An example of degenerate hyperbolicity in a cellular automaton with 3 states
AAn example of degenerate hyperbolicity ina cellular automaton with 3 states
Henryk Fuk´s and Joel Midgley-Volpato
Department of Mathematics and Statistics, Brock UniversitySt. Catharines, Ontario L2S 3A1, Canada
Abstract.
We show that a behaviour analogous to degenerate hyperbol-icity can occur in nearest-neighbour cellular automata (CA) with threestates. We construct a 3-state rule by “lifting” elementary CA rule 140.Such “lifted” rule is equivalent to rule 140 when arguments are restrictedto two symbols, otherwise it behaves as identity. We analyze the structureof multi-step preimages of 0, 1 and 2 under this rule by using minimal fi-nite state machines (FSM), and exploit regularities found in these FSM.This allows to construct explicit expressions for densities of 0s and 1safter n iterations of the rule starting from Bernoulli distribution. Whenthe initial Bernoulli distribution is symmetric, the densities of all threesymbols converge to their stationary values in linearly-exponential fash-ion, similarly as in finite-dimensional dynamical systems with hyperbolicfixed point with degenerate eigenvalues. In a linear continuous-time dynamical system given by ˙ x = A x , if x : R → R n and A is a real n × n matrix with all eigenvalues distinct and having negative realparts, x ( t ) tends to zero exponentially fast as t → ∞ . The same phenomenoncan be observed in nonlinear system ˙ x = f ( x ) (where f : R n → R n ) in a vicinityof hyperbolic fixed point, as long as the Jacobian matrix of f evaluated at thefixed point has only distinct eigenvalues with negative real parts. If, on the otherhand, the matrix A has degenerate (repeated) eigenvalues, the convergence tothe fixed point can be polynomial-exponential, that is, of the form P ( t ) e − bt ,where P ( t ) is a polynomial and b > (cid:20) x n +1 y n +1 (cid:21) = (cid:20) − (cid:21) (cid:20) x n y n (cid:21) (1)is defined by a matrix which has degenerate (double) eigenvalue , thus polynomial-exponential (linear-exponential in this case) convergence to the fixed point (0 , (cid:20) x n y n (cid:21) = (cid:18) (cid:19) n (cid:20) − n n − n n (cid:21) (cid:20) x y (cid:21) , (2) a r X i v : . [ n li n . C G ] J un nd we can clearly see the aforementioned linear-exponential convergence.Cellular automata are infinitely-dimensional dynamical systems, yet a be-haviour similar to hyperbolicity in finite-dimensional systems has been observedin many of them. In particular, in some binary cellular automata in one dimen-sion, known as asymptotic emulators of identity , if the initial configuration isdrawn from a Bernoulli distribution, the expected proportion of ones (or zeros)tends to its stationary value exponentially fast [3].Furthermore, an example of a probabilistic CA has been recently found [2]where the density of ones converges to its stationary value in a linear-exponentialfashion, just like in the case of degenerate hyperbolic fixed points in finite-dimensional dynamical systems. Could such behaviour be observed in deter-ministic CA as well? The purpose of this paper is to provide an example of suchdeterministic CA.We will consider 3-state nearest-neighbour CA obtained from elementarybinary CA by “lifting” them to 3-states. What we mean by this is the followingconstruction. Let g : { , } → { , } be a local function of elementary CAsatisfying g (0 , ,
0) = 0, g (1 , ,
1) = 1, and let f g : { , , } → { , , } bedefined by f g ( x , x , x ) = g ( x , x , x ) , x , x , x ∈ { , } g (cid:0) x , x , x (cid:1) , x , x , x ∈ { , } g ( x − , x − , x −
1) + 1 , x , x , x ∈ { , } x , otherwise . (3)This construction ensures that when f g is restricted to two symbols only, itbecomes equivalent to g , otherwise it behaves as identity. Conditions g (0 , ,
0) =0, g (1 , ,
1) = 1 ensure that there are no conflicts, so that, for example, f (2 , , f g for a number of elementary rules g . One of themost interesting of them is the case of g being elementary CA rule with Wolframnumber 140, defined as g ( x , x , x ) = x − x x + x x x , (4)where x , x , x ∈ { , } . As we will see, it actually exhibits degenerate hyper-bolicity. In what follows, we will refer to f g with g given by eq. (4) as “rule140”. We will also drop the index g and refer to f g simply as f . An example ofa spatio-temporal pattern produced by this rule is shown in Figure 1.Let us first introduce the notion of density polynomials. Let A = { , , } . Afinite sequence of elements of A , b = b b . . . , b n , will be called a block (or word )of length n . The set of all blocks of elements of A of all possible lengths will bedenoted by A (cid:63) .A block evolution operator corresponding to f is a mapping f : A (cid:63) (cid:55)→ A (cid:63) defined as follows. Let a = a a . . . a n ∈ A n where n ≥
3. Then f ( a ) is a blockof length n − f ( a ) = f ( a , a , a ) f ( a , a , a ) . . . f ( a n − , a n − , a n ) . (5)2 t Fig. 1.
Sample spatio-temporal pattern generated by 3-state rule 140. White, lightergray and darker gray cells (blue in color version) correspond, respectively, to 0, 1 and 2. If f ( b ) = a , than we will say that b is a preimage of a , and write b ∈ f − ( a ).Similarly, if f n ( b ) = a , than we will say that b is an n -step preimage of a , andwrite b ∈ f − n ( a ).Let the density polynomial associated with a string b = b b . . . b n be definedas Ψ b ( p, q, r ) = p ( b ) q ( b ) r ( b ) , (6)where i ( b ) is the number of occurrences of symbol i in b . If A is a set of strings,we define density polynomial associated with A as Ψ A ( p, q, r ) = (cid:88) a ∈ A Ψ a ( p, q, r ) . (7)One can easily show (in a manner similar as done in [3]) that if one startswith a bi-infinite string of symbols drawn from Bernoulli distribution whereprobabilities of 0 , p, q and r , then the proportion of sitesin state k after n iterations of rule f is given by Ψ f − n ( k ) ( p, q, r ). This quantitywill be called density of symbols k after n iterations of f .For 3-state rule 140 defined by eqs. (3) and (4), we generated sets of n -step preimages of 0, 1 and 2 for n varying from 1 to 7. Using AT&T FSMLibrary [1], we constructed minimal finite state machines (FSM) generating thesesets, and we found that these FSM exhibit regularities which can be exploitedto produce general expressions for Ψ f − n ( k ) ( p, q, r ). Results are described below.Formal proofs are omitted for lack of space, but they are available upon requestand will be published elsewhere. The set of n -step perimages of 1 can be described by a finite state machine(FSM) schematically shown in Figure 2. The FSM has four parts, denoted byP,Q,R and S. Parts Q and S are always the same, while parts P and R consists3 a) (b)
012 012 012012 111212 012211212 012211212 012211212 0122112 012121 n-2 n-4
P Q R Sd d dcba cba cba
Fig. 2.
Finite state machine for for 6-step (a) and (b) n -step preimages of 1 under therule 140. of repeated graph fragments, where the number of repetitions is, respectively, n − n − a , the only path with such property is 222,corresponding to density polynomial r . Similarly, if we want to end at b , thepossible paths are (cid:63)
11 and 221, yielding density polynomial ( p + q + r ) q + r q .For the node c , the possible paths are 021 , , ,
101 and 201, so the densitypolynomial is 2 pqr + q r + p q + q p . Finally, for node d , the paths are 022 and122, so that the polynomial is pr + qr . Let us, therefore, define a vector withentries corresponding to density polynomials of paths ending at a , b , c , and d , Q = r ( p + q + r ) q + r q pqr + q r + p q + pq pr + r q . (8)4n a very similar fashion, we can construct a vector holding density polynomialsfor paths starting at nodes of segment S labeled, respectively, a , b , c , and d , andending at the rightmost node, S = q + q r + 2 ( p + q + r ) qr + r qq + q r + ( p + q + r ) qr + r ( p + q + r ) ( p + q + r ) r q + ( p + q + r ) qr + ( p + q + r ) q . (9)It is easy to verify that the above components of S correspond to paths startingfrom a (111 , , (cid:63), (cid:63), b (111 , , (cid:63), (cid:63) (cid:63) ), from c ( (cid:63) (cid:63) (cid:63) ) andfrom d (221 , (cid:63), (cid:63) (cid:63) ).Let us now analyze segment R of the FSM shown in Figure 2. First let ussuppose that there is no repeated part in segment R , as it would be for the caseof n = 4, when the number of repetitions is n − R such that R i,j represents the density polynomial of all paths starting fromnode j of segment Q and ending in node i of segment S, where i, j ∈ { a, b, c, d } .This matrix has the form R = r q q r p + q + r q r . (10)As we can see, the only non-zero entries are diagonal ones and R b,a = q , R c,b = r , R c,d = q . This is because there are only three ways to finish at a different nodethat we started, namely if we start from a and finish at b (generating symbol 1along the way), if we we start from b and finish at c (generating 2), or if we startfrom d and finish at c (generating 1).Suppose now that the repeated fragment in segment R is repeated n − j of segment Q to node i of segment S will be represented by entries of matrix R n − . Furthermore, all paths from the beginning of segment Q to the end ofsegment S will be represented by the density polynomial given by S T R n − Q ,where T denotes transposition (row vector). Since the segment P is representedby ( p + q + r ) n − , the final expression for the density polynomial of n -steppreimages of 1 is Ψ f − n (1) ( p, q, r ) = ( p + q + r ) n − S T R n − Q. (11)5n order to obtain more explicit expression for Ψ ( p, q, r ), we will need to compute R n − . When r (cid:54) = q , R is invertible, and one can diagonalize it, R = L q r p + q + r
00 0 0 r L − , (12)where L = qr q − rrqq − r − q r ( q − r ) − qr − qr ( p + r )( q − r ) qr ( q − r )( p + q ) qrp + pq + pr + qr
00 1 0 1 . (13)This yields, after simplification, Ψ f − n (1) ( p, q, r ) = pq (cid:0) − pr + pq + q (cid:1) ( qλ ) n λ ( p + r ) ( q − r )+ qr (cid:0) − p r + p q + pq − pqr + r − q r (cid:1) ( rλ ) n λ ( p + q ) ( q − r )+ q (cid:0) p + p q + 2 p r + pr + 3 pqr + r + r q + q r (cid:1) λ n λ ( p + r ) ( p + q ) , (14)where we used λ = p + q + r .When r = q , matrix R becomes singular. In can be written in Jordan formas R = L ( M + N ) L − , (15)where L = q − − q ( p + q ) − q p + q − q ( p + q )
00 0 1 1 , M = p + 2 q q q
00 0 0 q , N = . (16)Matrices M and N commute, and matrix N is nilpotent, N = 0. Because ofthis, for any integer k ,( M + N ) k = M k + kN M k − = ( p + 2 q ) k q k q k
00 0 0 q k + kq k −
00 0 0 00 0 0 0 , (17)6nd finally Ψ f − n (1) ( p, q, q ) = ( p + q + q ) n − S T R n − Q = ( p +2 q ) n − S T L ( M + N ) n − L − Q = ( p + 2 q ) n − S T L ( p + 2 q ) n − q n − q n − ( n −
2) 00 0 q n −
00 0 0 q n − L − Q (18)After simplification this yields Ψ f − n (1) ( p, q, q ) = pq ( n + 1) ( qλ ) n λ ( q + p ) + q (cid:0) p + 4 p q + pq − q (cid:1) ( qλ ) n ( q + p ) λ + (cid:0) p + 3 p q + 4 pq + 3 q (cid:1) qλ n λ ( q + p ) , (19)where, as before, λ = p + q + r = p + 2 q .We shall add here that even though eq. (14) and (19) were derived assuming n ≥
4, they happen to be correct for n = 1 , p = 1 , q = 1 and r = 1, then Ψ f − n (1) (1 , ,
1) counts thenumber of preimages of 1. This yields a sequence exhibiting linear-exponentialgrowth, Ψ f − n (1) (1 , ,
1) = (cid:18) n
18 + 736 (cid:19) n + 1112 9 n . (20) For preimages of 0, FSM generating preimage sets are quite similar as for preim-ages of 1, thus we will omit details. Similar analysis as in the previous sectionyields, for r (cid:54) = q , Ψ f − n (0) ( p, q, r ) = (cid:0) − pr + pq + q (cid:1) pq ( qλ ) n λ ( p + r ) ( r − q ) + pr (cid:0) − r p + q p + q − r − qr (cid:1) ( rλ ) n λ ( p + q ) ( r − q )+ (cid:0) p + 2 p q + 2 p r + 2 r p + 3 qpr + 2 q p + r + 2 qr + q + q r (cid:1) pλ n ( p + q ) ( p + r ) λ , (21)and for r = q , Ψ f − n (0) ( p, q, q ) = (cid:0) p + 4 p q + 7 q p + 5 q (cid:1) pλ n λ ( p + q ) − pq ( n + 1) ( qλ ) n λ ( p + q ) − q p (cid:0) p + 8 pq + 6 q (cid:1) ( qλ ) n ( p + q ) λ . (22)7 ) b)
012 012 22 012 n-1 n+1 n .........
Fig. 3.
Finite state machines for 4-step (a) and n -step (b) preimages of 2 under therule 140. Again, when p = q = r = 1, the density polynomial counts preimages of 0, andwe obtain Ψ f − n (0) (1 , ,
1) = 1712 9 n − (cid:18) n
18 + 1936 (cid:19) n . (23)This sequence, similarly as the number of preimages of 1, exhibits linear-exponentialgrowth.For preimages of 2, preimage sets have much simpler structure, shown inFigure 3. Density polynomials for them are given by Ψ f − n (2) ( p, q, r ) = ( q + p ) rλ n − + r λ ( rλ ) n . (24)The number of preimages, obtained by taking p = q = r = 1, is in this case Ψ f − n (2) (1 , ,
1) = 3 n − + 23 9 n , (25)thus no linearity is present. As mentioned in the introduction, density polynomials Ψ f − n ( k ) ( p, q, r ) representprobability of occurence of k after n iterations starting from a Bernoulli distri-bution with probabilities of 0, 1 and 2 equal to, respectively, p , q , and r , where8 + q + r = 1. If we start with a symmetric Bernoulli distribution where r = q ,the probability of occurence of 1 after n steps, to be denoted by P n (1), will begiven by eq. (19) in which we substitute r = q and q = (1 − p ) /
2. This yields,after simplification, P n (1) = P ∞ (1) − ( p − p ) (cid:0) p n − p − pn − p − p + 1 (cid:1) (cid:18) − p (cid:19) n , (26)where P ∞ (1) = (1 − p ) (cid:0) p + 5 p − p + 3 (cid:1) p ) . (27)It is clear that for 0 < p < P n (1) tends to P ∞ (1) as n → ∞ , and thatthe convergence is linear-exponential in n . Such “degenerate” convergence takesplace for probability of occurence of 0 as well, as seen in eq. (22). When r (cid:54) = q in the initial Bernoulli distribution, the convergence is purely exponential, as ineq. (14) and (21). Probability of occurence of 2 is also always exponential, anddegeneracy is not possible in this case.The example of 3-state rule presented here is an interesting instance of aphenomenon similar to degenerate hyperbolicity in finite-dimensional dynamicalsystems. It is hoped that it stimulates further research on hyperbolicity in CA.The method of constructing density polynomials by using finite state machinesappears to be quite fruitful, and it should be applicable to many other cellularautomata rules. References
1. AT&T Finite-State Machine Library, version 4.0,
2. Fuk´s, H.: An example of computation of the density of ones in probabilistic cellularautomata by direct recursion (2015), submitted for publication3. Fuk´s, H., Soto, J.M.G.: Exponential convergence to equilibrium in cellular automataasymptotically emulating identity. Complex Systems 23, 1–26 (2014)2. Fuk´s, H.: An example of computation of the density of ones in probabilistic cellularautomata by direct recursion (2015), submitted for publication3. Fuk´s, H., Soto, J.M.G.: Exponential convergence to equilibrium in cellular automataasymptotically emulating identity. Complex Systems 23, 1–26 (2014)