On iteration schemes K0(n) and K1(n)

著者 "TOGASHI Akira, FUJINO Seiiti"journal orpublication title

鹿児島大学理学部紀要. 数学・物理学・化学

volume 23page range 111-124別言語のタイトル "K0(n), K1(n)の繰り返し模型"URL http://hdl.handle.net/10232/6476

On iteration schemes K0(n) and K1(n)

著者 TOGASHI Akira, FUJINO Seiitijournal orpublication title

鹿児島大学理学部紀要. 数学・物理学・化学

volume 23page range 111-124別言語のタイトル K0(n), K1(n)の繰り返し模型URL http://hdl.handle.net/10232/00007028

Rep. Fac. Sci., Kagoshima Univ., (Math Phys. & Chem.)

No. 23, p. Ill-124, 1990.

On iteration schemes Ko(n) and Kx(n)

by

Akira TOGASHI* and Seiichi FujlNO**

(Received September 7, 1990)

Abstract.

In this paper we shall analyze behaviors of finite cellular automata CA-90 (n) with

boundary conditions of 0-type and l-type, respectively, by using iteration schemes ^oW andK¥¥n), introduced in [2].

1. Introduction

In [1] Fujino has developed a theory of finite cellular automata CA-90 (n) with

boundary conditions of 0-type and l-type, respectively. In this paper we shall analyze

behaviors of the automata CA-90 (n) of boundary conditions of both types, by using

techiques developed in the paper [2].

Finite cellular automaton CA-90 (n) with boundary condition of 0-type is inter-

preted as a system Ko¥n) -<Z,/>, where Xis a set of all n-tuples with elements in Z2

-{0, l}, that is X-Z2, and/is a mapping from Xinto Xdefined by

f¥x)-Ax mod2　　:X),

where A is a matrix

A-

01

101　0101

●　●　●　　　　●　●　●

●　●　●　　　　●　●　●

0 1冒去

and additions and multiplications for computing Ax are mod 2-operations, respectively.

And the same automata with boundary condition ofトtype is represented as a system

K¥(n)-<X,g>, where Xis the set Z2, and g is a mapping from Xinto Xdefined by

Department of Mathematics, Faculty of Science, Kagoshima University, Kagoshima, Japan

Department of Mathemtaics, Faculty of Science, Kyushu University, Fukuoka, Japan

112 Akira Togashi and Seiichi Fujino

g(x)-Ax+d mod2　　UGZ)● ●

where A is the same matrix used in defining the mapping f, and d is a vector

r-i O O O i-I

mod 2-operations are also used in the definition of g.

We get immediately the following relation,

g(x)-f(x)+d mod2　(x∈X) 1.1

In the following we shall show existence of fixed points of f and g, and behaviors of iter-●

ated sequences {fp(x)} and {g?(x)} (p-0, 1, 2,�"�"�"), respectively.

2. Properties of schemes

We get the following propositions about schemes Kqin) and K¥ in).

Proposition 2.1. The mappingf: X-+X i∫ additive mod 2, that is,

f{x-¥-y mod2)-f(x)+/(jv)　mod2

for each x andy in X.

Proof It follows quickly from the definition of the mapping f.

Proposition 2.2　The mapping/: X-+X i∫ bijective if and only if the number n i∫ even.

Proof First we shall show the following result: The determinant detA mod 2, com-

puted by mod2 operations, is not zero if and only if the number n is even. Let us write

the determinant detA mod 2 by detnA mod 2 in order to clearly express the number n.

So we have

detnA mod 2-

01ioi0

101

●●●●

●●●●

●●●●

0101

10

On iteration schemes K｡(n) and K: (n)

-　detn-2A mod2

By using the above relation successively we get

detnA mod2 - detn-2A mod2

-　detn-4A mod2

detzA mod2 (ifniseven)

det&A mod2 (ifnisodd)

detnA mod2-

0　1

1　0

0 1 0

1 0 1

0 1 0

-1 (ifniseven)

-0 (ifnisodd)

So we have

113

and the above result holds.

By the definition of/ the mapping /: X-+X is bijective if and only if detA mod 2 is not

zero. The conclusion follows from combining the above two results.　　　　　　　□

Now we introduce a mapping, called 'code'from X into N (the set of all natural

numbers including zero) by

114 Akira Togashi and Seiichi Fujino

code(x) - ∑xj2n-J for each x-ノ=1

〟

れ　｣>....〟

Let us denote an arrow

code(x) -- code{y)

when fix)-y for x and y in X.

We get a diagram over code(x) by writing all arrows

code(x) -- code(y) for each x andy in Xwhen/U) -y holds.

*o3):1＼ノ＼ノ＼/

5

1oO

Type of scheme : {l2[2[2]] CO)

♂ll

†14

s7↓1

3

一三､ :こ9くユニ15

＼2/Typeofscheme : {6(2), 3(1), 1(1)}

l

J

ノ2＼29　　　20

-ォ1

†16

*o(5) :

ー

　

▼

1

J1H

U

8↓1 5

30

†12

On iteration schemes K｡(n) and Ki (n)

5ー23

3

1

_jー9→22　　　13ー8

＼ノ†24

野

4→10　　　　ー31ii璽寧l璽ィ

115

I.→27Q　21→.?

Type of scheme : {42(3), 22(1), 12(2)}

116

gi(3 :

5ノ1＼1*i(4):千 J

lU8｣0

Akira Togashi and Seiichi Fujino

il^^B^^^^E^^Hi

＼　/O

J

5〇Type of scheme : ¥l2[2[2]] (1)}

壬ォ*-^Aミ

†J

72

¥/

12

Typeofscheme:{6(2),3(1),1(1)}

人｡915　　　0

The diagram, thus constructed, is called a configuration diagram of scheme Kq in).

The configuration diagram of scheme K¥ ¥n) is constructed by using the mapping g in

place of the mapping f.

We shall show examples of configuration diagrams of Ko(n) and K¥ (n) for n-3, 4

and5.

1

119

ノ＼

＼2/

†11

On iteration schemes K｡(n) and Ki (n)

4

J

/2〔23→20　　　30←6　　　21→17　　10ー14

＼｡/↑3

1

2 9↓5

■

　

　

●

pnu

5(E<-

一　-

HU｢5日

山川に

＼ゝ　//

ffj鱈

00｣

｢

1　-

[山門

一一一一､d

9　　　　　　28　　18F J

117

229 24 13Q

Typeofscheme : {42(3),22 1), 12(2 }

In the above diagrams we show type of schemes by using notations defined in [2].

Proposition 2.3 Ifn i∫ odd, then there exi∫ts a nonzero vectory in X ∫uch that

f(x+y mod 2) -/(*)

and

g(x-¥-y mod 2) -g(x)

for every x in X.

118 Akira Togashi and Seiichi Fujino

Proof Let us take a vector

i-1　O i-I O t-I O rH

Then we have/(j>) -Ay mod 2-0, So we get, by additivity of the mapping f,

f(x-¥-y mod 2) -f(x) +/(j>) mod 2

-f(x),

and, by the formula (1.1)

g(x+y mod2) -f(x-¥-y mod2) +d mod2,

-f(x)+J¥v)+d mod2,

-/(*) +d mod 2,

-g(x)　　　　　mod 2,

The conclusion holds.

Next we shall study fixed points of the mapping f and g.■

Proposition 2.4　Fixedpoint∫ of the mappingf, that i∫ the vector x in X satisfying the rela-

tionf(x) -x, are as follows:

(ifn…0(tnod 3)) ,

(ifn…1(mod3)),

0 and

]　　]　　　　　　　　　　　]

r-I r-1　O i-I 1-I O

(ifn…2(mod 3)

On iteration schemes K｡(n) and Kx (n)

Proof The relationf(x) -x is written as follows:

*2-*1,

x¥+xs mod 2-^2,

x2+x4 mod 2-x%,

x卜i+#j+i mod 2-xi,

xn-2+xn mod 2-xn-i,

Xn-1　%n

(2.1)

119

By the equation (2.1), the zero vector is a fixed point of the mapping f for every n,

and the fixed point the first element of which is 1 is a vector in which the second ele-

ment is 1, the third element is 0, and the last two elements are equal.

Such a fixed point appears only if n-2(mod3), and the form of the fixed point is

]

]

]

r-H O i-I 1-I O i-I i-I O rH i-I

120 Akira Togashi and Seiichi Fujino

Proposition 2.5　Fixedpoints of the mapping g are a∫follow∫:

(i)

]　　　]　　　　　　　　　　　　　]

T-I O T 1 T-H O T-H

]　　　]　　　　　　　　　　　　]

O i-I i-I O r-I 1-I　　　　　　　0-1-I t-I O

]　　　]　　　　　　　　　　　　]

t-H O t-I i-I O H t-H O r-I rH O

lfn…2{mod 3)

lfn…0(mod 3),

lfn…1{mod3),

O　*-I i-I O r-I t-I

O　*-I r-¥　O t-I

On iteration schemes K｡(n) and Ki (n)

Proof The relation g(x) -x is written as follows:

1+^2 mod2-#i,

#i+#3 mod 2-#2,

#2+#4 mod 2-#3,

xj-i+xi+i mod 2-xi,

>サー2-r*nmod2-xn-i,,-2+xnxn-i+lmod2-xn

From the equation (2.2), we get

XI -X9,

where x2 is a mod 2-complement of #2, that is

(if　*2-0),

(if　*2-1).

Xn-2=

Finally we get

and

Xn-1 Xm

(if *サーi-0, that is, xn-l)

(if　*n-i-l, that is, xn-0)

2.2

121

Combining these results, it follows that●

(i) Fixed point the first and the last three elements of which are both 1, 0 and 1

appears only if n-0¥mod 3).′

(ii) Fixed point the first and the lastthree elements of which are 0, 1 and 1, and 1,

1 and 0, respectively, appears only if n-l (mod 3).

(iii) Fixed point the first and the last three elements of which are 1, 0 and 1, and 1,

1 and 0, or, 0, 1 and 1 and 1, 0 and 1, respectively, appears only ifrc-2 (mod3).

So we get the conclusion.

3. Characterization number h(n)

The mappings f and g have the following property.●

Proposition 3. 1

/(*) -fpGO +/(0) foreach x∈X(p-l, 2,-)

122 Akira Togashi and Seiichi Fujino

Proof (by induction)

Whenp-l,g(x)-/(*)+d mod2 (by definition)

-/(*) +*(o) mod2 (XGX)

Assume that the relation holds for β-1, that is,

サーI oc) -fp-1k) +f-1 (o)

Then, for each x in X we have

g/>(x) -g(/-1 (x) )

-/(/-'(x)) +d

(x∈x)

mod 2

-fifp~1(*)+/ (o))+rf　サmw/2

By additivity of the mappingJ;

n h(n)-　-■　-

^CM CO^UO O N flo O)O H CVJ CO^lO^O N OO Oi O H Ca CQ^f u5<D^

H H H H H H H H H H CM CM CM cM CM CM CM eM

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C

O

C

D

H

C

O

C

M5

cm to^r CM cM CM

r-4　　0　　0　^o

^V^^KX^^K^

H CM X CO^f O X^f o rH X CQ^f O X IO OO H tt<O CM t"~K CO O CM　><

h(n)-　　　　　　　　　　　　　　　　　　■m　蝣サ　�"　ォ�"

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On iteration schemes K｡(n) and Ki (n)

-fpW +/V~ (0).) +d mod2-/>(*) +*(♂-1 (o))　　mod 2

-/>(*) +/(0)The induction holds.

So we immediately get the following theorem.

Theorem 3.2. fp-g if and only iff(0) -0.

123

Let us denote the least positive number p such that ｣(0) -0, if exists, by h(n).

We call the number h(n), if exists, the characterization number of schemes Ko(n) and K¥

¥n). By using computer we get the following table of hin) (see p. 122.).

The table shows the following relation

(1) The number h(n) exists if and only ifn≠3{mod4).

(2) Ifniseven,thenwehave

h{2n+l) -2h(n)

the proofs of which are obtained in [3], using slightly changed form of CA-90(n).

4. Isomorphism between schemes Ko(n) and Ki(n).

The schemes Kqin) and K¥ (n) have the following relation.

Theorem 4.1 For two scheme∫ -KO(n) -<X,/> and K¥(n) -<Z, g>, fA*re exw^ a

bijection ≠ from X onto X ∫uch that the following diagram commute∫ and, ifp i∫ afixed point of

the mapping/, then ≠ (p) i∫ afixedpoint of the mapping g

f

Proof Let us take a mapping ≠ as follows:

≠(x)-x+qmod2　{x∈X)

where q is a fixed point of the mapping g, which exists for each n (proposition 2.5).

So wegetfor eachxinX

(≠　f) (x)-f(x)+q mod2,

-jt*) +gW mod 2,

-f(x) +f(q) +d mod 2,かりかり

加加)3n

u

ヴ　O.X

-/(* +′か′針′k

T二　二　二

十〟

-ゥ,

+d mod 2,

mod 2,

124 Akira Togashi and Seiichi Futino

And, ifp is a fixed point of the mapping/ that isf¥p) -p, then we have

g( ≠ (p)) - ≠ (f(p))　(the commutavity),

-≠(p)

So the vector ≠ ¥p) is a fixed point of the mappingg.

We can get similar results for schemes representing finite cellular automata

CA-150 (n) with both boundary conditions of 0-type and l-type, respectively.

References

[ 1 ] S. FujlNO: On the behavior oflinearfinite cellular automata CA-90(m) , RMC 63-03/ (1988), pp. 25

[ 2 ] T. Kitagawa and S. Fujino‥ On modular type iteration ∫cheme, RMC 64-09/ (1989), pp. 171

[ 3 ] Y. KAWAHARA: On the existence of Fujino's characterization number H (m) a∫∫ociated with finite cellular automata

CA-90(m-1), RMC 63-08J (1988), pp. 17