On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira,...
Transcript of On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira,...
![Page 1: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/1.jpg)
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
![Page 2: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/2.jpg)
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
![Page 3: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/3.jpg)
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
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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
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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
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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)}
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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)}
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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.
![Page 9: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/9.jpg)
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.
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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)
![Page 11: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/11.jpg)
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
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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
![Page 13: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/13.jpg)
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,-)
![Page 14: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/14.jpg)
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)- -■ -
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![Page 15: On iteration schemes K0(n) and K1(n)On iteration schemes K0(n) and K1(n) 著者 TOGASHI Akira, FUJINO Seiiti journal or publication title 鹿児島大学理学部紀要. 数学・物理学・化学](https://reader036.fdocuments.net/reader036/viewer/2022070803/5f0310917e708231d4075bc8/html5/thumbnails/15.jpg)
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,
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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