How does the neighborhood affect the global behavior of cellular - - PowerPoint PPT Presentation

How does the neighborhood affect the global behavior of cellular automata?

Hidenosuke Nishio (Kyoto)

11th Workshop on CA Gdansk University, September 3-5, 2005

Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 1/20

Summary

Motivation: In order to clarify the significance of the neighborhood, we investigate the global behavior of CA when the neighborhood is differ- ently chosen for a fixed local function. To formalize: Given a local function f : Qs → Q of arity s, investigate the global behavior of CA=(S, N, Q, f), where the neighborhood N = {n1, n2, ..., ns} of size s is differently chosen. Result: We report here a small result that a CA having the parity lo- cal function preserves the parity of global configurations in spite of any choice of neighborhoods within a simple algebraic constraint. We also consider subneighborhoods of Moore neighborhood in 2-dimensional Euclidean space—incomplete. This paper is along the line of the joint work with Maurice Margenstern and Fritz von Haeseler on the algebraic study of neighborhoods of CA.

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Neighborhood-sensitive and neighborhood-insensitive theories

• The historical and most fundamental neighborhood is one defined

by J. von Neumann which consists of 5 cells: Center, North, East, South and West. Using this neighborhood (called von Neumann) J.von Neumann [5] inaugurated the study of cellular automata by finding the 29 states local rule which results in a self-reproducing machine with computation universality. —Golden rule.

• The 9-cells neighborhood larger than von Neumann by 4 cells: North-

East, South-East, South-West and North-West is called Moore after

• E. F. Moore who proved one of the earliest theorems for CA (Garden
• f Eden theorem) [2]. —Neighborhood-insensitive.
• Game of Life [1] assumes Moore for the 2-dimensional binary state

CA, where the local rule is cleverly chosen and several interest- ing phenomena like construction and computation universality have been proved to emerge. —Neighborhood-sensitive.

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Simulation study of natural phenomena

When CA is used for the simulation in physics and biology, many au- thors assume the nearest neighbor like von Neumann and Moore. The lattice-gas theory is developed on the triangular grid with the nearest neighborhood of size 6. It is because not only of the intrinsic nature of the system to be simulated but also of the practical reason. It will be helpful to investigate each theory and application of CA with the naive question if the choice of the neighborhood affects the result or not.

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

• Cellular space S is expressed by a Cayley graph of a finitely gener-

ated group with generators G and relations R. S = G|R, where G = {g1, g2, ..., gr} is a set of finite number of generators (symbols) and R is a finite set of relations (equalities) of words over G and G−1, where G−1 = {g−1| g · g−1 = 1 for g ∈ G}: R = {wi = w′

i | wi, w′ i ∈ (G ∪ G−1)∗, i = 1, ..., n}.

Every element of S is presented by a word x ∈ (G ∪ G−1)∗. For x, y ∈ S, if y = xg, where g ∈ G ∪ G−1, then an edge labelled by g is drawn from point x to point y.

• Neighborhood (index) N = {n1, n2, ..., ns} ⊂ S.

For any cell x ∈ S, the information of cell xni reaches x in a unit

• f time.

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• Set of cell states Q = GF (q) where q = pn with prime p and

positive integer n.

• Local map f : QN → Q, where QN is the set of local configura-

tions.

• Global map F : C → C, where C = QS is the set of global
• configurations. F is uniquely induced by f and N;

F (c)(x) = f(c(xn1), c(xn2), · · · , c(xns)), where c(x) is the state of the cell at point x for any c ∈ C and x ∈

• S. When starting with a configuration c, the behavior (trajectory) of

CA is given by F t+1(c) = F (F t(c)) for any c ∈ C and t ≥ 0, where F 0(c) = c.

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Neighborhood and neighbors

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

• Let p ∈ S, then 1-neighbors of p, denoted as pN 1, is defined to be

pN 1 = {pn1, pn2, ..., pns}.

• m + 1-neighbors of p, denoted as pN m+1, is defined by

pN m+1 = pN m · N, m ≥ 0, where pN 0 = {p}. Note that when computing a word pni of new neighbors of p, the same relation R as in S is applied. The information of cells in pN m reaches p in m-units of time.

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

pN ∞ =

∞

• m=0

pN m.

• ∞-neighbors of 1 is simply called neighbors (of CA) and denoted

as N ∞. Here we note the following lemma, where generation of a semigroup is denoted by ·|·sg while that of a group by ·|·g. Lemma 1 g1, g2, ..., gr|Rg = g1, g2, ..., gr, g−1

1 , g−1 2 , ..., g−1 r |Rsg.

Example: Z2 = a, b| ab = bag = a, b, a−1, b−1| ab = basg

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Horse power problem

Definition 1 A neighborhood N is said to fill the space S, if N|Rsg = S. Typically the 3-horse is proved to fill the infinite chess board Z2 [3]. Theorem 1 The 3-horse N3H = {a2b, a−2, ab−2} ⊂ Z2 fills Z2. We have a theorem which provides a method to decide whether a neigh- borhood fills a space or not [4]. Theorem 2 Let x1, ..., xs ∈ Zd, where s ≥ d + 1. Then the neighbor- hood {x1, ..., xs} fills the space or x1, ..., xssg = Zd, if and only if the following two conditions hold. condition 1: gcd({det([xi1, ..., xid])|i1, ..., id ∈ {1, ..., s}}) = 1. condition 2: 0 ∈ int(conv({x1, ..., xs}). ( The zero of Rd should be in the interior of the convex hull of {x1, ..., xs}.)

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Parity function

This section is devoted to an example for showing how the neighborhood does not affects the global property of CA; The parity of global configura- tions is preserved by computation of CA having the parity local function in spite of arbitrary choice of neighborhoods besides a certain condition. Consider a CA=(S, N, Q, f), where Q is the binary set GF (2) = {0, 1} and S and N = {n1, n2, ..., ns} ⊂ S are arbitrary. As for the local rule f we consider the (binary) parity function defined by f = fBP,N(n1, n2, ..., ns) =

s

• i=1

c(ni) mod 2, (1) where c(ni) is the state of cell ni. The global map FBP,N : QS → QS is induced by fBP,N as usual and also called the (global) parity function. A configuration c ∈ QS is called finite if #{i | c(i) = 0, i ∈ S} < ∞. The finiteness of configurations is preserved by the parity function. In

Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 10/20

the following we assume finite configurations. The parity P (c) of a configuration c is defined by P (c) =

• x∈S

c(x) mod 2, (2) where c(x) is the state of cell x. Theorem 3 The parity function preserves the parity of configurations if the neighborhood size is odd. That is, P (FBP,N(c)) = P (c), (3) if and only if the neighborhood size is odd. Note that the parity function is not number conserving.

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Proof: This theorem is a special case of the following generalized theo- rem.

• Example 1 1-dimensional binary parity CA in Z = a|∅ with a neigh-

borhood of size 3 such as N3 = {a−1, 1, a}, N ′

3 = {a−2, 1, a2} and

N ′′

3 = {0, a, a2} preserves the parity, but one with N2 = {1, a} does

• not. The theorem holds for finite spaces like Z(n) = a|an = 1.

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Generalized parity function (Modulo p sum)

Let Q = {0, 1, ..., p−1, ...} = GF (q) and define the generalized parity function fGP,N by fGP,N(n1, n2, ..., ns) =

s

• i=1

c(ni) mod p. (4) The global map is denoted by FGP,N. The generalized parity GP (c)of a configuration c is defined by GP (c) =

• x∈S

c(x) mod p. (5) Theorem 4 GP (FGP,N(c)) = GP (c), (6) if and only if N ∈ Ns, where s = kp + 1, k ≥ 0.

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Proof: GP (FGP,N(c)) =

• x∈S

FGP,N(c)(x) =

• x∈S

fGP,N(xn1, xn2, ..., xns) (7) =

• x∈S

s

• i=1

c(xni) =

s

• i=1
• x∈S

c(xni). (8) We note here that

• x∈S

c(xni) =

• x∈S

c(x), 1 ≤ i ≤ s. (9) Then, if s = kp + 1, we have

s

• i=1
• x∈S

c(xni) =

• x∈S

c(x) = GP (c). (10) As for the necessity of the condition s = kp+1, we note the case where p = 2 and s = 2. Any CA with parity function maps all configurations having parity 1 (and 0) into those of parity 0 and does not preserve the parity.

• Neighborhood and global behavior of CA / H.Nishio

AUTOMATA 2005 14/20

3-Subneighborhoods of Moore

For a fixed space S, we consider the set of all finite neighborhoods rela- tive to S and denote it as NS. If N ⊂ N ′, where N and N ′ ∈ NS, N is called a subneiborhood of N ′. We concentrate here in the subneighborhoods of Moore neighborhood NM (Moore for short) in the 2-dimensional space Z2; NM = {1, a, a−1, b, b−1, ab, (ab)−1, ba, (ba)−1} ⊂ Z2.

• ✁

• ✎✑✏✓✒✕✔✗✖

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List of 3-subneighborhoods of NM

Here we show all 3-subneighborhoods of Moore up to the rotation of multiples of 90◦ and mirror reflection.

• ✂✁

• ✂✁

• ✂✁

• ✁

Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 16/20

• ✌✓✒

• ✗

• ✁

• ✁

• ✗

Neighborhood and global behavior of CA / H.Nishio AUTOMATA 2005 17/20

• ✁✄✂

• ✁

• ✁

Proposition 1 Three neighborhoods in Fig.5 of 3sub-360 fill Z2 but the

• thers not.

Proof: Theorem 2 holds for the neighborhoods in 3sub-360 with d = 2. However we can prove it without using the theorem.

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Representatives of 3-subneighborhoods

We notice here a representative from each class of 3-subneighborhoods listed above.

• ✂✁

• ✁

• ✁

• ✁

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Thank you for your attention !

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References

[1] Gardner, M.: The fantastic combinations of John Conway’s new game of ’life’, Scientific American, 223, 1970, 120–123. [2] Moore, E. F.: Machine models of self-reproduction, Proc. Symp. in Applied Mathematics 14, 1962, 17–33. [3] Nishio, H., Margenstern, M.: An algebraic Analysis of Neighborhoods

• f Cellular Automata, Technical Report (kokyuroku) vol. 1375, RIMS,

Kyoto University, May 2004, Proceedings of LA Symposium, Feb. 2004. [4] Nishio, H., Margenstern, M., von Haeseler, F.: On Algebraic Structure

• f Neighborhoods of Cellular Automata –full and one-way, Techni-

cal Report 253, CDMTCS, University of Auckland, December 2004, Proceedings of the International Workshop on Tilings and Cellular Automata, 2004. [5] von Neumann, J., Burks(ed.), A. W.: Theory of Self-reproducing Au- tomata, Univ. of Illinois Press, 1966.

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