The power of totalistic cellular automata and totalistic systolic networks

Mahi Comput. Modelling, Vol. 11, pp. 367-369, 1988

0895.7177/88 $3.00 + 0.00 Pergamon Press plc

Printed in Great Britain

The Power of Totalistic Cellular Automata and Totalistic Systolic Networks * Karel Department

Culik

II

of Computer

Science

University of Sou th Carolina Columbia,

S. C. 29208

Abstract A systolic network is an array of synchronized processors. It is totalistic if the states of its processors are integers and the next state of each processor is determined by the sum of all the states in its neighborhood including its own. Our main result is that every uniform regular network can be simulated by a totalistic systolic network. Keywords:

1

Systolic

network;

totalistic

totalistic one. Here, we introduce

process data in parallel by passing them from one processor to the neighboring ones in a regular rhythmical

rected

pattern. For the purpose of this paper we define a systolic network as an arbitrary synchronized network of finite automata. We then make precise the notion of regularity of such a network. A special case of a regular systolic network is an n-dimensional cellular automaton (CA). Interest in CA has been renewed since their

Figure

considered

of di-

to be regular is systolic

according to our definition. Our main results are that

every network

based

on bounded in-degree digraph can be simulated by a semitotalistic network and that every systolic network can be simulated by a uniform totalistic systolic network. To make our more general results in the last section more readable we first remind how to simulate a one-dimensional CA by a totalistic one [l]. Definition. A cellular automaton A with set of states Q and transition function f : Q x Q x Q + Q is called totalistic, if Q 2 N (positive integers) and there exists

CA are uni-

S-1 -

of coloring

f’(r

a function

f’

: U + .N such that ~(z,Y,z)

=

+ y + z) for all z, y, z in Q.

4

4

s-2 -

new types

and use them to define the regularity

network commonly

(an integer) of a cell depends on its own current state and the sum of the states of its neighbors, excluding the cell itself. The well known “Game of life” is an example of a semitotalistic 2-dimensional CA. totalistic

graphs

condition for systolic networks. Our definition is very broad so that every uniform (all processors identical)

application to the study of complex systems [5,6]. In this context S. Wolfram introduced the notion of totalistic CA. A CA is totalistic if its cells have integers as states and the next state of a cell only depends on the sum of all the cells in its neighborhood including itself, A CA is called semitotalistic if the next state

asked whether

oneThis

result has been strengthened in [11,where it was shown that every one-dimensional CA can be simulated by a

Systolic systems introduced by Kung and Leiserson [4] are regular arrays of synchronized processors which

-

graph coloring.

versal. D. Gordon [3] has shown that totalistic dimensional CA can simulate Turing Machine.

Introduction

S. Wolfram

automaton;

SO -

1: A configuration

of cellular

The transformation of an arbitrary one-dimensional CA whose states are the positive integers, see Figure 1, is accomplished by a cyclic “coloring” of cells. We use four different powers of a basis B such that in the number representation with base B (B-ary numbers) of the sum of any three neighboring states the

automaton

A

left neighbor, the right neighbor and the own cell still identifiable by the position of a missing entry. Consider e.g. a cellular automaton A with of states Q = 1,2,. . , n and transition function Q x Q x Q --) Q and let the Figure 1 depict part configuration of A. Let B = n + 1 be the basis

‘This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A-7403 MCM cod-”

367

are set d : of for

Proc. 6th In!. Conf

368

on Mathematical

ModrIling

s-x lOXS_l

lOOXS_~

Figure

so

-

2: A configuration

-

is

Q’ = {s x m 1 s E Q, m E {l,lO, where s x m is the product

100,1000}},

= =

10 x d(z, y, z) 100 x d(s, y,z)

d’(yzOz) d’(zOzy)

= =

1000 x d(z, y,z) y,z)

tem. It is now straightforward, that the cellular automaton A with set of states Q’ and totalistic transi-

A without many

A without

1 For every cellular automaton

ists a totalistic states

loss

cellular

automaton

of

and has at most

time

loss of

A there ez-

A’ which simulates

four

times

as

as A.

6( zo, 51,.

network,

is a triple

N = (G, p, d) where Q = (V, E) is a directed

ordered graph,

(ii) p : V -+ M, where M is a finite set of finite state machines, is a function called processor labeling, and d

:E

Here a (Ko,K1

,...,

-+ N is a function

finite

state

CA A

. ) 5”) = g(z0 + ‘. + z,).

(iii) there exists a function all ~0,.

if instead of the above

f : ,VN2--) N such that for

. , 2, for which 6 is defined Zn) =g(zo,zl+“.+z,).

6(x0,51,...,

that the underlying graph of a (semi)totalistic network need not be ordered. Further, a network is called unifomn if all of its processors are identical, i.e. card(M) = 1. Clearly, the underlying graph of a uniform network is of a constant in-degree, that is all of its vertices have the same in-degree. In order to derive our simulation results we will need the following notions. Let G = (V,E) be a directed graph and rw: V --t C, where C is a finite set of colors, its vertex that LYis

labeling

called

coloring.

We say

(i) proper if (U,ZI) E E implies O(U) # o(v), (ii) in-strong

if (u,w),

(zI,u)) E E with u # u imply

machine

nice if there exists a function f that for each 2) E V, a(u) = f(&(v)) (o(u)>”

In this section we briefly define notions needed in this

(iii)

-

(ii) there exists a function 9 : N + .N such that for all 2e,5rr . . , 5, for which 6 is defined

(iii)

note and fix our terminology. A systolic network, or briefly

6)

-

o(u) # o(v),

Definitions

2

-

10x3s

A network N is called totalistic or semitotalistic if all of its processors are so. It follows immediately

d’(zyz) d’(syz0)

for all z, y, z in Q. Again zyz, zyz0, yzOz, zOzy, 10,100, 1000 are to be interpreted as numbers in B-ary sys-

Theorem

CA simulating

--c

(i) K; c N for each i 5 12, and

contains 4n states and we can define now the (partial) totalistic transition function d’ : N + Q’ as follows

tion function d’ correctly simulates time and therefore we have

-

100 x ss

Further, a FSM is semitotalistic (ii) the following holds:

of integers s and m. So Q’

d(s,

s1

--

of the totalistic

the coloring factors 10, 100 and 1000 (in B-ary notation). Then the configuration in Figure 1 changes to the one in Figure 2. Note that our “coloring” is proper, in-strong and nice in the notation of Section 3. Thus our new set of states

1000 x

{o(u)I(%u)

: 2c + C such where &(v) =

E E:>.

Condition (i) is well known, while (ii) and (iii) seem to be new. Intuitively, (iii) means that the color of a vertex u is uniquely defined by a set of colors consisting of the color of u and its neighbors. Example and in-strong proper,

The directed graph in Figure 3 is proper but not nice; the one in Figure 4 is

in-strong

and nice.

called delay indicator. (FSM)

is a tuple A =

K,,h)wheren>Oand

(i) Ke is the state set of A, (ii) Kj, for i = 1,. neighbor (iii)

. . ,n, is the state set of the ith

of A, and

6 : PC0 X K1 X . . x K,, + transition function of A.

KO is the

(partial)

From the point of view of this note the following two types of FSM’s are important. Let N denotes the set of positive integers. A FSM A is totalistic if

Figure

3: A proper and in-strong

but not nice digraph.

Proc. 6th Int. Cmf.

on Mathematical

3

369

Modelling

Results

Now we generalize the simulation CA in the introduction. Lemma

1 For each directed

degree

there

Theorem (G,p,d)

4: A proper,

in-strong

a semitotalistic

network it is natural to require that, for each configuration the vertices with processors in nonquiescent states form a connected subgraph of G. Now, we say that network Nr = (Gl,p,,dl) strongly simulates network NZ =

(Gz,~2,&), if

(i) Gr = Gr and dr = d2, in other words, Nr and Nr differ at most through their processor labelings, and (ii) there exists a one-to-one mapping g : RI + R2, where f& denotes the set of configurations of N;, such that for each s(nezrr(w))

= nezrr(s(zu)),

(1)

where nezti defines the change of configurations in one step computations according to Ni. Mathematically, commutes.

(1) says that the diagram in Figure 5

network

such that fi

Definition. Finally, we define what we mean by saying that a network strongly simulates another one. First, a configuration w of network N is an assignment of states to each processor. In the case of an infinite

gmph

with

coloring

tolic if it satisfies

strongly

Lemma (i)

2 For

exists

every

e. all nodes in V

systolic

in-strong

(Y is a refinement

Theorem

= a(v)

network

a totalistic

ulates

N.

For examples

N

=

is

(G,p,d)

and nice;

of p, that is for

implies

p(u)

systolic

all u, u E V

= p(v).

3 For every systolic

struct

of G that

coloring (Y such that

a (vertex)

a is proper,

a(u)

N.

A network N = (G,p, d) is called J~Jthe following (regularity) conditions.

(ii) There exists a (vertex) coloring proper, in-strong and nice.

(ii)

in-

fi simulates

(i) G is of uniform in-degree,i. have the same in-degree.

there

bounded

that is in-strong.

2 For every (finite OT infinite) network N = that G has bounded in-degree zue can

= (G,$,d)

and nice digraph.

(vertex)

shown for

such

construct

Figure

exists

technique

network

network

N we can con-

sim-

N that strongly

and proofs of similar results see [Z].

References 111 J.Albert

and K. Culik II, Simple Universal

Cel-

lular Automaton and its One-way and Totalistic Version, Complex Systems 1 (1987) 1-16.

PI K.

Culik II and J. Karhumaki, On Totalistic Systolic Networks, Inf. Proc. Letters, to appear.

[31 D. Gordon, On the Computational talistic

Cellular

Automata,

Power of Tomanuscript.

[41 H.T. Kung and C.E. Leiserson, next2 c

Figure 5: Commutative

R2

diagram.

Our notion “strongly simulates” is very restricted, indeed. Namely, it is not important to specify how a network “computes” since in any natural case “strong simulation” means intuitive simulation. Further, the functions d (defining delays) do not play any role, that is the reason why we have not specified their behavior.

(for VLSI). Proceedings ence (1978).

[51 S. Wolfram, Automata, Physics

[61S.

Computation Theory of Cellular Communications in Mathematical

96 (1984) 15-57.

Wolfram,

Cellular

Systolic Arrays Sparse Matrix Confer-

Universality

Automata,

Physica

and

Complexity

10 D (1984)

l-35.

in