Hidenosuke Nishio- On the Growth of Neighborhoods of Cellular Automata

8/3/2019 Hidenosuke Nishio- On the Growth of Neighborhoods of Cellular Automata

1/33

On the Growth of Neighborhoods of Cellular

Automata

Hidenosuke Nishio (Kyoto)

12th Workshop on CA

Hiroshima, September 14, 2006

Growth of Neighborhoods of CA / H.Nishio AUTOMATA 2006 1/33


2/33

Outline

We investigate the growth of neighborhoods of cellular automata, follow-

ing the mathematical study of the growth of finitely generated groups,

particularly in connection with the Garden of Eden (GOE) theorem.

Background and Motivation

Definitions of CA, neighborhoods and neighbors

History and review of GOE theorem and related mathematics

Growth of Groups

Growth of Neighborhoods of CA

Some Results

Concluding Remarks



3/33

Background and Motivation

A cellular automaton (CA for short) is a uniformly structured information

processing system defined on a regular discrete space S, which is typ-

ically presented by a Cayley graph of a finitely generated group. Thesame finite automaton (cell) is placed at every point of the space. Every

cell simultaneously changes its state following the local function defined

on the neighboring cells. The neighborhood N is also spatially uniform.Most studies on CA assume the standard neighborhoods N

vNand N

Mafter John von Neumann and E. F. Moore, both neighborhoods being

defined in the 2-dimensional Euclidean grid Z2 = a, b | ab = ba.

Figure 1 The von Neumann neighborhoodHvN

Figure 2 The Moore neighborhoodNM



4/33


5/33

We are also investigating the problem : How does the Neighborhood

Affect the Global Behavior of Cellular Automata? , [11].

During such studies on the neighborhood, particularly concerning the

Garden of Eden theorem, the author encountered with the growth of

finitely generated groups.

Remarks

Difference between group theory vs. our theory of neighborhoods Group vs. semigroup: Cayley graph is undirected vs. directed.

Group identity element 1: Our neighborhood does not always con-

tain 1.



6/33

Cellular Automaton CA=(S, N, Q , f)

S : Cellular space is the Cayley graph (S) of a finitely generatedgroup S = G|R with generators G and relators R.

If G = {g1, g2,...,gr}, every element of S is presented by a wordx (G G1), where G1 = {g1| g g1 = 1, g G}. Theset Rof relators is written as

R = {wi = wi | wi, w

i (G G

1), i = 1,...,n}. (1)

For x, y (S), if y = xg, where g G G1, then an edgelabelled by g is drawn from vertex x to vertex y. Usually the cellularspace is simply denoted by S in stead of (S).

N : Neighborhood N = {n1, n2,...,ns} is a finite subset of S. Forany cell x S, the information of cell xni reaches x in a unit oftime. The set of all neighborhoods is denoted by N. The cardinality

#(N) of N is called the neighborhood size of a CA having theneighborhood N. The set of all neighborhoods of size s is denoted

by Ns.Growth of Neighborhoods of CA / H.Nishio AUTOMATA 2006 6/33


7/33

Q : Set of cell states is assumed to be a finite field Q = GF(q),where q = pn with prime p and positive integer n, see [15]. Thecase of the ring Q = Z/mZ is also considered.

f : Local function f is a map QN Q, where an element of QNis called a local configuration.

Global map F is a map C C, where an element of C = QS iscalled a global configuration. F is uniquely defined by f and N as

follows.F(c)(x) = f(c(xn1), c(xn2), , c(xns)), (2)

where c(x) is the state of cell x S for any c C. When startingwith a configuration c, the behavior (trajectory) of CA is given by

Ft+1(c) = F(Ft(c)) for any t 0, where F0(c) = c. (3)

Pattern: Let S be a finite subset of S. A finite configurationp = c|S

is called a pattern (for symbol, p c). S is called the support ofp.



8/33

Neighborhood and neighbors

Given a neighborhood (index) N = {n1, n2,...,ns} S for a cellularspace S = G | R, we recursively define the neighbors of CA.

Let p S, then 1-neighbors of p, denoted as pN1, is defined to be

pN1 = {pn1, pn2, ..., pns}.

m + 1-neighbors of p, denoted as pNm+1

, is defined by

pNm+1 = pNm N, m 0,

where pN0 = {p}.

Note that when computing a word pni of new neighbors of p, thesame relation Ras in S is applied. The information of cells in pNm

reaches p in m-units of time.



9/33

-neighbors of p, denoted as pN, is defined by

pN =

m=0

pNm.

-neighbors of 1 is simply called neighbors (of CA) and denotedas N.

The m-ball is defined by

Nm =m

k=0

Nk. (4)

Obviously, if 1 N then Nm = Nm.

The intrinsic m-neighbors : [Nm] = Nm \ Nm1. The informationof [Nm] can reach the origin in exactly m steps. Obviously, N =

m=0[Nm]. The intrinsic neighbor is called the boundary in group

theory.



10/33

A history

Moore and Myhill (1962-1963): GOE theorem

Milnor(1968): Growth of fundamental groups (differential geometry)

Gromov(1981, 1999): Polynomial growth and nilpotent subgroups,

surjunctivity

Grigorchuk(1983): Subexponential growth

Schupp(1986): Connecting GOE theorem with Milnors work

Machi and Mignosi(1993): Solution and generalization of Schupp

Nishio and Margenstern(2005-2006): Growth of neighborhoods of CA



11/33

Review of GOE theorem

The Garden of Eden (GOE) theorem was originally proved for Z2 byE.Moore(1962) [9] and J.Myhill(1963)[10].

Definition 1 Let S be a finite subset of S. Two different patterns p1and p2 are called mutually erasable, if, for all pair of configurations c1and c2 such that c1|S = p1, c2|S = p2 andc1|S\S = c2|S\S, we haveF(c1) = F(c2) .

A patternp is calledGarden of Eden (GOE), if there is not a configurationc such thatp F(c). (A GOE has not an ancestor).

Theorem 1 (Moore) If there are mutually erasable patterns, then there

are GOE patterns.

Theorem 2 (Myhill) If there are GOE patterns, then there are mutually

erasable patterns.



12/33

If there is no GOE patterns then F is surjective and if there is no mutuallyerasable patterns then F is injective when it is restricted to the finiteconfigurations.

Therefore these theorems together claim the following theorem, which is

called the GOE theorem today.

Theorem 3 (GOE theorem) F is surjective if and only if F is injectivewhen it is restricted to the finite configurations.



13/33

Sketch of the proof of Moores theorem

The proof principally relies on the following observation about the growth

of neighbors.

Let #([Nm]) be the cardinality of the intrinsic m-neighbors [Nm] and#(Nm) be that of m-ball. Then, we see

limm

#([Nm])/#(Nm+1) = 0. (5)

Then, if there are mutually erasable patterns, then there is a pattern in

#(Nm) which has no image in #(Nm+1).Q.E.D



14/33

Equation (5) is not always true.

For example, for the free group F2 = a, b|, we have

limm #([Nm])/#(Nm) = 2/3. (6)

There is a CA on F2 for which the GOE theorem does not hold.

After the seminal papers by Moore and Myhill, many mathematicians

have had interest in this problem and revealed that the GOE theoremholds for groups of polynomial and subexponential growth, but does

not for exponential growth, see Milnor(1968) [8], Schupp(1988)[16] and

Machi&Mignosi(1993) [6] and Gromov(1981,1999) [4, 5]. Also see Sil-

berstein and Coornaert (2006) [2].

Note that the mathematicians usually discuss the GOE theorem assum-

ing the generators of the group as the neighborhood. It is not our case.



15/33

Growth of groups

The growth function S of finitely generated discrete groups S = G|Rwas introduced by Milnor(1968) [8].

S(n) = #Bn = #{w | |w| n, w S}. (7)

For a free group F = a, b|, F = 2n. For 2-dimensional Euclideanspace S = Z2 = a, b|ab = ba, S = 2n2 + 2n + 1.

For most theory concerning the growth of groups, its asymptotic behavior

called the growth rate is of interest and Milnor defines it. Though there

are several definitions of the growth rate, they are all equivalent. The

following one is due to Babai (1997) [1].



16/33

Two monotone non-decreasing functions f1, f2 : N N are said to beequivalent (f1 f2), if there exist constantsc1, c2, C1, C2, n0 > 0 such that for all n n0,

C1f1(c1n) f2(n) C2f1(c2n). (8)

The relation is evidently an equivalence relation. The equivalenceclass of f is denoted by [f]. Let [f1] and [f2] be the equivalence classesto which f1 and f2 belong, respectively and define an order [f1] [f2]

if Cf1(cn) f2(n) for constants C,c,n0 0 and for all n n0.

Example 1 [n2] [n3], [an] [nb], andan bn for any positive integ1.

The growth rate [S] of a group S is an equivalence class to which Sbelongs. For Z2, [S] n2. The growth rate of groups is simply calledthe growth of groups.

The growth of groups is independent from the generators, Milnor(1968



17/33

Lemma 1 (Lemma 1 of [8]) LetG1 = {g1,...,gp} andG2 = {h1, ...,be two different sets of generators and let 1(n) and 2(n) bethe corresponding growth functions. Then, there exist positive con-

stantsk1 andk2 so that

2(n) 1(k1n)

and

1(n) 2(k2n)

for alln.

By Lemma 1 we have [1] [2] in Babais sense.



18/33

There are three kinds of growth of groups;

polynomial subexponential exponential.

Gromov(1981) shows that a group has polynomial growth if and only

if it has nilpotent subgroups [4].

Grigorchuk (1983) gives a group Swhich has subexponential growth

[3].


G th f i hb h d


19/33

Growth of neighborhoods

The growth function (N,S) of neighborhood N in S = G|R is definedby

(N,S)(m) = #{w | w Nm}, (9)

where Nm is the m-ball of radius m defined by Equation (4). Note thatNm is generally different from Bm. For example, in case of a 3-horseN3H, the word a of length 1 is included by N

3 (and N3) but not by N1.

Moreover the identity 1 of length 0 first appears in N12

[13].

The growth rate [(N,S)] of a neighborhood N S is similarly definedto be an equivalence class to which (N,S) belongs.

We discuss here the growth function/rate of neighborhood N comparedwith that of group S itself. The problem is not trivial even though we haveLemma 1 by Milnor. First, it is seen that if N = G G1 which is thecase of NV, then (N,S) = S. Then, the growth of neighborhoods isequal to that of spaces.


E l 2 F th N d th M i hb h d i S


20/33

Example 2 For the von Neumann and the Moore neighborhoods inS =Z2, we have(NV,S)(m) = 2m

2+2m+1 = S(m) and (NM,S)(m) =4m2 + 4m + 1 > S(m), respectively. Both neighborhoods have thesame growth ratem2, which is equal to the growth rate ofZ2.

Example 3 A calculation of the growth function of N3H vs. that of NVinZ2 is shown below. The growth rate of 3-horse is also polynomial.

n 1 2 3 4 5 6 7 8 9 10

NV(n) 5 13 25 41 61 81 113 145 181 221N3H(n) 3 9 18 35 62 100 147 208 277 353


B i ti f th th f i hb h d


21/33

Basic properties of the growth of neighborhoods

First we notice some basic properties of the growth of neighborhoods.

Lemma 2 IfN N then

(N,S) (N,S)

.

From this lemma, we have

Lemma 3 For anyN S,

(N,S) (NN1,S) and [(N,S)] [N], (10)

whereN is the growth function of the groupN | Rg.

Then we have the following theorem.

Theorem 4 For a cellular space S = G|Rg and any neighborhoodN S,

[(N,S)] [S], (11)

where the equivalence holds if and only ifN fillsS.

Proof: By Lemmas 2 and 3 we have the theorem. Growth of Neighborhoods of CA / H.Nishio AUTOMATA 2006 21/33

GOE theorem for arbitrary neighborhoods


22/33

GOE theorem for arbitrary neighborhoods

We show the GOE theorem for CAs having the neighborhood, which is

not necessarily the generators of group.

Theorem 5 The GOE theorem holds for a CA which has a neighborhood

of polynomial growth.

Proof: Moores proof shown above generally applies to such a CA. The

converse is also proved in the same way.


Growth of Hyperbolic Horse


23/33

Growth of Hyperbolic Horse

CA on hyperbolic spaces has been investigated by M.Margenstern and

other authors, see [7] for his latest literature.

For showing the growth of hyperbolic horses in {5, 4}, we utilize the fol-lowing figures drawn by M. Margenstern for the paper by Nishio,Margenste

and von Heaselar(2005) [14].



24/33

Figure 4 The neighborhood associated to one move of the hyperbolic

horse



25/33

2

5 3

1

4

5

5

3

3

5

5

4

4

2

2

1

1

451

234

4

45

1

5

1

2

2

3

45

1

2

3

1

3

3

4

5

4

4

5

5122

2

3

3

44

1

1

2

2

3

3

3

2

1

5

3

2

4

5

12 3 4 5

5 1

1 2

3

2

1


horse in the tree representation of the pentagrid.



26/33

Figure 6 The motions of the horse from a tile to its immediate neigh-

bours of the pentagrid.



27/33

5

5

3 2

451

234

45

1

5

1

2

2

3

45

1

2

3

1

3

3

4

5

4

4

5

5122

2

3

3

44

1

2

3

12 3 4 5

5 1

1 2

3

2

1

2

5 3

1

4

3

5

5

4

4

2

1

1

4

1

2

3

3

2

1

5

3

2

4

5


horse


Generally the hyperbolic space does not allow a Cayley graph presenta-


28/33

Generally, the hyperbolic space does not allow a Cayley graph presenta

tion of a group. However, the pentagrid {5, 4} can be treated by meansof its dual hyperbolic grid {4, 5}, which is seen to be a Cayley graph ofthe group H{4,5} as is shown below.

H{4,5} = 1, 2, 3, 4, 5 | 12 = 21, 23 = 32, 34 = 43, 45 = 54, 51 = 15,i = i1, 1 i 5,where {1, 2, 3, 4, 5} is the symbol set of generators.

First, as for the growth of the group H{4,5}, the following propositionholds.

Proposition 1 The growth rate of the pentagrid{4, 5} is exponential.

Next, we investigate the horse power problem on the hyperbolic plane:for a neighborhood N, decide if N fills the space S or not. If N consistsof s elements, it is called an s-horse.

First we give the following Theorem 3.8 of [14], which has been rewritten

in my formulation.Growth of Neighborhoods of CA / H.Nishio AUTOMATA 2006 28/33


29/33


30/33

References


31/33

[1] Babai, L.: The growth rate of vertex-tansitive planer graphs, 8th

Annual ACM-SIAM Symposium on Discrete Algorithms, 1997.

[2] Cecceherini-Silberstein, T., Coornaert, M.: The Garden of Eden

theorem for linear cellular automata, Ergod. Th. & Dynam. Sys.,

26, 2006, 5368.

[3] Grigorchuk, R. I.: On Milnors problem of group growth, Soviet Math.

Dokl., 28, 1983, 2326.

[4] Gromov, M.: Groups of Polynomial Growth and Expanding Maps,

Publ. Math. I.H.E.S., 53, 1981, 5373.

[5] Gromov, M.: Endomorphisms of symbolic algebraic varieties, J.Eur. Math. Soc., 1, 1999, 109197.

[6] Machi, A., Mignosi, F.: Garden of Eden Configurations for Cellular

Automata on Cayley Graphs of Groups., SIAM J. Discrete Math.,

6(1), 1993, 4456.Growth of Neighborhoods of CA / H.Nishio AUTOMATA 2006 31/33

[7] Margenstern, M.: A New Way to Implement Cellular Automata on


32/33

[ ] g , y p

the Penta- and Heptagrids, Journal of Cellular Automata, 1, 2006,

124.

[8] Milnor, J.: A note on curvature and fundamental groups, J. Diff.Geometry, 2, 1968, 17.

[9] Moore, E. F.: Machine models of self-reproducution, Proc. Sympo-

sium in Applied Mathematics, 14, 1962.

[10] Myhill, J.: The converse to Moores Garden-of-Eden theorem, Proc.Amer. Math. Soc., 14, 1963, 685686.

[11] Nishio, H.: How does the Neighborhood Affect the Global Behavior

of Cellular Automata?, Proceedings of ACRI2006, eds. El Yacouybi,

B. Chopard and S. Bandini, LNCS 4173, 2006.[12] Nishio, H., Margenstern, M.: An algebraic Analysis of Neighbor-

hoods of Cellular Automata, Submitted to JUCS, 2005.

[13] Nishio, H., Margenstern, M.: An algebraic Analysis of Neighbor-

hoods of Cellular Automata, Technical Report (kokyuroku) vol.Growth of Neighborhoods of CA / H.Nishio AUTOMATA 2006 32/33

1375, RIMS, Kyoto University, May 2004, Proceedings of LA Sym-


33/33

y y y g y

posium, Feb. 2004.

[14] Nishio, H., Margenstern, M., von Haeseler, F.: On Algebraic

Structure of Neighborhoods of Cellular Automata Horse PowerProblem, To appear in Fundamenta Informaticae, 2006.

[15] Nishio, H., Saito, T.: Information Dynamics of Cellular Automata I:

An Algebraic Study, Fundamenta Informaticae, 58, 2003, 399420.

[16] Schupp, P. E.: Array, Automata and GroupsSome Interconnec-tions, Proceedings of the LITP Spring School on Theoretical Com-

puter Science, C. Choffrut, ed, LNCS 316, 1988.


Documents

Hidenosuke Nishio- On the Growth of Neighborhoods of Cellular Automata