n p ( − p ) . Exponential convergence toward limiting values can clearly be observed in all ofthese block probabilities.
Another example is rule 140A, defined as w ( | b ) = b ∈ { , , , } , b ∈ { , , } , − α if b = , (71)and w ( | b ) = − w ( | b ) for all b ∈ { , } . For this rule the set supp w n ( |· ) hasthe same structure as for rule 200A, except that the values of w n ( | b ) are different.For blocks b ∈ A n + having the structure (cid:63) · · · (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n − · · · (cid:124) (cid:123)(cid:122) (cid:125) k − (cid:63) · · · (cid:63) (cid:124) (cid:123)(cid:122) (cid:125) n − k , (72)where 0 ≤ k − ≤ n , it has been demonstrated (Fuk´s and Skelton, 2011a) that w n ( | b ) = β n if k = , β n (cid:16) αβ (cid:17) k − (cid:0) n − k − (cid:1) + β n − k + k − ∑ j = (cid:0) n − k + jj (cid:1) α j if 2 ≤ k ≤ n , k = n + . (73)For all other blocks in supp w n ( |· ) one has w n ( | b ) =
1. Using this result, proba-bility P n ( ) can be computed assuming initial measure µ p , although the summationin eq. (62) is rather complicated. The end result, shown below, is nevertheless sur-prisingly simple. P n ( ) = − ρ ( − ρ ) − ρ ( − ( − ρ ) α ) n . (74)Corresponding formulae for P n ( a ) for all | a | ≤ Another case when eq. (14) becomes solvable is when there exists a subset of blockswhich is called complete . A set of words A (cid:63) ⊃ C = { a , a , a , . . . } is complete withrespect to a CA rule F if for every a ∈ C and n ∈ N , P n + ( a ) can be expressed asa linear combination of P n ( a ) , P n ( a ) , P n ( a ) , . . . . In this case, one can write eqs.(14) for blocks of the complete set only, and the right hand sides of them will alsoonly include probabilities of blocks from the complete set. This way, a well-posedsystem of recurrence equations is obtained, and (at least in principle) it should besolvable.This approach has been recently applied to a probabilistic CA rule defined by w ( | ) = , w ( | ) = α , w ( | ) = , w ( | ) = , (75) w ( | ) = β , w ( | ) = γ , w ( | ) = , w ( | ) = , and w ( | b ) = − w ( | b ) for all b ∈ { , } , where α , β , γ ∈ [ , ] are fixed pa-rameters. This rule can be viewed as a generalized simple model for diffusion ofinnovations on one-dimensional lattice (Fuk´s, 2016a). The complete set for this ruleconsists of blocks 101, 1001, 100001, . . . . Equations (14) for blocks of the completeset simplify to P n + ( ) = ( − γ ) P n ( ) + ( α − αβ + β ) P n ( ) + αβ P n ( ) , (76)and, for k > ontents 25 P n + ( k ) = ( − α )( − β ) P n ( k )+( α − αβ + β ) P n ( k + )+ αβ P n ( k + ) . (77)The above equations can be solved, and, using the cluster expansion formula (Stauf-fer and Aharony, 1994), P n ( ) = ∞ ∑ k = kP n ( k ) , (78)one obtains, assuming that the initial measure is µ p , P n ( ) = (cid:40) E (( p β − ) ( p α − )) n + F ( − γ ) n if αβ p − ( α + β ) p + γ (cid:54) = , ( G + Hn )( − γ ) n − if αβ p − ( α + β ) p + γ = , (79)where E , F , G , H are constants depending on parameters α , β , γ and p (for detailedformulae, see Fuk´s, 2016a). For αβ p − ( α + β ) p + γ =
0, this is an example ofa linear-exponential convergence of P n ( ) toward its limiting value, the only oneknown for a binary rule. Both approximate and exact methods for computing orbits of Bernoulli measuresunder the action of cellular automata need further development.Regarding approximate methods, although some simple classes of CA rules forwhich local structure approximation becomes exact are known, it is not known ifthere exist any wider classes of non-trivial rules for which this would be the case.This is certainly an area which needs further research. There seems to be someevidence that orbits of many deterministic rules possessing additive invariants arevery well approximated by local structure theory, but no general results are known.Regarding exact methods, the situation is similar. Although methods for comput-ing exact values of P n ( a ) presented here are applicable to many different rules, it isstill not clear if they are applicable to some wider classes of CA in general. Somesuch classes has been proposed, but formal results are still lacking. For example,there is a number of rules for which convergence of P n ( ) to its limiting value P ∞ ( ) is known to be exponential, and it has been conjectured that for all rules known as asymptotic emulators of identity this is indeed the case. However, there seems tobe some recent evidence that for rule 164, which belongs to asymptotic emulatorsof identity, the convergence is not exactly exponential (A. Skelton, private commu-nication). Are then other classes of CA rules for which the convergence is alwaysexponential? And, more importantly, are there any wide classes of non-trivial CAfor which exact formulae for probabilities of short block are obtainable?Another interesting question is the relationship between exact orbits of CA rulesand approximate orbits obtained by iterating local structure maps. Which featuresor exact orbits are “inherited” by approximate orbits? It seems that often existenceof additive invariants is “inherited” by local structure maps, yet more work in this direction is needed. On a related note, such behavior of P n ( ) as observed in rules172 or 140A (discussed earlier in this article) strongly resembles hyperbolicity infinitely-dimensional dynamical systems. Hyperbolic fixed points are common typeof fixed points in dynamical systems. If the initial value is near the fixed point andlies on the stable manifold, the orbit of the dynamical system converges to thefixed point exponentially fast. One could argue that the exponential convergenceto P ∞ ( ) observed in such rules as rule 172 or 140A is somewhat related to finitely-dimensional hyperbolicity. Since local structure maps which approximate dynamicsof a given CA are finitely-dimensional, one could ask what is the nature of theirfixed points – are these hyperbolic for CA exhibiting hyperbolic-like dynamics? Ishyperbolicity of orbits of CA rules somewhat “inherited” by local structure maps?If so, under what conditions does this happen? All those questions need to be inves-tigated in details in future years.Finally, one should mention that both theoretical developments and examplespresented in this article pertain to one-dimensional cellular automata. Higher-dimensional systems have been studied in the context of the local structure theory(Gutowitz and Victor, 1987), and some examples or two-dimensional cellular au-tomata with exact expressions for small block probabilities are known (Fuk´s andSkelton, 2011b), yet the orbits of Bernoulli measures under higher-dimensional CAare still mostly an unexplored terrain. Given the importance of two- and threepice-dimensional CA in applications, this subject will likely attract some attention in thenear future. References
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