ELSEVIER Physica D 110 (1997) 67-91

PHYSICA

Additive cellular automata and global injectivity

R.A. Dow School of Mathematical Sciences, Queen Mary and Westfield College, University of London, Mile End Road, London E1 4NS, UK

Received 5 November 1996; received in revised form 4 April 1997: accepted 21 April 1997 Communicated by J.D. Meiss

Abstract

We employ algebraic methods to investigate the global injectivity of additive cellular automata when the state alphabet is a finite commutative ring. The set of local rules of such additive cellular automata in D > 1 dimensions itself forms a ring. The invertible elements in this ring correspond to the globally injective additive cellular automata. We indicate how one determines the invertible elements and thus the globally injective additive cellular automata. In particular we show that, in contrast to the general situation, the global injecfivity of additive cellular automata in D > 1 dimensions is decidable.

Construction of local rules for the inverse cellular automata is illustrated through examples. A strict upper bound on the radius of the inverse cellular automata is obtained. In the special case of linear cellular automata with state alphabet the ring of integers modulo m we count the globally injective cellular automata of up to a given radius.

We show that in one dimension a non-globally injective additive cellular automaton is either not injective on spatially periodic configurations of any period N ~ N + or the set of N > 0 such that the cellular automation is injective on spatially periodic configurations of period N has positive density in N+.

Keywords: Cellular automata; Reversibility; Algebraic dynamics

1. Introduct ion

In this paper we make extensive use of ring theoretical methods to investigate the injectivity properties of additive

cellular automata. Cellular automata (CA) are a class of extended discrete dynamical systems that have attracted

considerable attention in recent years. Loosely speaking a CA consists of a lattice, where each site of this lattice has

a state from some state alphabet A, a finite set. Locally the state of a site in the lattice evolves according to a local

rule which assigns to the site in the lattice a new state. The new state depends only upon the states of some finite set

,t" of neighbouring sites, i.e. the local rule is a map f : A X ~ A. The local dynamics induces the global dynamics

of the whole CA, the global rule (this is made precise in Section 2). In this paper we always take the lattice to be

the D-dimensional integer lattice 77 D, D > 1, thus the global rule is a map F : A 7/o ", A 7/o .

CA can provide simple models of complex systems occurring in physics, chemistry, biology, computation and

so on (see [1,2]). Globally injective CA are of particular interest (see [3]). In view of their combination of physical

interest and analytical tractability, globally injective additive CA seem to be interesting objects for study. A CA is

globally injective if its global rule is injective. Richardson [4] has shown that a CA is globally injective if and only

0167-2789/97l$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00074-2

68 R.A. Dow/Physica D 110 (1997) 67-91

if it is reversib!e (and hence the global rule is bijective). Whilst global injectivity is implied by local injectivity (the injectivity of the local rule as a map A X > A, which can only happen if 12(I = 1) global injectivity does not in

general imply local injectivity. In one dimension there is an algorithm for deciding whether a given CA is globally

injective in a finite number of steps [5,6]. The global injectivity of higher-dimensional CA is undecidable, i.e. there is no such algorithm [5,6]. However, Sato [7] has shown that if the state alphabet is a finite commutative ring then the global injectivity of linear CA is decidable in higher dimensions.

Additive CA are a class of CA (which includes the linear CA) which are particularly amenable to theoretical

analysis, for this reason they have (at least in the linear case) been extensively studied for instance see [8-16]. A

CA with global rule F is additive if for any two configurations P, Q ~ A Z° one has F ( P + Q) = F(P) + F(Q). Clearly this statement is meaningless unless one has some notion of addition in A go . I f one can define addition

in A then this will extend to A eD, thus one could choose A to be a finite commutative group. However, we find it convenient to take A to be a finite commutative ring where we can exploit the advantage of having multiplicative as well as additive structure. We show that, with suitable choices of state alphabet, one can easily construct examples of globally injective additive CA. We also show, again with suitable state alphabets, that the global injectivity of

additive CA in two or more dimensions is decidable. As dynamical systems CA over rings naturally fall into the realm of algebraic dynamics. In this developing area

one considers dynamical systems defined over rings other than the familiar real and complex fields [17-21]. Of

particular relevance are [22] where CA are associated with polynomial dynamics over a finite field and [8,13]. A n y C A with a state alphabet with a prime power number of states pn can be identified with a CA over the finite

field Fpn in a particularly pleasing way. Any map 1 cxp~ > IZp,, can be written as a polynomial map in 12([ variables

(see [23, p. 100]). Thus the local rule can be written as a function of the states of the neighbouring sites using only

the ring operations in IZp,. Throughout the document state alphabets are always taken to be finite commutative rings. For a general reference

on algebra see [24]. The basic idea of the paper is simple. Let the state alphabet be A, we merely note that for all

D > !, the set of linear local CA rules over A and more generally the set of additive local CA rules over A form rings, the D-dimensional linear and additive CA rings over A. The multiplication in these rings is composition of

local rules and the units (the invertible elements in the ring) must correspond to globally injective CA. Thus, our approach is to identify the units: We do this by finding ring isomorphisms from the CA rings to better known rings

where the units are easily identifiable. In any ring R the set of units forms a group under multiplication, denoted by R*. The D-dimensional linear CA ring over A is denoted by LD(A) and the D-dimensional additive CA ring over A is denoted by Add D (A). Thus, our task is to identify Add D (A)* (Sato [71 has already identified L D (A)*).

In the rest of this section we outline the results to be found in the remainder of the paper. In Section 2 terminology is introduced and definitions made. The distinction between linear and additive rules is made clear. Any finite commutative ring A is a Y_/mZ-algebra for some integer m > 1 (i.e. A has the structure of a 77/m71-module; recall

that a module is a generalisation of a vector space where the scalars can lie in any ring) where m is the characteristic of A. We show that a CA with state alphabet A is additive if and only if its local rule is 77/m77-1inear whereas a CA over A is linear if and only if its local rule is A-linear. As far as we are aware nearly all the earlier studies of additive CA have in fact been confined to linear CA; however, Moore [16] has recently considered the problem of

predicting additive CA of the more general type. We define and discuss global injectivity. When a CA has an inverse, that inverse is a CA (see [4]). As a globally

injective additive CA over A with local rule f and global rule F is reversible there is an inverse CA with global rule G and local rule g. The global rule G is the inverse map to F and the local rule g must be the inverse of f under composition of rules (note that this is not the same as composition of maps). It follows from elementary linear algebra that g must be 77/m2e--linear and hence G is additive. Thus, the globally injective D-dimensional additive CA over A are precisely those whose local rules are units in Add D (A).

R.A. Dow/Physica D 110 (1997) 67-91 69

In Section 3 we discuss the representation of additive CA that we will be using. The state of posit ion (ii . . . . . iD) in 27D is deno tedby aft ..... iD. The elements of LD(A) are the local rules of the form

kl kD

f:aix ..... iD ~ ~ "'" ~ Otjl ..... jDail+Jl ..... iD+jo, (1 .1)

Jl =--ll jD =--ID

where each aJl ..... jD C A and will be referred to as the rule coefficinets of f . Turning to AddD(A) , the set of

27/mZ-linear maps from A to itself, Homz/mZ (A, A), is a ring. The general element of A d d D ( A ) has the form

kl kD

f : a i l ..... iD ~ ~_, "'" ~_~ Xjl ..... jD(ail+j, ..... iD+JD), (1.2) Jl =--ll JD=--ID .

where each Xjl ..... JD E Hom2_/m~_(A, A). In (1.1) and (1.2) the d-radii of f is Ld = max(lid[, ]kd[), 1 < d < D,

and the radius of f is the maximum of the d-radii .

For the additive rules we find that a special case of the state alphabet suffices for most of our needs. I f A has a basis

of n elements over 27/m27 for some n, so that A is an n-dimensional 27/m27-algebra, we find that the representation

of A d d D (A) is fairly straightforward. The following result characterises the additive CA ring in this special case,

it is a colloquial form of Theorem 3.2.

Theorem 1.1. Let A be commutative n-dimensional 27/m27-algebra. Then for any integer D > 1, the D-dimensional

additive CA ring Over A, Add D (A), is isomorphic to Mn (27/m27[Xl, x l 1 . . . . . XD, XDI]), the ring of n × n matrices

with entries in the Laurent polynomial ring 27/mZ[xl, x~ 1 . . . . . XD, XD1].

Thus additive CA can be thought of as polynomials with matrix coefficients.

In Section 4 we consider global injectivity. In the linear case the choices of state alphabet A = Fq and A = 77/pkZ, k > 1, are interesting. In these two cases one can determine whether a given linear CA is globally injective just by

inspection of its local rule, as Sato has shown (in the proof of his Theorem 1 1, part (2), [7]) that a linear CA over

such A is globally injective if and only if exactly one of the rule coefficients is a unit. Thus in the finite field case

the CA is globally injective if and only if :exactly one of the rule coefficients is non-zero. In the Z/pI 'Z case the CA

is globally injective if and only if exactly one of the rule coefficients is a unit in 77/pkZ and the others are nilpotent.

Given a positive integer m we write m as a product of powers of distinct primes,

t

H kr. (1.3) m -= Pr r=X

Then for linear CA over 7/ /mZ we have the following, a form of Theorem 4.7.

Theorem 1.2. The number S(m, D, ll . . . . . lD) of local rules f ~ LD(77/m27) of d-radi i up to Id, 1 < d < D, giving globally injective CA is

) S(m, D, ll . . . . . lD) = (2ld + 1) H ( P r - i)p~ kr-1)i-t~=l(2ld+l).

\ d = l r = l

The numbers of globally injective l inear CA over Z/m27 for 2 __ m _< 48 and 1 < D < 3 are listed in Table 1.

The next result is a consequence of Theorem 1.2.

70 R.A. Dow/PhySica D 110 (1997) 67-91

Theorem 1.3. For fixed dimension D and d-radii ld, 1 < d < D, let m be given by (1.3). Then the fraction of linear CA over 77/m77 that are globally injective depends only on the distinct primes occurring in (1.3), not upon the powers to which they occur.

For example the fraction of one-dimensional, radius 1, linear CA over 77/2k77 which are globally injective is 3/8 for all k _> 1.

We show that the global inj ectivity of D-dimensional additive CA over n-dimensional 77/m77-algebras is decidable.

We indicate how one determines the local rule of the inverse CA when it exists and give examples. A more general form of state alphabet is also considered. The main result for additive CA in the finite field case can be summed up in the following special case of Theorem 4.2.

Theorem 1.4. Let f be the local rule of a D-dimensional additive CA over £q, q = pn. Then f gives a globally

injective CA if and only if the image of f in Mn (~p [x l , X l 1 . . . . . XD, XD1]) has determinant of the form otx~' . . , x JD ,

where c~ is a non-zero element of Yp and j l . . . . . jD E 7/.

We are able to determine a strict upper bound on the radius of the inverse local rule of a globally injective additive

CA, which is independent of the dimension D. We have the following result for two important cases (derived from Theorem 4.5).

Theorem 1.5. Let f be the local rule of a globally injective D-dimensional additive CA of radius l over A.

(i) I f A = Z / m Z where m is given by (1.3) then the radius of the local rule of the inverse CA is at most

1 m a x ( 2 p r k r -1 - 1). l < r < t

(ii) If A = ~:q where q = pn then the radius of the local rule of the inverse CA is at most

l(2n - 1).

In Section 5 the failure of global injectivity is discussed for one-dimensional additive CA. When a CA one- dimensional is not globally injective it fails to be injective on certain spatially periodic configurations (see Section 2).

The conclusions of this section can be illustrated by the following result, a special case of Theorem 5.3.

Theorem 1.6. Let f be the local rule of a one-dimensional linear CA over 77/m77 or a one-dimensional additive CA over a finite field such that the CA is not globally injective. Then one can determine a finite set of integers, I ( f ) ,

such that the CA fails to be injective on spatially periodic configurations of period N if and only if N is divisible

by one or more of the elements of I ( f ) . I f 1 is not an element of I ( f ) then the set of integers N such that the CA is injective on spatially periodic configurations of period N has positive density in N +.

2. Preliminaries

We shall be concerned with CA on regular lattices, where by a regular lattice £ we mean a regular array of sites (or cells) in one or more dimensions. We shall always take/3 to be the D-dimensional integer lattice 2 D. We shall follow Toffoli and Margolus [3] in our general definition of a CA. To each site in 7/D one assigns a state from a state alphabet A, a finite set. Each assignment of an element of A to each site of 7/D is called a configuration of A on 77 ° . Thus, the set of configurations of A on 770 is the Cartesian product of copies of A indexed by 7/D, A ZD . Note that

R.A. Dow/Physica D 110 (1997) 67-91 71

if A has the structure of an R-module for some ring R then A ~° is also an R-module (the direct product of copies

of A indexed by 779).

Let S be the abelian group of translations of 7/o onto itself, then S is i tself isomorphic to 779. The elements of

S will be called displacements. The action of a displacement s ~ S on a site i c 770 yields a new site s + i. A

neighbourhood is a finite set of displacements, one applies a neighbourhood &" as an operator to a site i to yield a

set of sites, the &'-neighbourhood of i , i + &" = {i + x: x 6 X}, y c i + Pc" is called a neighbour of i. Its size, [XI,

is the number of elements in X. The radius of &" is the length of the longest displacement.

Let a ~ A 2-°, then the ith component of a is the state of site i in a, denoted ai. The neighbourhood projection

operator [i + 2(] extracts from a the collection of states of the neighbours of i:

[i + &'](a) ~ A x . (2.1)

Thus for each i, [i + &'] may be thought of as a map A 2 > A x .

Let f be a mapping,

f : A pe > A, (2.2)

then f can be applied at each site i of configuration a to yield a new configuration a ' :

a I = f ( [ i + &'](a)). (2.3)

Formally, the map f is known as the rule table and the pair (X, f ) is the local rule. We shall consistently abuse this

terminology and refer to f as the local rule: The radius of the local rule is the radius of X. The local rule induces a

global map F , often called the globa ! rule, F : A zD > A 7-D via Eq. (2.3). One can now define a CA formally as

a 4-tuple (77 D, A, &', f ) where the finite set A is the state alphabet, f is a rule table, 77D is a regular lattice and X

is a neighbourhood. However, this definition is somewhat cumbersome and we shall nearly always just refer to the

CA by the local rule f or the global rule F . We shall some times refer to a CA with state alphabet A as a CA over

A. An important feature of CA is that they commute with the translations or shifts of the lattice 7/°, i.e. if s E S

and a c A 2-° then if F is the global rule of a CA then

f ( s ( a ) ) = s ( f ( a ) ) ,

where s acts on a by ai ~ a s + i .

If A is a ring and f is the map APe > {0} then we shall call f the trivial rule or zero rule.

The time evolution of a CA is defined at the local level by

a~ +1 = f ( [ i + &'](at)) , (2.4)

where a t is the configuration at t ime t in the evolution from some initial condition a = a ° and a~ is the state of the

i th site at t ime t in the evolution. At the global level we have

a t+l = F ( a t ) . (2.5)

We shall also be interested in finite CA with periodic boundary conditions. In the one-dimensional case on N

cells one chooses sites labelled 0, 1 . . . . . N - 1. Then site N is identified with site 0 etc. and site - 1 is identified

with site N - 1 and so on. Thus, for a configuration a of a CA with these boundary conditions one has

ai+rnN = a i , 0 < i < N , m :~ ~. (2.6)

One can think of the sites of a CA with periodic boundary conditions as being arranged on a circle or cylinder (such

CA are sometimes called cylindrical CA, see for instance [25]). Alternatively one can think of such CA as being the

72 R.A. Dow/Physica D 110 (1997) 67-91

infinite system but with configurations restricted to those consisting of infinite repetitions of the same block of N sites. Thus, the study of finite CA with periodic boundary conditions is equivalent to the study of CA on spatially periodic configurations. Note that CA with periodic boundary conditions retain the property of commutativity with

the shifts of the lattice. In more than one dimension CA with periodic boundary conditions are defined in the obvious similar way.

Let f be the local rule of a CA, we shall denote the corresponding global rule in the periodic boundary condition case on N cells by FN.

A CA rule is additive if for any P1, t>2 E A ZD the global rule F satisfies

F(P1 +/'2) = F(P1) + F(P2). (2.7)

For this definition to be meaningful one must have at least some notion of addition for the state alphabet. We shall require that the state alphabet be a finite commutative ring.

One may consider CA whose local rules are linear combinations, i.e. in D > 1 dimensions

k

ai ~ f(ai-I . . . . . ai+k) = E ~jai+j , (2.8) j=-I

where o~j c A, i, j , l, k 6 Z ° , and the notation ~ = - l , which we will use frequently, means the sum over all

D-tuples (jl . . . . . j o ) e Z ° with --ld _< jd --< kd, 1 < d _< D. We shall sometimes refer to the o~j as the rule coefficients. Such local rules are A-linear maps A 11+t2+ 1 ~ A and are referred to as linear rules. The corresponding

global rules are also A-linear and thus clearly additive. It is easy to show that every D-dimensional A-linear global

rule arises from a local rule of the form (2.8). An immediate question is: Are the only additve CA rules those arising from A-linear local rules? To answer this question we look more closely at what additivity means when A is a finite ring.

Any finite ring A has characteristic m, an integer greater than 1 and is thus naturally a Z/mZ-module (indeed a 77/mZ-algebra) with (left) action

( n + m Z , a)~-~na for a l l a 6 A , n 6 7 7 .

Such a ring A contains an isomorphic copy of 77/m77 within its centre. It is clear that if the local rule f is 77/m77-1inear then the global rule F will be Z/mY_-linear and thus additive. Conversely, suppose that F is additive, then given

any P 6 A ZD one has

F ( 2 P ) = F(P) + F(P) = 2 F ( P )

and so on, it follows that F is 77/m77-1inear. In particular one must have

F(P1)i + F(P2)i = F(P1 + P2)i, nF(P)i = F(nP)i ,

for all i ~ 77D, p , P1, P2 E A ZD and n E 77/m7/. Hence, the local rule f is Z/mZ-linear and we have proved the following result, the corollary is immediate once one recalls that the A-linear maps from A to itself form a ring

isomorphic to A (see [26, p. 83]).

Lemma 2.1. A CA with state alphabet a finite ring A of characteristic m is additive if and only if its local rule is Z / m77-1inear.

Corollary 2.1. When A = 77/m77, a CA rule is additive if and only if it is linear.

R.A. Dow/Physica D 110 (1997) 67-91 73

Thus, a CA is additive but not l inear when it is formed from 7/ /mZ-l inear maps of A to itself which cannot be

expressed as multiplication by an element of A. When A is not Z/mY_ there are additive CA over A which do not

arise from A-linear local rules. The smallest A for which this occurs are Y4, the field with four elements, and F 2 )< [tz2,

both of which a r e iz2 ~ Z/27/a lgebras .

Let A be a finite commutative ring of characteristic m, then A is a Z /mZ-a lgebra . The set of Z/mY_-linear maps

from A to A, Hom~/ml_(A, A), is a ring ([23, p. 125]). As with the linear case, it is easy to see that the local rules

of additive CA over A in D-dimensions are of the form

k

f :ai ~ ~ Xj(ai+j), (2.9)

where Xj ~ Homz/mz(A, A), i, j , l, k ~ 7/D and we have used the same notation in the summation as in (2.8).

Definition 2.1. Let f be of the form (2.9). For 1 < d < D the d-radius of f is

Ld = max(lid[, Ikdl).

The radius of f is then max(L1 . . . . . LD). If A has a basis of n elements over 77/m77 so that A is an n-dimensional 2 / m Z - a l g e b r a then (see [27, Ch. vii])

Hom~_/m~_(A, A) ~- Mn(77/m77), where Mn(7//mT/) is the ring o f n x n square matrices with entries in Y_/mZ. It

follows immediately from this and (2.9) that there are rn (21+1)n2 additive rules of radius up to l over A, for each

l ~ N, as for each ai+j one can choose Xj in one of IM~(77/m77)l = m n2 ways.

The state alphabets most commonly considered have been the rings 77/mZ and the finite fields. When A =

lZq, q = pn, n > 1, the finite field with pn elements, an n-dimensional IZp-algebra, one can easily see that (2.9) can

be written in the alternative form:

k n - 1 pS f : ai ~-+ Z Z Olsjai+j ' (2.10)

j=--l s=0

where each C~sj ~ ~q. A CA is said to be globally injective if its global rule is injective. Formal ly a CA with global rule F is said to be

reversible if there is another CA with global rule F I such that for any pair of configurations c and c I, F(c) = c ~ i f

and only if F~(c ~) = c. If the CA with local rule f is globally injective shall say that f gives a globally injective

CA. Richardson [4] showed that a CA is globally injective if and only if it is reversible. There are procedures to

determine whether or not any one-dimensional CA is injective (and hence reversible). The following result may be

found in Culik II et al. [5,6].

Theorem 2.1 (CulikII). For a one-dimensional CA the global rule F " A ~

injective on all spatially periodic configurations.

A Z is injective if and only if it is

As noted earlier one can thank of CA with local rule f and periodic boundary conditions on N cells as being the

infinite CA with local rule f but with the configurations restricted to those that consist of infinite repetitions of the

same block of N sites. Thus, the study of finite CA with periodic boundary conditions is equivalent to the study

of the action of finite CA on configurations consisting of infinite repetitions of a finite configuration. The result of

Culik II et al. can thus be restated as: a one-dimensional CA is reversible if and only if the corresponding CA on N

cells with periodic boundary configurations is reversible for all N > 0.

74 R.A. Dow/Physica D 110 (1997) 67-91

In more than one dimension the reversibility of CA is undecidable in general (see [3] or [5]); however, Sato

[7] has shown that this is not the case for linear CA over a finite commutative ring. We show in Section 3 that, more generally, reversibility is decidable for additive CA at least when the state alphabet is in a large class of finite commutative rings. Whilst clearly if a higher-dimensional CA fails to be injective on the subset of spatially periodic configurations then it cannot be globally injective, the converse is no longer true.

For an additive CA over A where A has characteristic m we know that the local and global rules are Z/m77-linear. If the CA is globally injective and hence invertible, the inverse global rule and hence also the local rule of this inverse

CA must be Z/mY-linear (by elementary linear algebra). Thus, the inverse CA of a globally injective additive CA

is also an additive CA. We shall denote the inverse local rule of a CA with local rule f by f - 1 . This should not be confused with an inverse for f as a map A x > A, for as we shall see f may well not be bijective as a map.

Let f be the local rule of globally injective CA, global rule F. It follows that there must be a local rule f - 1 such

that, with o denoting composition of local rules, f o f - 1 = f - 1 o f :ai ~ ai. Let u c A then there is a globally injective CA, global rule Su, with local rule cru : ai ~ ai -}- u. Clearly Cru-1 = Cr_u. For any CA F with local rule f one can form the CA Fu, u ~ A, with local rule f~ = cr~ o f . Clearly Fu = Su o F. Such rules have been considered in [8,1 6]. The proof of the following theorem is straightforward.

Theorem 2.2. Let F be the global rule of a CA over A with local rule f . Let u ~ A. Then F is globally injective if and only if Fu is globally injective. If F is globally injective with inverse local rule f - 1 then fu -1 = f - 1 o o-_ u and if Fu is globally injective with inverse local rule fu -1 then f - 1 = fu-1 o cru.

By means of Theorem 2.2, results on global injectivity of additive rules can be extended to a larger class of rules,

for instance if f is the local rule of an additive, globally injective CA radius 1 over a finite commutative ring A. Let f - 1 have radius k. Then the CA over A with local rule fu, u ~ A\{0}, has inverse local rule

f£-I :ai ~ f - l ( a i - k . . . . . ai+k) -- f - l ( u . . . . . U). (2.11)

3. Representation of additive cellular automata

The additive CA most commonly studied have been those with linear local rules, we begin by considering these rules, We are concerned with CA rules whose state alphabet is a finite commutative ring A. We shall refer to such

CA as CA over A. Let L ° (A) be the set of local linear D-dimensional CA rules of the form (2.8) over the finite commutative ring

A. We can consider L v (A) as a ring with operations defined as follows. If

12 k2 f " ai ~ Z otjai+j, ll,12 c 77 D and g " ai ~ ~ flsai+s, k l , k2 E 7/D,

j=--ll s=-kl

then f ÷ g is defined in the obvious way and f g is defined by composition of rules:

12 /2 +g2 oej fls (3.1) f g : a i I---> ~ ogj t~sai+s-kJ = Z ai+I.

j=-l l s 1 l=- l l -k l ~ s+j+l

The zero element is the zero or trivial rule ai k--> 0 and the identity is the identify rule ai ~ ai. It is clear that LD(A) is commutative. We call LD(A) the D-dimensional linear CA ring over A.

R.A. Dow/Physica D 110 (1997) 67-91 75

It is well known that linear CA can be represented by polynomials. We represent the ring LD(A), by showing

that LD(A) as the ring of Laurent polynomials over A in D indeterminates, A[xl, x~ 1, ., XD, XD1], i.e. the polynomials in 2D indeterminates Xd and x~ -1 such that XdX2 t = xff~Xd = 1, 1 < d < D.

The following result is easily proved.

Theorem 3.1. For any finite commutative ring A and all D > 1

LD(A) -~ A[xl, X l I . . . . . XD, XD1],

via

12 ) 12 r : f : a i ~ + E ~;ai+j " E ~ jXl j l " ' ' x D :D

J=--/l j=--ll

It is immediate that units in A[xl, Xa I . . . . . XD, XD 1 ] correspond to linear local rules with inverses in L D (A) and that such rules must correspond to the globally injective CA.

Turning to the general additive case, let the set of all D-dimensional additive rules over A, where A has charac- teristic m and is thus a 7//m7/-algebra, be Add D (A), these are the rules of the form (2.9). Like L D (A), Add D (A) is

a ring, the composition of additive rules

12 k2

f : a i ~-> E Xj(ai+j) and g : ai ~ E Ys(ai+s) j=- l l s=-kl

is given by

f g :a, e--> E Xj Y,(a,+j+s) = E E XjYs (ai+,). (3.2) j=--ll s 1 l=--ll -kl j+s=I

The ring Add D (A) is clearly non-commutative in general. We shall call Add D (A) the D-dimensional additive CA

ring over A. Clearly L D (A) is a commutative subring of Add D (A). When A has a basis o fn elements over 7//mZ and

is thus an n-dimensional Z/mT/-algebra we can find a useful representation of Add(A). Let ¢ be the isomorphism from Homy_/my_ (A, A) to Mn(7//m7/).

Theorem 3.2. Let A be a finite commutative 7//mT/-algebra of dimension n. Then

AddD(A) ~-- Mn(7//m7/[Xl, x~ 1 . . . . . XD, XD1]).

Proof It is easy to see that the map crp:AddO(A) > Mn(7//m7/)[Xl, x { 1 . . . . . XD, XD1],

( 2 ) 2 f " ai ~ E Xj(ai+j) ~-> E +(Xj)Xl j' "''XD jD (3.3)

j=--ll j=--ll

is a ring isomorphism. It is well known that for any ring R

M n ( R [ x l , X l 1 . . . . . XD, XD1]) ~ M n ( R ) [ x l , X l 1 . . . . . XD,XD1]. []

The image of LD(A) in Mn (R[xl, Xl I . . . . . XD, XD1]) is a commutative subring. Whilst one could deal with the linear rules over A as a subring of Add ° (A), if one is concerned only with the linear case it is often easier to deal with L D (A) separately, using the representation of Theorem 3.1.

76 R.A. Dow/Physica D 110 (1997) 67-91

Given f ~ Add D (A) it is important to bear in mind the distinction between f as a C A rule and f as a map A x > A. Frequently f will be a unit in Add(A) but will not be a bijective as a map (i.e. whenever 12(I > 1).

4. Global injectivity

We shall begin by considering linear CA (i.e. those CA with local rules of the form (2.8)) when A is either a finite field or the ring of integers modulo pk (equivalently the ring Z / p k 2 ) for any prime p and integer k > 1. It is easy

to see that if an element of LD(A) gives a globally injective CA then the inverse CA must have a local rule which is an element of L D (A). Thus, the reversible linear CA are the units in L D (A) and hence correspond to the units in

A[xl , X l 1 . . . . . XD, XD1]. Sato [7] showed that the global injectivity of D-dimensional linear CA over finite commutative rings is decidable.

It is instructive to consider Sato's results here. In the proof of his Theorem 11, part (2), Sato showed that if A is a

finite commutative local ring of characteristic pk, k > 1, then a linear rule of the form (2.8) is globally injective if and only if exactly one of the rule coefficients oej is a unit in A (the others are necessarily nilpotent). In particular, this applies to A = Z / p k Z , k > 1 and A = Fq, q = ph, h _> 1, the finite field with q elements. In the finite field

case one must then have

f ~ LD(gZq) * if and only if f : ai ~ otjai+j, o~j 7£ O, (4.1)

corresponding to the fact that the units in Fq[Xl,X~ 1 . . . . . XD, XD 1] are the monomials oex~ 1 ...XJD D (see

Appendix A). Clearly in this case one has

f - 1 : ai ~ oFflai- j . (4.2)

In the 77/pk77 ~ 7lip k one has the following surjective ring homomorphism whose existence follows from the

factor theorem for rings:

)~k : Z / P kZ > Z / p Z , a + pkZ ~ a + pZ. (4.3)

The kernel of )~k consists of the non-units in 2_/pk2_, i.e. those of a + pk77 such that pla. The following induced homomorphism is surjecfive:

-ilk : Z /PkZ[x l , X l 1 . . . . . XD, XD 1] > Z / p Z [ x l , X l 1 . . . . . XD, XD~],

k2 k2 Z OejX(I"''XJDD ~ Z )~k(Oej)X~I'''XJDD" (4.4)

j=--k l j = - k l

Note that in contrast to the finite field case there are linear local rules corresponding to globally injective CA which are not injective as maps (Z /pkZ ) x > Z/pk7/.

From Sato's result it is clear that f ~ LD(7//pkT/) is globally injective if and only if exactly one of the rule coefficients is a unit and the others are in the kernel of )~k. Thus, an alternate characterisation of the elements of L D (?7 / pkZ) * i s :

Lemma 4.1. Let f E L ( Z / p k Z ) , k > 1. Then the CA corresponding to f is globally injective if and only if ~k (~ ( f ) ) is a monomial.

Lemma 4.1 corresponds to the fact that the units in 2_/pkT/[xl, X l 1 . . . . . XD, X~ 1 ] are the elements of the form

(see Appendix A)

R.A. Dow/Physica D 110 (1997) 67-91 77

D + P ( x l . . . . . x l), . ( 4 . 5 )

where P (Xl . . . . . XD 1) is in the kernel of~k. Sato did not construct explicit inverses for the linear CA he considered; we do so for f c LD(7//pk7/) *.

Theorem4.1. Let f c LD(7//pkT/) *, k > 1. Let a j , j c 7/D, be the only unit amongst the rule coefficients of f .

Then

_pk-1 jlpk-1 . xJDPk-I f - 1 ---- .c-t(o~j Xl .. ~:(f)pk-l--1).

Proof. r ( f ) must be of the form

g(xl . . . . . XD 1) = ~ j X l j' "" "XD jD + P(Xl . . . . . XD1),

where P(x l . . . . . XD 1) is in the kernel of Pk. One can show (the proof is essentially that of Sato,s Lemma 3, see [7]) that

-1 pk-1 pk-1 _jlpk--1 _jDpk-I g(xl . . . . . X D ) = a) X 1 ' ' ' X D

pk-1 jlpk-1 . . . xJDpk-1 Hence 1 = o t f x 1 g(x l . . . . . XD1) pk-~ and the result isproved. []

Example4.1. Consider f c L(7//47/), f : a i i--+ 2ai + ai-3. Then z ( f ) = 2 + x 3 hence, by Theorem 4.1, f - 1 = z - l ( x - 6 ( 2 + x3)). Hence

f - 1 . ai ~ ai+3 + 2ai+6.

One can also use Sato's result to count the linear local rules in L D (Z/pkT?) of up to a given radius l.

Lemma 4.2. The number of linear local rules over Z/pky_ of d-radii Id or less, 1 _< d < D, which correspond to globally injective CA is

(2ld + 1) (p -- 1)p(k-1)I-I~=l (21d+1).

Proof. In 7//pkZ there are (p - 1)p k-1 units and pk-1 nilpotent elements (see for instance [24, p. 96]). Fix

j , - - ld <_ Jd < ld, 1 < d <_ D , for ~j to be a unit and all other rule coefficents nilpotent, then there are

(p _ 1)p(k-l) 1-[~=~ (2ld+1) possibilities, thus letting j vary there are

(21d + 1) (p -- 1)p(k-1)l-[~=~ (2ld+l)

distinct elements of L (~_/pk7/) of d-radii up to ld giving globally injecfive CA. []

From (4.1) the number of elements of L(Fq) of d-radii Id or less, 1 < d < D, that give globally injective CA is

exactly (q -- 1)I-[ff_l (21d + 1). Turning to the additive case, let A be an n-dimensional 7//m7/-algebra. The units in A d d D ( A ) are the local

rules giving globally injective CA and these correspond to units in Mn (Z/m7/[xl , x ~ 1 . . . . . XD, XD1]) under the

78 R.A. Dow/Physgca D 110 (1997) 67-91

isomorphism q). The following theorem, which characterises globally injective additive CA over finite-dimensional

Z/mZ-algebras, follows immediately.

Theorem 4.2. Let A be an n-dimensional Z/mZ-algebra and let f ~ A d d D (A). Then f gives a globally injective

CA if and only if Det (cI)(f)) is a unit in Z / m Z [ x l , x~ 1 . . . . . XD, XD 1 ].

Let A be an n-dimensional Z/pkZ-algebra, k > 1, then f c A d d D (A) gives a globally injective CA if and only

if

D e t ( ~ ( f ) ) = otx~ I .. . x jD + P ( x l , . . . , XD1), (4.6)

where ot ~ (Z/p~77) *, j ~ y_D, P(x l . . . . . XD 1) in the kernel of~k. In particular if A = ~Cq, q = pn, n > 1, one

has that rule f E AddD(gZq) gives a globally injective CA if and only if

Det(CI)(f)) = otx~ 1.. .x j ° , ot # O, j E 7/D. (4.7)

The isomorphism q5 : Hom~,/m~(A, A) ~ Mn(77/mT/) for an n-dimensional algebra over Z / m Z is a standard construction. However, for the reader unfamiliar with this construction we give some of the details here. For more

detail see [27, Ch. vii]. Let X E Hom~_/m~_(A, A) and let {o91 . . . . , con} be an ordered basis for A over Z / m Z . Then for 1 < i < n

n

X (coi) -=- ~ aijo)j, aij ~ Z/mY_. (4.8) j=l

Then

4) (X) = {aij }. (4.9)

Note that if A ~ b n then = ~ i = 1 bi°)i

X (b) = bi aij°oJ = Z biaij ¢oj. i=1 j = l j = l \ i=1 /

Thus, if one writes b as a column vector, (bl . . . . . bn) T, where T denotes transpose, then to obtain the image of b

under X one operates on b with the transpose of ~b (X), q5 (X)t:

b ~+ ~b(X)tb.

The following example illustrates the technique.

Example 4.2. Let A = Z/4Z[z] / ( ( z 2 + z + 1)Z/4Z[z]), a two-dimensional Z /4Z algebra. A basis for A over Z /4Z is {1 + (z 2 + z + 1)Z/4Z[z], z + (z 2 + z + 1)Z/4Z[z]}. From now on we shall abbreviate a + (z 2 + z + 1)Z/4Z[z]

to a. Let X E Hom2_/4~_(A, A), then, by (4.8) and Z /4Z linearity, X is of the form

X : o e + f i z ~ - ~ o t a + o t b z + f i c + f l d z

for all ~, fi E Z /4Z and some a, b, c, d E Z/4Z. Then

X(1) = a + bz, X ( y ) = c + dz.

R.A. Dow/Physica D 110 (1997) 67-91

From (4.9) it follows that

It is easy to verify that if XL 6 Homz/4z (A , A) is A-linear then q~(XL) is of the form

gd(XL) = ( a b ) 3b a + 3b "

Let f E AddZ(A), say f :ai w-~ X - 1 , - l ( a i - l , j - a ) q- X- l ,o(ai -a , j ) -4- Xl,o(ai+l,j) where

X_l,_l(Ot q- flz) :~x , X_l,O(Olq-flz) =f ig , Xl,o(ot-}-flz) = fi + 2c~z.

Then

~b(f) = ~b(Xl,0)x -1 -}- ¢(X-I ,O)X -4- (o(X-1, .1)xy -~ x_ 1 x

=> Detclo(f) = x2y + 2x -2 c Z /4g [x , x -1, y, y - i ] . ,

hence ( D e t ~ ( f ) ) -1 = x -Zy -1 + 2 x - 6 y -2 so

{ x - l y -1 + 2 x - 5 y "2 2 x , 3 y -1 ) qo(f) -1 : I 3 x - 3 y - 1 q -2x-Yy -2 x - l q _ 2 x - 5 y - 1

/

: qh(y7,z)x-Vy -2 q- dp(Y5,2)x-5y -2 -4- dd(Y5,1)x-5y -1

q-qS(Y3,1)x-3 y -1 -}- ~ ( Y l , 1 ) x - l y -1 q- q~(Y1,0)x -1.

One finds that

Y7,2(ot q- flZ) = 2/3, Ys,z(ot + flZ) = 2oe , I15,1(~ + f lZ) = 2flZ,

Y3 , l ( o t+~ z )=3 f i+2o tZ , Yl , l (Ot+fiz) =or , YI,0(c~ + flZ) = f l z .

Hence,

f - 1

79

Definition 4.1. let f c AddD(A) be a unit, where A is an n-dimensional 77/mZ-algebra. let

Det(crp(f))-I = ~ otixil . .. xiD D C 7/Im7/[Xl, X l 1 . . . . . XD, xDll • i=r

: ai ~-+ Y7,2(ai+v,j+2) q- Y5,2(ai+5,j+2) q- Y5,1(ai+5,j+l)

q-Y3,1(ai+3,j+l) q- Yl,a(ai+l,j+l) q- Yl,0(ai+l,j).

Using Theorem 2.2 we see that the rule fu also gives a globally injective CA for each u c A. For instance, using (2.11),

f l I : ai t-+ Y7,2(ai+7,j+2) q- Y5,2(ai+5,j+2) -+- Y5,1(ai+5,j+l)

q- Y3,1 (ai+3,j+l) q- YI,1 (ai+l,j+l) q- Yl,o(ai+l,j) -}- 3 + 2z.

Given f E Add D (A), a unit and of radius l, we can find an upper bound on the radius of f - 1. We need the following definition.

80

Then

R.A. Dow/Physica D I10 (1997) 67-91

rid(f) = max(lrdl , ISdl), 1 < d < D, r i ( f ) = max (rid(f)) . l < d < D

Theorem 4.3. Let A be an n-dimensional 7/ /mZ-algebra. Let f c A d d D ( A ) be a unit of radius I and d-radius

let, 1 < d < D. Then f - I has d-radius at most rid(f) + (n -- 1)ld and radius at most r i ( f ) + (n - 1)/.

Proof If f is a unit in A d d D ( A ) then D e t ( ~ ( f ) ) is a unit in Mn(77/mZ[xl, X l I . . . . . XD, XD1]). Now,

r b ( f ) -1 = D e t ( ¢ p ( f ) ) - l g p ( f ) a,

where ~ ( f ) a is the classical adjoint of q~(f) (see [27, Ch. vii]). The i j term of q~(f)a is ( - 1 ) i+j Det (dP( f ) ) j i )

where D et (q5 (f)ji) is the minor of q> ( f ) at j i . For 1 < d < D the maximum of the moduli of the maximum and

minimum degrees in Xd of this entry is (n -- 1)ld. It follows that the maximum of the moduli of the maximum and

minimum degrees in Xd of any entry of q ~ ( f ) - I is rid(f) + (n -- 1)ld, thus the result for the d radius follows and

hence that the radius of f - 1 is at most

mr~<~ (Od(f) + (n -- 1)/d) < r i ( f ) + (n -- 1)/. [] l < d D

It follows from Theorem 4.1 and (4.6) that for A an n-dimensional 77/pkZ-algebra, k > 1, and f E A d d D ( A ) *

of radius l that one must have

nlp k-1 + n l (p k-1 - 1) > r i ( f ) . (4.10)

Thus we have:

Corollary 4.1. Let A be an n-dimensional Z/pk77-algebra, k >_ 1 and f ~ A d d D ( A ) * of d-radius ld, 1 < d < D

and radius I. Then the d radius of f - l l s at most (2np k=l - 1)ld and the radius of f - 1 is at most (2np k-1 - 1)/.

This upper bound on the radius of the local rule of the inverse of a globally injective CA cannot be improved

in general, i.e. the upper bound is obtained for some rules, for instance the rule of Example 4.2 whose inverse has

radius 7, the upper bound predicted by Corollary 4.1.

Let A be an n-dimensional 77/m77-algebra where the unique factorisation of m into powers of primes is m = t [Ir=l pkr. Then, as is well known,

x . . . x

and one can show that

A ~ A1 x . . . x At, (4.11)

where Ar is an n-dimensional 77/p~r 7Z-algebra (see Theorem A.1 in Appendix A). We consider a more general

case. Suppose that A ~ A1 x . . . x A, where Ar is an nr-dimensional 77/p~r 7/-algebra, 1 < r < t. Let 69 be the

i somorphismand0r , 1 < r < t, be the natural projection homomorphism A > Ar ,a ~ 6)(a)r. The isomorphism O induces isomorphisms

LD(A) ~-- LD(A1) x . . . x LD(At) , (4.12)

A d d D ( A ) -~ AddD(A1) x . . . x AddD(At ) . (4.13)

R.A. Dow/Physica D 110 (1997) 67--91 8t

Here one has f w-~ (.fl . . . . . . /~) where, recal l ing (2.3),

.]r : ai ~ 0r ( f ( [ / -q - ,)¢']A/Tt))), 1. < r < t. (4.14)

l In part icular with m = Iq , .= l P~' where each p,- is pr ime and kr > 0 then

A d d O ( Z / m Z ) ~-- LD(7/ /mT/) ~-- LD(7//p/[~7/) x . . . x LD(Z /p~ tT / ) . (4.15)

k~ . . . . X - " l Let c-/, r bc the isolnorphisrn from Hom~..~/,p]?.//(A,.,. ~ A t ) to Mn,. (7//Pr 7/Ix, ]) and let ~r&. be the bomomorph i sm

of (4.4). The fol lowing theorem is immedia te from the above and Theorem 4.2.

Theorem4.4 . Let A -~ AI x - - . x At where Ar is an n r -d imens iona l 7//pr~7/-algebra, 1 < r _< t. Then f E

A d d O ( A ) gives a global ly injective C A i:f and only il' D e t qS,-()r) is a uni t in 7/ /p~/7/[xt , x I t . . . . . Xo , Xot] ,

1 < r < t .

t k,. in par t icu larTheorem 4.4 holds when m = 1-Ir--I Pr where each Pr is pr ime and kr > 0 and A is an n -d i lnens iona l

7//mT/-algebra.

From Corollaries 4.1 and Theorem 4.4 one obtains:

Theorem 4.5. Let m ~ L. : ~Ir=l Pr where each p,. is pr ime and kr > 0 and let A be an n-d imens iona l /7 /mT/-a lgebra .

Then if f 6 A d d D(A)* has d radius ld, 1 < d < D and radius l then t h e d radius and radius of of f - t are at most

ld max ( 2 n r p k~-I -- 1) and 1 max (2n,.p~ ' - 1 - - 1 ) , I. < r -~:t 1 -< r < t

respectively,

One would like to be able to use Theorem 4.4 to construct the inverse local rule when it exists. Let m = ]-I t k~ r=t Pr ~- 1~I'=1 qr. Then, as the q,. are pairwise coprime, tlaere exist integers sl . . . . . st, s l ,2 , . . . . . st,t such that

s lq i + s2q2 = 1,

S l ,zqtq2 + s3q3 : 1, : (4 .16 )

s l , t - t q t " ' q ~ + stqt = 1.

Theorem 4.6. Let A be an n-d imens iona l Z/mT/-a lgebra , where m = I-[ t k,. t r=t Pr = I~r.-I qr. Let f E A d d l ) ( A ) * and suppose one has found the .])Y t ~ A d d O ( A r ) *, 1 < 1" < t. Then

f - 1 = f t - I + (.ft~.ll _ ,f~-l)s, t qt + (fl-_ I -- .f~-~ ) s t - , st q,. .. qt

q - ' " q- ( f l 1 -- ¢2 -1 ) s Z " ' s t q 2 " ' " qt,

where the ari thmetic is performed rood m.

P l w ~ This is jus t a matter of verifying, with the aid of (4.16), that Or(.f -1 ) = .fr -1 , 1 < r < t, for the stated f - I El

Example 4.3. Let A = 7 / / 6 0 7 / . Then

L(7//607/) ~ L(7//47/) x L ( 7 / / 3 Z ) x L(7//57/) -= Lj x L2 x L3.

82 R.A. Dow/Physica D 110 (1997) 67-91

Let f ~ L(7//6OZ), say f : ai w-> 24ai_1%- 5ai %- 30ai+1. One finds that

f l : ai ~ ai %-2ai+1, f2 : ai ~ 2ai, f3 : ai w-~ 4ai-1

and that

f l 1 : ai w-~ ai %- 2ai+1, f2 -1 : ai ~ 2ai, f 3 1 : ai ~+ 4ai+1.

Using Theorem 4.6 one has

f - 1 = 4a i+1%- 25 (2a i -- 4 a i + l ) -- 75(a i %- 2 a i + l -- 2ai) -= 5ai %- 54a i+1 .

Lemma result. The next theorem is the generalisation of 4.2, its proof follows exact ly the same lines as that

Theorem 4. 7. Let m t kr = I-Ir=l Pr where pr i sp r ime andkr > 0, 1 < r < t. Then the number S(m, D, L1 . . . . . LD) of elements of LD(7//m7/) *, D > 0, with d-radi i up to L d, 1 < d < D, is

S(m, D, L 1 . . . . . LD) = (2La + 1) (Pr -- 1)pr r = l

The corresponding result for linear rules over Dzq is

D

S(~q, D, L1 . . . . . LD) = (q -- 1) I--[ (2La + 1). (4.17) d=l

The function S(m, D, L1 . . . . . LD) has some useful properties. Let ml , m2 be coprime integers, it is easy to

verify that

S ( m l m 2 , D , L1 . . . . . L D ) = S ( m l , D , L1 . . . . . L D ) S ( m 2 , D, La . . . . . LD) . (4.18)

t t kr+vr If m I-Ir=l Pr ~r and m' ~-- = I-Ir=l Pr , Ur >__ 0 , 1 < r < t, then

1-I~=1(2L~+~) S(mI, D, L1 . . . . . O . . . . . (4.19)

\ ] r = l

In particulax

S(rn ~, D, L1 . . . . . L D ) = m (k--1) I-ID=l(2Ce+ l) S(m, D, L1 . . . . . L D ). (4.20)

In Table 1 we give, with the aid of Theorem 4.7, the number of globally injective linear rules of radius 1 over

2~/m77 for 2 < m < 48 and in one, two and three dimensions.

Using S(m, D, L1 . . . . . L o ) we can find the fraction of D-dimensional linear CA rules over 7//m77 with d-radii

up to La, 1 < d < D, that are globally injective. This fraction is, with ~ = 1-[ff=l(2La + 1),

t S(m, D, L1 . . . . . LD) = 6t I -I Pr -- 1

rna r=l Pra (4.21)

Note that (4.21) is independent of the powers kr of the primes Pr, 1 < r < t, occurring in the prime factorisation

of m. Thus we have:

R.A. Dow/Physica D 1t0 (1997) 67-91

Table 1 The number of D-dimensional globally injective linear CA of radius 1 over 77/m77 for D -- 1, 2, 3 and 2 < m < 48

83

D m 1 2 3 2

2 3 9 27 3 6 18 54 4 24 4608 3 623 878656 5 12 36 108 6 18 162 1458 7 18 54 162 8 192 2359296 486388759756013 568 9 162 354294 411782264189298

10 36 324 2916 11 30 90 270 12 144 82944 195 689447424 13 36 108 324 14 54 486 4374 15 72 648 5832 16 1536 1207959552 65281994259 189975434133 504 17 48 144 432 18 486 3 188 646 11 118 121 133 111046 19 54 162 486 20 288 165 888 391378:894848 21 108 972 8748 22 90 810 7290 23 66 198 594 24 1152 42467 328 26264993026 824732672 25 1500 70312500 804662704467773437500 26 108 972 8748 27 4374 6973568 802 3 140085798 164 163223 281069126 28 432 248 832 587 068342272 29 84 252 756 30 216 5832 157464 31 90 270 810 32 12288 618475 290624 8762000948777 521623 145212555 558912 33 180 1620 14580 34 144 1296 11 664 35 216 1944 17496 36 3888 1632586752 1492248 958 114950165 823488 37 108 324 972 38 162 1458 13 122 39 216 1944 17496 40 2304 84934656 52529986053 649465344 41 120 360 1080 42 324 8748 236 196 43 126 _ 378 1134 44 720 414720 978447237120 45 1944 12754584 44472484532444184 46 198 1782 16038 47 138 414 1242 48 9216 21743 271 936 3 525 227 689996258673443209216

Theorem 4.8. For f ixed D and d - rad i i Ld, 1 < d <_ D, the f rac t ion o f l inear C A over 77/m7/which are g loba l ly

in jecf ive d e p e n d s on ly u p o n the d i s t inc t p r i m e s occu r r i ng in the p r i m e fac to r i sa t ion o f m, no t u p o n the p o w e r s to

w h i c h these p r i m e s occur.

84 R.A. Dow/Physica D 110 (1997) 67-91

It is clear that the fraction (4.21) goes to zero in the limit of large 8.

Example 4.4. The fraction of radius 1, one-dimensional linear CA rules over Z/2kT? that are globally injective is 3/8 for all k > 1. For 77/2k31z this fraction is 1/12 for all k, l > 1.

5. Failure of global injectivity

Having established how to determine whether a given additive local rule yields a globally injective CA we give a

method for determining the lengths of the spatially periodic configurations on which a non-globally injective additive CA fails to be injective. Our approach is similar to that of Voorhees [28], where a representation as polynomials over C was employed to deal with one-dimensional linear CA over finite fields. We feel that our method has

some advantages over that of Voorhees, in that the construction for the linear case is simpler and that the method extends easily to the general additive case and to state alphabets other than finite fields. We deal only with the

one-dimensional case. To consider spatially periodic configurations of period N we use periodic boundary conditions on N cells. Let A

be a finite commutative ring. For each integer N > 1 there is a ring homomorphism from L ( A ) onto the quotient ring A [ X ] / ( x N -- 1)A[x]:

A xl (m A xl 2 i z 2 K N : L ( A ) > (x N f : aiv-+ Z otjai+j ~ Z ° t J x - J ÷ (XN -- 1)A[x]. (5.1) j=--II j=--ll

Given a linear local rule f we shall denote K N ( f ) by q]-N ( f ) or just ~-N when no confusion can arise. The represen- tation given by (5.1) is that used to represent the global dynamics of a linear CA in [8], equivalent to that in [13].

In [8] it is shown that the configurations of such CA on N cells with periodic boundary conditions can be identified bijectively with the elements of A[x] / (x N - 1)A[x]:

N-1 aoal . . . a N ~ ~ aix i + (x u -- 1)A[x]. (5.2)

i=0

The global dynamics of such CA are then represented in A [ x ] / ( x N - 1)A[x] by multiplication:

~ U : a ~ ~ N ( f ) a . (5.3)

One says that the local rule f has representative ~-N on N cells. It is clear that the CA corresponding to linear local rule f is reversible on N cells if and only if ~-N is a unit in A[x] / (x N -- 1)A[x] and hence the CA corresponding to f is injective on spatially periodic configurations of period N if an only if ~-N is a unit in A[x] / (x N -- 1)A[x].

When A is a finite field one has

TN is auni t if and only if gcd(q]-N(X), X N - - 1) ---- 1. (5.4)

Let A be any ring. Let A[x] ~ R(x ) = ao + a lx + . . . + a n - i x n-1 + x n, where a0 is a unit and n > 0. Consider x -1 + R ( x ) A [ x , x - l ] , this coset contains the element

x -1 -- a o l x - 1 R ( x ) = - a o a (am + a2x ÷ . . . ÷ an-1 x n - 2 ÷ x n - l ) = R1 (x) E a [ x ] , (5.5)

thus for every integer k > 0

x -k ÷ R ( x ) A [ x , x -1] --- Rk(x) + R ( x ) A [ x , x -1] ---- R1 (x) k + R ( x ) A [ x , x - l ] . (5.6)

R.A. Dow/Phys ica D 110 (1997) 67-91 85

Lemma 5.1. Let A be a finite ring (not necessarily commutative). Let R(x) ~ A[x] C A[x, x -1] be monic with

degree greater than zero and f (0) a unit in A, then

a[x, x -1] ~ A[x]

R(x)A[x, x -1] -- R(x)A[x]

via

tR " a(x) + R(x)A[x] ~-+ a(x) + R(x)A[x, x- l ] .

Proof. It follows from (5.6) that every coset or(x, X -1) -}- R(x)A[x, x -1 ] contains elements of the form a(x) + R (x)A[x, x -1] with a(x) E A[x], thus the map LR is surjective. Clearly tR is well defined and a ring homomorphism.

It remains to show that LR is an isomorphism. We consider the kernel of LR, suppose this is non-trivial, i.e. there is some a(x) c A[x] such that a(x) + g(x)A[x] # R(x)A[x] but a(x) + R(x)A[x, x -x] = R(x)A[x, x - i ] . We can

choose the representative a(x) to have degree less than that of R(x), with such a choice of a(x) we must have, in A[x, x - l ] , that

a(x) = b ( x , x - 1 ) R ( x ) = Z bjxJ R(x). j =--li ]

If 12 > 0, one contradicts degree of a(x) less than that of R(x) and if ll > 0, one contradicts a(x) c A[x], hence there can be no such b(x, x -1) and hence no such a(x), thus tR has trivial kernel and is an isomorphism. []

Note that t~ 1 is given by

12 -- 1 12

Z °tJxJ + R ( x ) A [ x ' x - 1 ] ~ Z otjRj + ~ o t j x j +R(x )A[x l , (5.7) j=-- I 1 j=-- l l j = 0

where the Ri (x) are given by (5.5) and (5.6). It is easy to show this once one notes that

Rs(x)x k + R(x)A[x, x -1] = x k-s + R(x)A[x, x - I ] , 0 < s < k,

Rs(x)x k + R(x)A[x, X - 1 ] = Rs-k(x) + R(x)A[x, X - I ] , S > k. (5 .8 )

There is a relation between the isomorphism r and the homomorphism K N for each N > 0. For each N > 0 there

is a projection homomorphism TxU_ 1 : A[x, x -1] ) A[x, x -1] / (x N - 1)A[x, x - l ] :

a(x, x -1) ~ a(x, x -1) + (x N - 1)A[x, x - l ] . (5.9)

Lemma 5.2. For each integer N > 0

K N ~ l -lxN_l 0 ]'~xN_l. 0 T.

Proof Let f ~ L(A) be given by (2.8). For each N > 0 one has, using (5.5) and (5.6),

- 1 12

t -1 o TxN_ 1 o v ( f ) = ~ otjX - j + ~ a j ( x N - - 1 ) j + (X N -- 1)A[x] xN--1 j=-- I 1 ;' j = 0

li 12 = _jxJ + u j - j + (x N - 1)a tx] ,

j = l j=O

86 R.A. Dow/Phys ica D 110 (1997) 67-91

- 1 12

~- Z O l j x U - J -I- ~ OtjX u - j -]- (X u - - l ) a [ x ]

J=-h j=o

= t e N ( f ) .

In the above we have used the fact that X u - j -[- (X N -- 1)A[x] = x N j - j + (X N -- 1)A[x], as the difference of these

two elements is

- - (X N j - j - - X N - j " ~- (X u -- 1)A[x] = - x u - j (x u - 1 ) ( x N ( j - 2 ) -1- x N ( j - 3 )

+ . . . + x N + 1) + (x u -- 1)A[x]

= (x N -- 1)A[x]. []

The advantage of T over KN is that T, by (5.10), associates to each linear local rule f a fixed polynomial g(x) ,

independent of the number N of cells.

Let f ~ L ( A ), then

12 Ii Ii

j=-- l l j=--12 j=--12

hence

r ( f ) = x- t2 g(x) , g (x ) ~ A[x]. (5.10)

Let A = Fq and f E L(lZq). From (5.10) we have r ( f ) = x-12g(x) where Igq[X] ~ g (x ) ¢ olx s. Thus

n

g(x) = o~x v 1--I gi (x) ei , (5.11) i=1

where ei c N\{0} and gi(x) is monic and irreducible, 1 < i < n. For each linear local rule f we have r ( f ) = x - Zg ( x ) and we can write g(x) in the form (5.11). Each of the gi (X) divides a cyclotomic polynomial Qdi (x) (see

[29, Ch. 2, Section 4]) for exactly one positive integer di coprime to p. We define

I ( f ) = {d: gcd(g(x), Qd(x)) > 1}, (5.12)

then we have the following result.

Theorem 5.1. A CA with local linear rule f over IZq is reversible on N cells if and only if d # N for all d 6 1 ( f ) .

Proof With g(x) given by (5.11) let d c I ( f ) then gcd(g(x), Qd(x)) ~ 1, hence as Qd(x)l xN - 1 whenever

d iN , TN ( f ) is not a unit whenever diN. Suppose conversely that d # N for any d c I ( f ) but TN = -~N ( f ) is not a unit and hence gcd(~-n (x), x N - 1) ¢ 1. It follows that there is some irreducible polynomial h (x) I gcd(]-N (x), x N --

1), h ( x ) 7 ~ x . N o w ~-N = x l 2 N - 1 2 g ( x ) + ( x N - - 1)IZq [X], thus h(x)]g(x) , but then h(x) is one of the gi (x ) a n d t h u s

h (x)[Qe (x) for some e lN, thus e ~ I ( f ) , a contradiction. Hence TN must be a unit and the rule is reversible on N

cells. []

As an example we list in Table 2 those numbers of cells for which the radius 1 linear CA over 1:3 are irreversible (we exclude those rules which are always globally injective). The information required for the reproduction of the

table is the following:

Q l ( x ) = x - 1 , Q2(x) = x + l , Q4(x) = x2 + l , Q8(x) = (x2 + x + 2)(x2 + 2x + 2).

R.A. Dow/Physica D 110 (1997) 67-91 87

Table 2 Local rules of radius 1 over F 3, f given in the form et_l~0Cel, meaning f is air---> Ol_lai_ 1 q- etoa i q- etlai+ 1 (the CA corresponding to f is not reversible on N cells whenever n IN and hence fails to be injective on spatially periodi c configurations of period divisible by n)

f v ( f ) n f v ( f ) n

011 x - l ( x + 1) 1 102 x - l ( x - 1)(x + 1) 101 X--I(x 2 + 1) 4 201 2X-I(X -- 1)(X + 1) 110 X + 1 2 210 2(X -- 1) 111 X-l(X -- 1) 2 1 120 X -- 1 022 2X-1 (X + 1) 2 121 x - l ( x + 1) 2 202 2X--1 (X 2 + 1) 4 112 X-I(x 2 +X + 2) 220 2(X + 1) 2 211 2x- l (x 2 +2X +2) 222 X-I(x -- 1) 2 1 122 x--l(x 2 + 2X + 2) 012 X-I(x -- 1) 1 212 2X-I(x ÷ 1) 2 021 2X-I(x = 1) 1 221 2X--I(X 2 +X + 2)

1 1 1 1 2 8 8 8 2 8

T u m i n g now to f ~ L ( Z / p k Z ) , k > 1. In

fol lowing surjective homomorphisms .

Pk " L (7 / / PkZ) > L ( 7 / / p g ) ,

( )( ) 12 12

f : ai ~ Z oqai+j ~ - f : ai ~ Z )~k(oq)ai+j ; j=--ll j=--ll

addit ion to the h o m o m o r p h i s m s )~k, Pg defined earler we have the

(5.13)

Y / p k Z [ x , x - I ]

Ak " (x N _ 1)Z/pk7/[x ' x _ l ]

a(x , x -1 ) -t- (x N -- 1 ) Z / p k Z [ x , x -1]

Z / p Z [ x , x -1] >

(x u - 1 ) Z / p Z [ x , x -1]

~-~ Vk (a (x , x - l ) ) ÷ (x u -- 1)Z/pT/[x , X - l ] ; (5.14)

~_/p~y_[x] Z/p77[x] Ak " (x u _ 1 ) Z / p k Z [ x ] > (x n -- 1 ) Z / p Z [ x ] '

n n ~ _ a i x i + (x N - 1 ) Z / p k Z [ x ] ~ ~_ )~k (a i ) x i + (x N -- 1)Z/pT/[x].

i=0 i=0

It is easy to verify that the diagram (5.1 6) commutes , where the i somorphism r : L (g /pk7 / )

denoted zk for clarity when k > 1. Similarly, we write TklxN_ i and tk,xN_ 1 when k > 1.

(5.15)

> 7 / /pkZ[x , X ' l ] is

g,xN-l> Z / p k g [ x ] L ( g / p k Z ) ~> 7 / /pkT /[x ,x -1] Tk.xN 17 g / P k g [ x , X -1] t -1 (X n -- 1 ) Z / p k Z [ x , x -1] (x N -- 1 ) Z / p k Z [ x ]

L(Z/pZ) r> Z//p7/[X,X_]] YxN-1 Z / p Z [ X , X -1 ] L-1 xN-1 Z/pZ[x] > >

(x n -- 1 ) Z / p Z [ x , x -1] (x N - 1 ) g / p Z [ x ]

One has the fo l lowing result.

(5,16)

Theorem 5.2. Let f c L (Z /pk7 / )kL(77 /pk /77) * be such that P k ( f ) ~ O. Then the C A corresponding to f fails to

be inject ive on spatially periodic configurat ions of per iod N if and only if N is divisible by an e lement of I (Pk ( f ) ) .

88 R.A. Dow/Physica D 110 (1997) 67-91

Proof Denote t -1 k,xU_l o Tk,xN_ 1 o r k ( f ) by fN. If some element of I ( p ~ ( f ) ) divides N then fly i s not a unit in

77/pl~Z[x]/(xU -- 1)Z/pkZ[x] and hence one does not have injectivity on spatially periodic configurations of period

N.

Suppose that no element of I (Pk ( f ) ) divides N but f u is not a unit. Then Ak ( fN) is a unit and Ak is surjective so

there is some a such tha ta = Ak(b) and A k ( f x ) a = 1. Hence f u b = 1 +c where c E ker A-~. Then ( fNb) pk-1 = 1 and thus fN is a unit, a contradiction. []

Becasue of Theorem 5.2 we define I ( f ) for f ~ L(7//pk77)\L(Z/plC/77) * to be I ( p k ( f ) ) unless p ~ ( f ) = 0 in

which case we define l ( f ) = {1}. From our earlier results the case A = 2-/pky-, k _> 1, clearly suffices to handle

the case A = Z / m Z any positive integer m > 1.

In general one can represent the configurations of an additive CA over an n-dimensional 77/m77-algebra A

with periodic boundary conditions on N ceils (equivalently spatially periodic configurations of period N) by the

elements of A[x, X--1] / (X u -- 1)A[x, X - 1 ] ~ A[x] / (x N -- 1)A[x], via a o a l ' " a N - 1 w-~ a = ao + a lx + " " +

aN lX N-a + (X N -- 1)A[x]. Let {o91 . . . . . ogn} be a basis for A over Z / m Z , writing aj = ~ i ~ 1 b j i w i , 0 <

j < N -- 1, as a column vector one can then represent the action of the CA corresponding to f E A d d ( A ) by

multiplying a by the transpose of 4~ ( f ) and reducing modulo x N - 1. Equivalently let (PN ( f ) be the image of

4~ ( f ) in MN((7//mT/[x]/(x N -- 1)77/m77[x])) under the obvious natural homomorphism. Then the CA dynamics

is represented by

a ~-+ ~ N ( f ) t a . (5.17)

Now

Det ~ N ( f ) = T N ( D e t @ ( f ) ) ,

where TN is the projection homomorphism A[x, x -1] ~ A[x, x - 1 ] / ( x N - 1)A[x, x - l ] . Then CA fails to be

injective on N cells if and only if D e t ~ N ( f ) is not a unit. For instance, when A = Fq, q = pn, this happens

if and only if Det ~ ( f ) = x -V h (x ) for some v c ~ and h(x) not coprime to x N - 1 i f and only if d i N where

d ~ I (7 - l ( D e t crp ( f ) ) ) . For an n-dimensional Z/pkT/-algebra A and f ~ A d d ( A ) \ A d d ( A ) * we de fne

I ( f ) = l ( r - l ( O e t q a ( p k ( f ) ) ) , (5.18)

with Pl the identity, unless P k ( f ) = 0 when we define I ( f ) = {1}. It follows from earlier results that for A -------

A1 × . . . × At, where Ar is an nr-dimensional Z/prkr 7/-algebra, 1 < r < t, that we can define

t

I ( f ) = U I ( f r ) . r=l

(5.19)

Example 5.1. Let f :ai ~ o92a2+1 + coa2_1 c Add(Y-4). Then

cp( f ) = ( x + x - I x ) X - 1 x - 1 + x '

and thus

Det ~ ( f ) = x - 2 ( 1 + X 2 -[-X 4) = x - 2 Q 3 ( x ) 2,

where Q3 (x) is the third cyclotomic polynomial over I:2. Hence I ( 7 - l (Detcb ( f ) ) ) = {3} and the CA corresponding

to f fails to be injective on spatially periodic configurations of period divisible by 3.

R.A. Dow/Physica D 110 (1997) 67-91

Given a set A C N + let An = {m < n : m E A}. Then the density of A in N + can be defined as

IAn] /x(A) = lim

n-----+ ~ /~

89

(5.20)

Theorem 5.3. Let A ~ A 1 x - - • x At, where Ar is an nr,dimensional 77/p~:rr77-algebra, 1 < r < t. Then for any

f ~ A d d ( A ) \ A d d ( A ) * there is a finite set of positive integers, I ( f ) such that the CA corresponding to f fails to be injective on spatially periodic configurations of period N if and only if N is divisible by an element of I ( f ) .

For such rules f , if 1 ¢ I ( f ) then the set of integers N > 0 such that the CA corresponding to f is injective on

spatially periodic configurations of period N has density greater than zero in N +.

Proof The existence of I ( f ) follows from the discussion preceding the theorem. For f with i ~ I ( f ) and x 6 N +

let A ( f ) be the set of integers N > 0 such that the CA corresponding to f is injective on spatially periodic configurations of period N and let

A x ( f ) = {y < x " d . fy , d E l ( f ) } .

Let Ux = lAx ( f ) ] and let L be the lowest common multiple of the elements of I ( f ) . It is not hard to see that, with n = m L + j , O < _ j < L ,

/ z ( A ( f ) ) = lim u j + m u L n - ~ j + m L

As the j and uj are a finite set of finite integers it is clear that

U L I x ( a ( f ) ) = - - > 0. [] (5.21)

L

Acknowledgements

I would like to thank Franco Vivaldi for useful discussion during the course of this work. This work was supported by the UK Engineering and Physical Sciences Research Council under grant no. G R / K 17026.

Appendix A

We have assumed that the reader is familiar with the basics of algebra, i.e. that they know what groups, rings,

algebras, etc. are and understand what homomorphisms of groups, rings, algebras, etc. are. The reader who is not, should consult a text such as Lang [24]. However, we will emphasise one simple feature here, the characteristic of a ring. First, we shall briefly discuss the units in a Lanrent polynomial ring.

It is well known that the polynomial ring R [x] over an integral domain (ID) i s itself an integral domain and hence,

as R[x, y] -~ R[x][y], by induction one sees that R[xl . . . . . XD] is an ID. It is easy to see that this result extends to R[x l , x ~ 1 . . . . i X D, X D 1 ] . Every element of R[x, x -1] can be written a s x - J a (x ), j c N and a(x ) ~ R[x ], a

simple argument involving the degree o fa (x ) shows that the units in R[x, x -1] are of the form o~xJ, ot c R*, j ~ 77.

A simple induction on the number of variables shows that the units in R[x l , x { 1 . . . . . XD, XD 1 ] are of the form

~jx~ ~ . . . x jD, o~ E R*. As any field is an ID we have found the units in the finite field case. To see that the units in the 77/pk77 case, k > 1, are of the form shown in Section 4 one employs the homomorphism Pk (see (4.4)), the fact that ring homomorphisms map units to units and the argument of Theorem 4.1.

90 R.A. Dow/Physica D 110 (1997) 67-91

Let R be a ring, then under the ring addit ion R is an addit ive group. The charac ter i s t ic o f R, Char R, is the

addit ive order o f the mul t ipl icat ive identi ty 1R, i.e. the least integer m > 0 such that m 1R = 0 i f any such integer

exists, 0 otherwise. For example the characteris t ic o f Z is 0, the characteris t ic o f 2~/m77 is m. It is easy to show

that any finite r ing has non-zero characteristic. In a commuta t ive ring R the b inomia l theorem holds. I f the ring has

p r ime characteris t ic p then for any a, b 6 R one has (a + b) p = a p q- b p.

Let the characteris t ic o f a finite r ing R be m > 1, then there is a natural inject ive ring h o m o m o r p h i s m f rom 2 / m Z

into the centre o f R, namely n + m77 ~ n l e (the centre o f a r ing R is the set of e lements that commute with everY

e lement o f R, a subring o f R). This h o m o m o r p h i s m makes R into a Z/mTZ-algebra.

An e lement e o f R is idempotent i f e 2 = e, for example OR and 1R are always idempotent , the trivial idempotents.

Two idempoten t e lements e l , e2 are orthogonal i f e le2 ----= 0. The principal ideal generated by a central idempotent

e lement e in R, eR, is a ring, wi th operat ions those in R but mul t ip l icat ive identi ty e. A r ing R is i somorphic to a

direct product o f rings, R1 x • • • × Rt, i f and only i f there is a set o f t pa i rwise or thogonal non-tr ivial idempotents

ei, 1 < i < t, in the centre o f R t such that ~ i = l ei = 1 and Ri ~ Rei for 1 < i < t. This is the Peirce Decomposition (see [26, p. 37]), a fo rm of the Chinese remainder theorem.

t Theorem A.1. Let A be an n -d imens iona l 77/mZ-algebra where m ~I r= l Pr kr- Then A -~ t = [-Ir=l Ar where Ar is an n-d imens iona l Z/pkrrZ-algebra, 1 < r < t.

t krn Proof Clear ly [A I = m n = I-[r=1 Pr . As 77/m7/ ~-- Z/p~llZ x . . . x 7//pkt t there are t pa i rwise or thogonal

idempotents el . . . . . e2 in 7//m7/such that Z/p~rr7/~- ~rT//mZ, 1 < r < t. For 1 < r < t let er be the image of er

under the natural h o m o m o r p h i s m 7/ /mZ > A. It fo l lows that the er are a set o f pai rwise or thogonal idempotents

in the centre of A, thus

t

A ~- I - I e r A . r = l

Let COl , . . . , COn be a basis for A over Z/mY_. Clear ly every e lement o f era can be writ ten as a sum ~lelCOl + • • ' +

c~tetcot with ~r in the i somorphic image of g~rT_/mT/ in erA. Hence lerAI < pkrrn, 1 __< r _< t. Suppose that for n Ot ksn some s, 1 < s < t, that Y~4=Â ierCOi = 0 with some oti ~& 0. Then [esA[ < p= , it fo l lows that there must be

krn an r, 1 < r < t, such that lerAI > Pr , a contradict ion. Hence erCOl . . . . . erCOn is a basis for erA over Z/p~rZ, l < r < t . []

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