15 Pages
English

Reversible Cellular Automaton Able to Simulate

-

Gain access to the library to view online
Learn more

Description

Reversible Cellular Automaton Able to Simulate Any Other Reversible One Using Partitioning Automata Jérôme Olivier Durand-Lose ? Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile. e-mail: LATIN '95, LNCS 911, pp. 230244 Abstract. Partitioning automata (PA) are dened. They are equivalent to cellular automata (CA). Reversible sub-classes are also equivalent. A simple, reversible and universal partitioning automaton is described. Finally, it is shown that there are reversible PA and CA that are able to simulate any reversible PA or CA on any conguration. 1 Introduction The main interest of reversibility in computation is backtracking a phenomenon to its source and in relation with physics, isoentropic phenomena modelization and saving energy, and had have various interests in relation to physics as ex- plained by Tooli and Margolus in [15]. It is well known that, given any d- dimensional cellular automata (CA), it can be simulated by one (d+1)-dimensio- nal CA which is reversible [12]. It is still an open problem if it can be simulated by a reversible CA of the same dimension. For example, Morita showed in [7] that this is true in dimension one but only over nite congurations.

  • both nite

  • simulate any reversible

  • can simulate

  • called nodes

  • partition

  • reversible pa

  • transition function


Subjects

Informations

Published by
Reads 12
Language English

Exrait

b
Rev
the
ersible
the
Cellular
des
Automaton
out
Able
imensional
to
no
Sim
CA
ulate
v
An
b
y
exists
Other
P
Rev
ysical
ersible
a
One
lo
Using
is
P
is
artitioning
pap
Automata
sim
Je
and
Olivier
d
Durandose
CA
?
v
Departamen
duced
to
of
de
they
Ingenier
the
Matemica
is
F
a
acultad
from
de
mapp
Ciencias
This
Ficas
Co
y
one
Matemicas
In
Univ
and
ersidad
A
de
automata
Chile
of
San
A
tiago
deduced
Chile

eail
ersible
jdurandl
ulate
aim
A
a
ite
im
ere
c
and
hi
80's
le
other
cl
Lik
LA
an
TIN
p
'95,
a
LNCS
states
911,
v
pp
the
230244
deed
Abstract
elementary
P
set
artitioning
orho
automata
a
(P
diren
A
as
are
the
deed
Chile
They
in
are
only
equiv
ite
alen
presen
t
some
to
results
cellular
automata
automata
m
A
with
Rev
sho
ersible
an
sublasses
rev
are
ersal
also
detailed
equiv
result
alen
an
t
with
A
,
simple
imensional
rev
A
ersible
to
and
y
univ
ersible
ersal
CA
partitioning
b
automaton
inite
is
automata
describ
in
ed
y
Finally
oli
,
of
it
mo
is
gases
sho
ersible
wn
nomena
that
cellular
there
ork
are
lattice
rev
is
ersible
t
P
and
A
alue
and
set
CA
A
that
h
are
of
able
e
to
function
sim
A
ulate
y
an
rule
y
ansition
rev
to
ersible
tiles
P
neigh
A
d
or
in
CA
The
on
in
an
regular
y
h
conuration
supp
1
ECOS
In
renc
tro
eration
duction
this
The
true
main
dimension
in
but
terest
o
of
er
rev
conurations
ersibilit
the
y
t
in
er
computation
deitions
is
basic
bac
ab
ktrac
partitioning
king
(P
a
and
phenomenon
utual
to
ulations
its
cellular
source
are
and
wn
in
efore
relation
example
with
a
ph
ersible
ysics
univ
iso
P
en
is
tropic
Then
phenomena
main
mo
is
delization
for
and
y
sa
,
ving
2
energy
d
,
there
and
d
had
rev
ha
P
v
and
e
able
v
sim
arious
an
in
d
terests
rev
in
P
relation
or
to
o
ph
er
ysics
oth
as
and
ex
conurations
plained
artitioning
b
w
y
st
T
tro
oli
b
and
Margolus
Margolus
T
in
in
[15].
middle
It
the
is
as
w
dels
ell
lattice
kno
and
wn
rev
that
ph
giv
phe
en
[14].
an
e
y
automata
d
w
-
on
dimensional
inite
cellular
A
automata
de
A
a
it
oin
can
of
b
lattice
e
has
sim
v
ulated
in
b
ite
y
of
one
.
(
tile
d
a
dimensio

nal
rectangle
CA
no
whic
Lik
h
CA
is
global
rev
of
ersible
P
[12].
is
It
b
is
a
still
cal
an
called
op
tr
en
function
problem
and
if
the
it
of
can
or
b
a
e
b
sim
o
ulated
is
b
ed
y
to
a
cell
rev
plane
ersible
cut
CA
to
of
t
the
?
same
researc
dimension
w
F
partially
or
orted
example
y
Morita
and
sho
F
w
h
ed
op
in
in
[7]
thatIt
partitions
b
of
w
tiles
y
(a
The
partition
;
is
er
fully
,
determined
results
b
a
y
(
h
built
,
onservative
v
rev
and
wn
its
CA
origin
can
An
ork
elementary
a
tr
)
ansition
o
is
of
the
h
parallel
It
replacemen
oli
t
ersible
of
[6]
all
or
the
this
tiles
ulate
of
it
a
an
giv
univ
en
is
parti
2
tion
e
b
no
y
states
their
h
images
1.
b
and
y
Q
the
v
elemen
follo
tary
y
transition
1
function
ignals
The
sim
glob
F
al
logicunctionalit
tr
all
ansition
the
function
K
is
do
the
without
sequen
can
tial
logic
comp
constructions
osition
is
of
ersible
v
of
arious
there
elemen
able
tary
P
transitions
w
A
in
P
y
A
an
is
extended
rev
automata
ersible
a
i
Z
its
of
global
h
function
a
is
c
in
v
v
(
ertible
of
and
y
is
n
the
b
global
tile
function
,
of
,
some
(
P
2
A
of
It
(
is
)
equiv
;
alen
and
t
arc
to
and
bijectivit
able
y
an
of
olean
the
and
elemen
a
tary
called
transition
gic
function
are
whic
k
h
um
is
1
decidable
demonstrated
emma
it
8).
y
Cellular
com
automata
t
A
P
are
ulate
the
of
most
results
famous
and
mo
is
del
P
of
to
parallel
y
phenomena
A
and
the
arc
oth
hitectures
concluded
They
rev
ha
P
v
sim
e
rev
b
or
een
explained
widely
turn
studied
P
for
one
decades
ulate
and
one
there
rev
is
more
a
b
lot
higher
of
P
results
A
ab
v
out
inite
them
(
[16].
2
After
oin
a
are
brief
.
deition
de
of
alue
CA
set
b
.
oth
ation
sim
with
ulations
in
b
de
e
Q
t
the
w
Deiti
een
x
CA
e
and
nonero
P
b
A
and
and
t
b
tegers
et
;
w

een
co
rev

ersible
(
CA
and
and
y
rev
conuration
ersible
is
P
rectangle
A
:
are
c
constructed
+
Th

us
[0
as

far
1]
as
signals
computation
to
is
the
concerned
hitecture
the
routing
class
gates
of
is
P
to
A
ulate
esp
y
rev
o
ersible
circuit
P
redkin
A
T
is
studied
equiv
binary
alen
y
t
c
to
lo
the
where
one
functions
of
rev
CA
and
esp
eep
rev
n
ersible
b
CA
of
and
[13,2].
the
Morita
class
in
of
that
P
can
A
an
is
ite
com
ersible
putational
puting
univ
constan
ersal
inputs
ble
drop
to
u
sim
sim
ulate
an
an
circuit
y
this
T
Using
uring
of
mac
logic
hine
some
The
it
fact
sho
that
that
CA
u
and
able
P
sim
A
an
can
rev
sim
P
ulate
Finally
eac
gathering
h
result
other
b
w
parts
as
is
already
that
men
exist
tioned
ersible
b
and
y
A
T
to
oli
ulate
and
y
Margolus
ersible
in
A
[14].
CA
Here
is
full
ho
constructiv
to
e
this
demonstrations
ersal
that
A
care
to
ab
that
out
sim
conserv
an
ation
other
of
but
rev
not
ersibilit
ersible
y
y
are
Those
giv
can
en
e
In
to
the
dimensions
second
Deitions
part
artitioning
an
(P
example
w
of
o
a
er
simple
dimensional
and
lattic
rev
L
ersible
=
P
Z
A
).
is
p
giv
ts
en
L
Mar
called
golus
des
and
Eac
T
no
oli
has
describ
v
ed
in
a
ite
more
of
simple
Q
one
A
ith
onur
only
is
t
lattice
w
a
o
alue
states
eac
and
no
t
C
w
=
o
L
tiling
is
and
set
sho
conurations
w
on
ed
Let
b
,
oth
b
rev
t
ersibilit
o
y
natural
and
um
univ
ers
ersalit

y

in
e
[14,4].
w
The
in
one
The
presen
t
ted
2
here
h
has
v
the
of
in
ordinates
terest
;
of
)
not
size
ha
h
ving
)
virtual
origin
0
x
signals
)
and
a
someho
c
w
C
needs
the
less
wing
space
part
This
c
P
t
A
=
noted
j
P
x
u
h
,
+
has
:v
four
+
states
;
for
1]
the
[0
ether
v
0
:
and;
Deiti
t
on
partition
2.
x
The
size
p
y
artition
1
of
)
size
v
(
.
h
y
v
t
)
(
and
2
origin
f
(
in
x
Q
y
A
)
t
is
ansition
the
all
partition
tary
of
(
L
t
with
e
the
t
tiles
;
of
x
size
partitioning
(
n
h
e
v
to
)
n
and
!
origin
a
(
v
x
)
y
A
)
An
.
for
It
parallel
is
y
rectangular
Fig
and
is
regular
x
as
@
sho
t
wn
t
in
t
Fig
;
1.
0
b
)
b
e
b
(
b
!
b
transition
b
Deiti
b
b
b
h
b
i
b
i
b
eral
b
information
b
are
b
1
b
e
b

b
h
b
F
b
en
b
partitions
b
the
b
h
b
a
b
the
b
its
b
on
b
T
b
T
b
giv
b
sim
b
t
b
tiles
b
images
b
as
b
An
b
T
b
determined
b
origin
b
)
b
y
b
@
b
;
b
;
b
;
b
;
b
;
b
;
b
t
b
)
b
1
b
e
b
;
b
(
b
1
b
t
b
)
b
1
b
T
b
2.
b
elemen
b
origin
b
)
b
5.
b
is
b
P
b
;
b
)
b
(
b
y
b
1
b
n
b
:
b
are
b
to
b
from
b
Their
b
x
b
i
b
i
b
(
b
:
b
h
b
v
b
Q
b

b
).
b
or
b
giv
b
P
b
all
b
ha
b
e
b
same
b
(
b
v
b
is
b
constan
b
of
b
P
b
called
b
size
b
Deiti
b
4.
b
Elementary
b
r
b
(e.t.)
b
is
b
a
b
en
b
the
b
ultaneous
b
replacemen
b
of
b
its
b
b
b
their
b
b
b
e
b
in
b
2.
b
elemen
b
transition
b
x
b
fully
b
b
b
its
b
(
b
y
b
.
b
x
b
)
b
@
b
R
b
0
b
0
b
1
b
0
b
0
b
1
b
1
-
1
h
0

2
-
1
h
2

(
6
0
v
0
?
e
6
t
v
;
?
)
@
(
@
0
R
1
(
e
x
t
y
;
)
)
=
(
t
0
0
2
;
e
0
t
t
;
1
)
;
x
0
Fig
t
T
0
:
;
tary
1
of
t
(
1
y
;
.
1
on
Fig
A
1.
automaton
(6
deed
;
y
4)
=
partition
Q
of
(
origin
v
(
;
x
(
y
x
)
;
.
i
Deiti
)
on

3.

The
;
Elementary
g
T
Sev
r
partitions
ansition
used
F
order
unction
let
(e.t.f.)
spread
e
tile
is
tile
a
origins
function
(
from
i
and
y
to
)
the

set

of
.
tiles
Deiti
b
on
called
6.
t
The
e
up
is
dating
pro
function
:
of
and
a
the
P
of
A
Deiti
the
1
glob
ersibilit
al
g
tr
inexp
ansition
instead
G

,
.
maps
of
conurations
states
in
:
to
neigh
conurations
)
It
b
is
(
the
that
sequen
A
tial
F
comp
exists
osition
!
of
,
all
natural
the
The
parallel
z
elemen
is
tary
other
transitions
global
asso
automata
ciated
).
with
their
eac
d
h
x
of
x
the
h
partitions
function
G
y
:
set
C
,
!
=
C
2
=
ab
T
as
x
surjectivit
n
alen
;y
F
n
f

G
T
real
x
enco
n
and
1
e
;y
to
n
f
1


can
:
iterating
:
str
:
v

;
T

x
and
1
=
;y
also
1
ulates
:
the
Deiti
to
on
on
7.
=
A
ts
P
and
A
a
P
neigh
is
y
r
ordinates
eversible
2
i
g
its
).
global
v
function
the
G
a
is
n
in
CA
v
Q
ertible
Q
and
ations
G
The
1
!
is
c
the
(
global
(
function
:
of
n
a
v
P
it
A
sho
the
y
inverse
and
P
are
A
for
Rev
on
ersible
an
P
o
A
F
are
g
noted
G
R
f
A
i
Lemma
w
8.
functions
A
!
P
:
A
,
is
and
r
e
eversible
and
i
h
its
=
elementary
n
tr
all
ansition
.
function
e
is
f
invertible
global
Pr
ulation
o
i
of
are
By
and
deition
Z
if
z
the
g
P
.
A
i
P
are
is

rev
or
ersible
ords
G
ulates
is
automaton
in
i
v
sim
ertible
function
If
3
G
Automata
is
w
in
same
v
L
ertible
Z
then
p
since
L
G
el
=
tak
T
alues
x
set
n
.
;y
orho
n
deed

ite
T
e
x
=
n
;
1
:
;y
x
n
8
1
2

an
:
hanging
:
of
:
according

of
T
ors
x
c
1
:
;y
Q
1
10.
,
deed
T
=
x
;
1
C
;y
is
1
c
m
Let
ust
a
b
al
e
:
injectiv
is
e
F
Because
x
of
(
the
+
construction
;
of
+
T
;
,
;
the
+
elemen
)
tary
o
transition
e
function
of
e
w
m
also
ust
wn
b
injectivit
e
,
injectiv
y
e
rev
and
y
as
equiv
it
t
w
P
orks
Deiti
o
9.
v
or
er
y
a
w
ite
functions
set
:
it
!
is
and
bijectiv
:
e
!
Con
.
v
simulates
ersely
n
,
time
if
there
e
t
is
o
bijectiv
ding
e

let
F
T
G
0