<s>
A	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
is	O
a	O
cellular	O
automaton	O
in	O
which	O
every	O
configuration	O
has	O
a	O
unique	O
predecessor	O
.	O
</s>
<s>
That	O
is	O
,	O
it	O
is	O
a	O
regular	O
grid	O
of	O
cells	O
,	O
each	O
containing	O
a	O
state	O
drawn	O
from	O
a	O
finite	O
set	O
of	O
states	B-Application
,	O
with	O
a	O
rule	O
for	O
updating	O
all	O
cells	O
simultaneously	O
based	O
on	O
the	O
states	B-Application
of	O
their	O
neighbors	O
,	O
such	O
that	O
the	O
previous	O
state	O
of	O
any	O
cell	O
before	O
an	O
update	O
can	O
be	O
determined	O
uniquely	O
from	O
the	O
updated	O
states	B-Application
of	O
all	O
the	O
cells	O
.	O
</s>
<s>
The	O
time-reversed	O
dynamics	O
of	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
can	O
always	O
be	O
described	O
by	O
another	O
cellular	O
automaton	O
rule	O
,	O
possibly	O
on	O
a	O
much	O
larger	O
neighborhood	O
.	O
</s>
<s>
Several	O
methods	O
are	O
known	O
for	O
defining	O
cellular	O
automata	O
rules	O
that	O
are	O
reversible	O
;	O
these	O
include	O
the	O
block	O
cellular	O
automaton	O
method	O
,	O
in	O
which	O
each	O
update	O
partitions	O
the	O
cells	O
into	O
blocks	O
and	O
applies	O
an	O
invertible	O
function	O
separately	O
to	O
each	O
block	O
,	O
and	O
the	O
second-order	O
cellular	O
automaton	O
method	O
,	O
in	O
which	O
the	O
update	O
rule	O
combines	O
states	B-Application
from	O
two	O
previous	O
steps	O
of	O
the	O
automaton	O
.	O
</s>
<s>
Reversible	O
cellular	O
automata	O
form	O
a	O
natural	O
model	O
of	O
reversible	B-Application
computing	I-Application
,	O
a	O
technology	O
that	O
could	O
lead	O
to	O
ultra-low-power	O
computing	O
devices	O
.	O
</s>
<s>
A	O
cellular	O
automaton	O
is	O
defined	O
by	O
its	O
cells	O
(	O
often	O
a	O
one	O
-	O
or	O
two-dimensional	O
array	O
)	O
,	O
a	O
finite	O
set	O
of	O
values	O
or	O
states	B-Application
that	O
can	O
go	O
into	O
each	O
cell	O
,	O
a	O
neighborhood	O
associating	O
each	O
cell	O
with	O
a	O
finite	O
set	O
of	O
nearby	O
cells	O
,	O
and	O
an	O
update	O
rule	O
according	O
to	O
which	O
the	O
values	O
of	O
all	O
cells	O
are	O
updated	O
,	O
simultaneously	O
,	O
as	O
a	O
function	O
of	O
the	O
values	O
of	O
their	O
neighboring	O
cells	O
.	O
</s>
<s>
If	O
the	O
update	O
rule	O
for	O
such	O
an	O
automaton	O
causes	O
each	O
cell	O
to	O
always	O
remain	O
in	O
the	O
same	O
state	O
,	O
then	O
the	O
automaton	O
is	O
reversible	O
:	O
the	O
previous	O
state	O
of	O
all	O
cells	O
can	O
be	O
recovered	O
from	O
their	O
current	O
states	B-Application
,	O
because	O
for	O
each	O
cell	O
the	O
previous	O
and	O
current	O
states	B-Application
are	O
the	O
same	O
.	O
</s>
<s>
Despite	O
its	O
simplicity	O
,	O
the	O
update	O
rule	O
that	O
causes	O
each	O
cell	O
to	O
copy	O
the	O
state	O
of	O
a	O
neighboring	O
cell	O
is	O
important	O
in	O
the	O
theory	O
of	O
symbolic	O
dynamics	O
,	O
where	O
it	O
is	O
known	O
as	O
the	O
shift	B-Algorithm
map	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
if	O
the	O
left	O
neighbor	O
's	O
state	O
is	O
and	O
the	O
right	O
neighbor	O
's	O
state	O
is	O
,	O
the	O
new	O
state	O
of	O
a	O
cell	O
is	O
the	O
result	O
of	O
combining	O
these	O
states	B-Application
using	O
a	O
pairwise	O
operation	O
defined	O
by	O
the	O
equation	O
.	O
</s>
<s>
The	O
operation	O
used	O
to	O
combine	O
pairs	O
of	O
states	B-Application
in	O
this	O
automaton	O
forms	O
an	O
algebraic	O
structure	O
known	O
as	O
a	O
rectangular	O
band	O
.	O
</s>
<s>
Multiplication	O
of	O
decimal	O
numbers	O
by	O
two	O
or	O
by	O
five	O
can	O
be	O
performed	O
by	O
a	O
one-dimensional	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
with	O
ten	O
states	B-Application
per	O
cell	O
(	O
the	O
ten	O
decimal	O
digits	O
)	O
.	O
</s>
<s>
In	O
Rule	O
90	O
,	O
the	O
state	O
of	O
each	O
cell	O
is	O
the	O
exclusive	O
or	O
of	O
the	O
previous	O
states	B-Application
of	O
its	O
two	O
neighbors	O
.	O
</s>
<s>
This	O
use	O
of	O
the	O
exclusive	O
or	O
makes	O
the	O
transition	O
rule	O
locally	O
invertible	O
,	O
in	O
the	O
sense	O
that	O
any	O
contiguous	O
subsequence	O
of	O
states	B-Application
can	O
be	O
generated	O
by	O
this	O
rule	O
.	O
</s>
<s>
Rule	O
90	O
is	O
not	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
rule	O
,	O
because	O
in	O
Rule	O
90	O
every	O
assignment	O
of	O
states	B-Application
to	O
the	O
complete	O
array	O
of	O
cells	O
has	O
exactly	O
four	O
possible	O
predecessors	O
,	O
whereas	O
reversible	O
rules	O
are	O
required	O
to	O
have	O
exactly	O
one	O
predecessor	O
per	O
configuration	O
.	O
</s>
<s>
Because	O
this	O
function	O
is	O
invertible	O
,	O
the	O
automaton	O
defined	O
by	O
these	O
rules	O
is	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
.	O
</s>
<s>
In	O
this	O
way	O
,	O
Toffoli	O
showed	O
that	O
many	O
features	O
of	O
irreversible	O
cellular	O
automata	O
,	O
such	O
as	O
the	O
ability	O
to	O
simulate	O
arbitrary	O
Turing	B-Architecture
machines	I-Architecture
,	O
could	O
also	O
be	O
extended	O
to	O
reversible	O
cellular	O
automata	O
.	O
</s>
<s>
As	O
Toffoli	O
conjectured	O
and	O
proved	O
,	O
the	O
increase	O
in	O
dimension	O
incurred	O
by	O
Toffoli	O
's	O
method	O
is	O
a	O
necessary	O
payment	O
for	O
its	O
generality	O
:	O
under	O
mild	O
assumptions	O
(	O
such	O
as	O
the	O
translation-invariance	O
of	O
the	O
embedding	O
)	O
,	O
any	O
embedding	O
of	O
a	O
cellular	O
automaton	O
that	O
has	O
a	O
Garden	O
of	O
Eden	O
into	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
must	O
increase	O
the	O
dimension	O
.	O
</s>
<s>
Using	O
this	O
method	O
it	O
is	O
possible	O
to	O
show	O
that	O
even	O
one-dimensional	O
reversible	O
cellular	O
automata	O
are	O
capable	O
of	O
universal	B-Architecture
computation	I-Architecture
.	O
</s>
<s>
The	O
second-order	O
cellular	O
automaton	O
technique	O
is	O
a	O
method	O
of	O
transforming	O
any	O
cellular	O
automaton	O
into	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
,	O
invented	O
by	O
Edward	O
Fredkin	O
and	O
first	O
published	O
by	O
several	O
other	O
authors	O
in	O
1984	O
.	O
</s>
<s>
Specifically	O
,	O
the	O
transition	O
function	O
of	O
the	O
automaton	O
maps	O
each	O
neighborhood	O
at	O
time	O
to	O
a	O
permutation	B-Algorithm
on	O
the	O
set	O
of	O
states	B-Application
,	O
and	O
then	O
applies	O
that	O
permutation	B-Algorithm
to	O
the	O
state	O
at	O
time	O
.	O
</s>
<s>
The	O
reverse	O
dynamics	O
of	O
the	O
automaton	O
may	O
be	O
computed	O
by	O
mapping	O
each	O
neighborhood	O
to	O
the	O
inverse	O
permutation	B-Algorithm
and	O
proceeding	O
in	O
the	O
same	O
way	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
automata	O
with	O
binary-valued	O
states	B-Application
(	O
zero	O
or	O
one	O
)	O
,	O
there	O
are	O
only	O
two	O
possible	O
permutations	B-Algorithm
on	O
the	O
states	B-Application
(	O
the	O
identity	O
permutation	B-Algorithm
and	O
the	O
permutation	B-Algorithm
that	O
swaps	O
the	O
two	O
states	B-Application
)	O
,	O
which	O
may	O
themselves	O
be	O
represented	O
as	O
the	O
exclusive	O
or	O
of	O
a	O
state	O
with	O
a	O
binary	O
value	O
.	O
</s>
<s>
In	O
this	O
way	O
,	O
any	O
conventional	O
two-valued	O
cellular	O
automaton	O
may	O
be	O
converted	O
to	O
a	O
second-order	O
cellular	O
automaton	O
rule	O
by	O
using	O
the	O
conventional	O
automaton	O
's	O
transition	O
function	O
on	O
the	O
states	B-Application
at	O
time	O
,	O
and	O
then	O
computing	O
the	O
exclusive	O
or	O
of	O
these	O
states	B-Application
with	O
the	O
states	B-Application
at	O
time	O
to	O
determine	O
the	O
states	B-Application
at	O
time	O
.	O
</s>
<s>
However	O
,	O
the	O
behavior	O
of	O
the	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
determined	O
in	O
this	O
way	O
may	O
not	O
bear	O
any	O
resemblance	O
to	O
the	O
behavior	O
of	O
the	O
cellular	O
automaton	O
from	O
which	O
it	O
was	O
defined	O
.	O
</s>
<s>
Any	O
second-order	O
automaton	O
may	O
be	O
transformed	O
into	O
a	O
conventional	O
cellular	O
automaton	O
,	O
in	O
which	O
the	O
transition	O
function	O
depends	O
only	O
on	O
the	O
single	O
previous	O
time	O
step	O
,	O
by	O
combining	O
pairs	O
of	O
states	B-Application
from	O
consecutive	O
time	O
steps	O
of	O
the	O
second-order	O
automaton	O
into	O
single	O
states	B-Application
of	O
a	O
conventional	O
cellular	O
automaton	O
.	O
</s>
<s>
Patt	O
performed	O
a	O
brute	B-Algorithm
force	I-Algorithm
search	I-Algorithm
of	O
all	O
two-state	O
one-dimensional	O
cellular	O
automata	O
with	O
small	O
neighborhoods	O
;	O
this	O
search	O
led	O
to	O
the	O
discovery	O
of	O
this	O
automaton	O
,	O
and	O
showed	O
that	O
it	O
was	O
the	O
simplest	O
possible	O
nontrivial	O
one-dimensional	O
two-state	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
.	O
</s>
<s>
In	O
this	O
technique	O
,	O
a	O
change	O
to	O
the	O
state	O
of	O
a	O
cell	O
is	O
triggered	O
by	O
a	O
pattern	O
among	O
a	O
set	O
of	O
neighbors	O
that	O
do	O
not	O
themselves	O
change	O
states	B-Application
.	O
</s>
<s>
Rules	O
using	O
this	O
model	O
can	O
simulate	O
the	O
billiard	B-Application
ball	I-Application
computer	I-Application
,	O
</s>
<s>
A	O
cellular	O
automaton	O
consists	O
of	O
an	O
array	O
of	O
cells	O
,	O
each	O
one	O
of	O
which	O
has	O
a	O
finite	O
number	O
of	O
possible	O
states	B-Application
,	O
together	O
with	O
a	O
rule	O
for	O
updating	O
all	O
cells	O
simultaneously	O
based	O
only	O
on	O
the	O
states	B-Application
of	O
neighboring	O
cells	O
.	O
</s>
<s>
The	O
update	O
rule	O
of	O
the	O
automaton	O
is	O
a	O
bijection	B-Algorithm
;	O
that	O
is	O
,	O
a	O
function	O
that	O
is	O
both	O
one-to-one	B-Algorithm
and	I-Algorithm
onto	I-Algorithm
.	O
</s>
<s>
This	O
condition	O
is	O
obviously	O
implied	O
by	O
the	O
assumption	O
that	O
the	O
update	O
rule	O
is	O
a	O
bijection	B-Algorithm
.	O
</s>
<s>
In	O
the	O
other	O
direction	O
,	O
the	O
Garden	O
of	O
Eden	O
theorem	O
for	O
cellular	O
automata	O
implies	O
that	O
every	O
injective	O
update	O
rule	O
is	O
bijective	B-Algorithm
.	O
</s>
<s>
Clearly	O
,	O
for	O
this	O
to	O
be	O
possible	O
,	O
the	O
update	O
rule	O
must	O
be	O
bijective	B-Algorithm
.	O
</s>
<s>
In	O
the	O
other	O
direction	O
,	O
if	O
the	O
update	O
rule	O
is	O
bijective	B-Algorithm
,	O
then	O
it	O
has	O
an	O
inverse	O
function	O
that	O
is	O
also	O
bijective	B-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
it	O
is	O
a	O
homeomorphism	O
that	O
commutes	O
with	O
the	O
shift	B-Algorithm
map	I-Algorithm
,	O
as	O
the	O
Curtis	O
–	O
Hedlund	O
–	O
Lyndon	O
theorem	O
implies	O
.	O
</s>
<s>
Most	O
of	O
these	O
turn	O
out	O
to	O
be	O
equivalent	O
either	O
to	O
injectivity	O
or	O
to	O
surjectivity	B-Algorithm
of	O
the	O
transition	O
function	O
of	O
the	O
automaton	O
;	O
however	O
,	O
there	O
is	O
one	O
more	O
alternative	O
that	O
does	O
not	O
match	O
either	O
of	O
these	O
two	O
definitions	O
.	O
</s>
<s>
This	O
class	O
turns	O
out	O
to	O
be	O
distinct	O
from	O
both	O
the	O
surjective	B-Algorithm
and	O
injective	O
automata	O
,	O
and	O
in	O
some	O
subsequent	O
research	O
,	O
automata	O
with	O
this	O
property	O
have	O
been	O
called	O
invertible	O
finite	O
automata	O
.	O
</s>
<s>
Alternative	O
algorithms	O
based	O
on	O
automata	B-Application
theory	I-Application
and	O
de	O
Bruijn	O
graphs	O
were	O
given	O
by	O
and	O
,	O
respectively	O
.	O
</s>
<s>
He	O
defines	O
a	O
nondeterministic	O
finite-state	B-Architecture
transducer	I-Architecture
that	O
performs	O
the	O
transition	O
rule	O
of	O
the	O
automaton	O
on	O
periodic	O
strings	O
.	O
</s>
<s>
Sutner	O
defines	O
a	O
directed	O
graph	O
(	O
a	O
type	O
of	O
de	O
Bruijn	O
graph	O
)	O
in	O
which	O
each	O
vertex	O
represents	O
a	O
pair	O
of	O
assignments	O
of	O
states	B-Application
for	O
the	O
cells	O
in	O
a	O
contiguous	O
sequence	O
of	O
cells	O
.	O
</s>
<s>
Edges	O
are	O
only	O
included	O
in	O
the	O
graph	O
when	O
they	O
represent	O
compatible	O
state	O
assignments	O
on	O
the	O
overlapping	O
parts	O
of	O
their	O
cell	O
sequences	O
,	O
and	O
when	O
the	O
automaton	O
rule	O
(	O
applied	O
to	O
the	O
neighborhood	O
determined	O
by	O
the	O
potential	O
edge	O
)	O
would	O
give	O
the	O
same	O
results	O
for	O
both	O
assignments	O
of	O
states	B-Application
.	O
</s>
<s>
A	O
related	O
algorithm	O
of	O
determines	O
whether	O
a	O
given	O
rule	O
is	O
surjective	B-Algorithm
when	O
applied	O
to	O
finite-length	O
arrays	O
of	O
cells	O
with	O
periodic	O
boundary	O
conditions	O
,	O
and	O
if	O
so	O
,	O
for	O
which	O
lengths	O
.	O
</s>
<s>
His	O
construction	O
uses	O
the	O
von	O
Neumann	O
neighborhood	O
,	O
and	O
cells	O
with	O
large	O
numbers	O
of	O
states	B-Application
.	O
</s>
<s>
In	O
the	O
same	O
paper	O
,	O
Kari	O
also	O
showed	O
that	O
it	O
is	O
undecidable	O
to	O
test	O
whether	O
a	O
given	O
cellular	O
automaton	O
rule	O
of	O
two	O
or	O
more	O
dimensions	O
is	O
surjective	B-Algorithm
(	O
that	O
is	O
,	O
whether	O
it	O
has	O
a	O
Garden	O
of	O
Eden	O
)	O
.	O
</s>
<s>
In	O
a	O
one-dimensional	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
with	O
states	B-Application
per	O
cell	O
,	O
in	O
which	O
the	O
neighborhood	O
of	O
a	O
cell	O
is	O
an	O
interval	O
of	O
cells	O
,	O
the	O
automaton	O
representing	O
the	O
reverse	O
dynamics	O
has	O
neighborhoods	O
that	O
consist	O
of	O
at	O
most	O
cells	O
.	O
</s>
<s>
For	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
,	O
if	O
the	O
initial	O
configuration	O
is	O
chosen	O
uniformly	O
at	O
random	O
among	O
all	O
possible	O
configurations	O
,	O
then	O
that	O
same	O
uniform	O
randomness	O
continues	O
to	O
hold	O
for	O
all	O
subsequent	O
states	B-Application
.	O
</s>
<s>
Making	O
a	O
change	O
to	O
the	O
initial	O
state	O
of	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
may	O
cause	O
changes	O
to	O
later	O
states	B-Application
to	O
remain	O
only	O
within	O
a	O
bounded	O
region	O
,	O
to	O
propagate	O
irregularly	O
but	O
unboundedly	O
,	O
or	O
to	O
spread	O
quickly	O
,	O
and	O
lists	O
one-dimensional	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
rules	O
exhibiting	O
all	O
three	O
of	O
these	O
types	O
of	O
behavior	O
.	O
</s>
<s>
When	O
run	O
on	O
a	O
finite	O
array	O
of	O
cells	O
with	O
periodic	O
boundary	O
conditions	O
,	O
starting	O
from	O
a	O
small	O
seed	O
of	O
random	O
cells	O
centered	O
within	O
a	O
larger	O
empty	O
neighborhood	O
,	O
it	O
tends	O
to	O
fluctuate	O
between	O
ordered	O
and	O
chaotic	O
states	B-Application
.	O
</s>
<s>
Another	O
way	O
to	O
formalize	O
reversible	O
cellular	O
automata	O
involves	O
abstract	O
algebra	O
,	O
and	O
this	O
formalization	O
has	O
been	O
useful	O
in	O
developing	O
computerized	O
searches	O
for	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
rules	O
.	O
</s>
<s>
As	O
Boykett	O
argues	O
,	O
any	O
one-dimensional	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
is	O
equivalent	O
to	O
an	O
automaton	O
in	O
rectangular	O
form	O
,	O
in	O
which	O
the	O
cells	O
are	O
offset	O
a	O
half	O
unit	O
at	O
each	O
time	O
step	O
,	O
and	O
in	O
which	O
both	O
the	O
forward	O
and	O
reverse	O
evolution	O
of	O
the	O
automaton	O
have	O
neighborhoods	O
with	O
just	O
two	O
cells	O
,	O
the	O
cells	O
a	O
half	O
unit	O
away	O
in	O
each	O
direction	O
.	O
</s>
<s>
If	O
a	O
reversible	O
automaton	O
has	O
neighborhoods	O
larger	O
than	O
two	O
cells	O
,	O
it	O
can	O
be	O
simulated	O
by	O
a	O
reversible	O
automaton	O
with	O
smaller	O
neighborhoods	O
and	O
more	O
states	B-Application
per	O
cell	O
,	O
in	O
which	O
each	O
cell	O
of	O
the	O
simulating	O
automaton	O
simulates	O
a	O
contiguous	O
block	O
of	O
cells	O
in	O
the	O
simulated	O
automaton	O
.	O
</s>
<s>
That	O
is	O
,	O
every	O
semicentral	O
bigroupoid	O
defines	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
in	O
rectangular	O
form	O
,	O
in	O
which	O
the	O
transition	O
function	O
of	O
the	O
automaton	O
uses	O
the	O
operation	O
to	O
combine	O
the	O
two	O
cells	O
of	O
its	O
neighborhood	O
,	O
and	O
in	O
which	O
the	O
operation	O
similarly	O
defines	O
the	O
reverse	O
dynamics	O
of	O
the	O
automaton	O
.	O
</s>
<s>
Every	O
one-dimensional	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
is	O
equivalent	O
to	O
one	O
in	O
this	O
form	O
.	O
</s>
<s>
When	O
researchers	O
design	O
reversible	O
cellular	O
automata	O
to	O
simulate	O
physical	O
systems	O
,	O
they	O
typically	O
incorporate	O
into	O
the	O
design	O
the	O
conservation	O
laws	O
of	O
the	O
system	O
;	O
for	O
instance	O
,	O
a	O
cellular	O
automaton	O
that	O
simulates	O
an	O
ideal	O
gas	O
should	O
conserve	O
the	O
number	O
of	O
gas	O
particles	O
and	O
their	O
total	O
momentum	B-Algorithm
,	O
for	O
otherwise	O
it	O
would	O
not	O
provide	O
an	O
accurate	O
simulation	O
.	O
</s>
<s>
The	O
typical	O
type	O
of	O
conserved	O
quantity	O
measured	O
in	O
these	O
studies	O
takes	O
the	O
form	O
of	O
a	O
sum	O
,	O
over	O
all	O
contiguous	O
subsets	O
of	O
cells	O
of	O
the	O
automaton	O
,	O
of	O
some	O
numerical	O
function	O
of	O
the	O
states	B-Application
of	O
the	O
cells	O
in	O
each	O
subset	O
.	O
</s>
<s>
For	O
instance	O
,	O
recall	O
the	O
one-dimensional	O
cellular	O
automaton	O
defined	O
as	O
an	O
example	O
from	O
a	O
rectangular	O
band	O
,	O
in	O
which	O
the	O
cell	O
states	B-Application
are	O
pairs	O
of	O
values	O
(	O
l	O
,	O
r	O
)	O
drawn	O
from	O
sets	O
and	O
of	O
left	O
values	O
and	O
right	O
values	O
,	O
the	O
left	O
value	O
of	O
each	O
cell	O
moves	O
rightwards	O
at	O
each	O
time	O
step	O
,	O
and	O
the	O
right	O
value	O
of	O
each	O
cell	O
moves	O
leftwards	O
.	O
</s>
<s>
Any	O
one-dimensional	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
may	O
be	O
placed	O
into	O
rectangular	O
form	O
,	O
after	O
which	O
its	O
transition	O
rule	O
may	O
be	O
factored	O
into	O
the	O
action	O
of	O
an	O
idempotent	O
semicentral	O
bigroupoid	O
(	O
a	O
reversible	O
rule	O
for	O
which	O
regions	O
of	O
cells	O
with	O
a	O
single	O
state	O
value	O
change	O
only	O
at	O
their	O
boundaries	O
)	O
together	O
with	O
a	O
permutation	B-Algorithm
on	O
the	O
set	O
of	O
states	B-Application
.	O
</s>
<s>
The	O
first-order	O
invariants	O
for	O
the	O
idempotent	O
lifting	O
of	O
the	O
automaton	O
rule	O
(	O
the	O
modified	O
rule	O
formed	O
by	O
omitting	O
the	O
permutation	B-Algorithm
)	O
necessarily	O
behave	O
like	O
the	O
ones	O
for	O
a	O
rectangular	O
band	O
:	O
they	O
have	O
a	O
basis	O
of	O
invariants	O
that	O
flow	O
either	O
leftwards	O
or	O
rightwards	O
at	O
a	O
constant	O
rate	O
without	O
interaction	O
.	O
</s>
<s>
The	O
first-order	O
invariants	O
for	O
the	O
overall	O
automaton	O
are	O
then	O
exactly	O
the	O
invariants	O
for	O
the	O
idempotent	O
lifting	O
that	O
give	O
equal	O
weight	O
to	O
every	O
pair	O
of	O
states	B-Application
that	O
belong	O
to	O
the	O
same	O
orbit	O
of	O
the	O
permutation	B-Algorithm
.	O
</s>
<s>
However	O
,	O
the	O
permutation	B-Algorithm
of	O
states	B-Application
in	O
the	O
rule	O
may	O
cause	O
these	O
invariants	O
to	O
behave	O
differently	O
from	O
in	O
the	O
idempotent	O
lifting	O
,	O
flowing	O
non-uniformly	O
and	O
with	O
interactions	O
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
different	O
regions	O
of	O
the	O
automaton	O
have	O
different	O
average	O
values	O
of	O
some	O
conserved	O
quantity	O
,	O
the	O
automaton	O
's	O
rules	O
may	O
cause	O
this	O
quantity	O
to	O
dissipate	O
,	O
so	O
that	O
the	O
distribution	O
of	O
the	O
quantity	O
is	O
more	O
uniform	O
in	O
later	O
states	B-Application
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
total	O
momentum	B-Algorithm
within	O
any	O
single	O
line	O
is	O
conserved	O
.	O
</s>
<s>
Three-way	O
collisions	O
are	O
also	O
possible	O
in	O
the	O
FHP	O
model	O
and	O
are	O
handled	O
in	O
a	O
way	O
that	O
both	O
preserves	O
total	O
momentum	B-Algorithm
and	O
avoids	O
the	O
unphysical	O
added	O
conservation	O
laws	O
of	O
the	O
HPP	O
model	O
.	O
</s>
<s>
Therefore	O
,	O
if	O
a	O
cell	O
has	O
equal	O
numbers	O
of	O
neighbors	O
in	O
the	O
two	O
states	B-Application
,	O
it	O
may	O
flip	O
its	O
own	O
state	O
without	O
changing	O
the	O
total	O
energy	O
.	O
</s>
<s>
This	O
defines	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
which	O
can	O
be	O
used	O
to	O
investigate	O
the	O
Ising	O
model	O
.	O
</s>
<s>
proposed	O
the	O
billiard-ball	B-Application
computer	I-Application
as	O
part	O
of	O
their	O
investigations	O
into	O
reversible	B-Application
computing	I-Application
.	O
</s>
<s>
A	O
billiard-ball	B-Application
computer	I-Application
consists	O
of	O
a	O
system	O
of	O
synchronized	O
particles	O
(	O
the	O
billiard	O
balls	O
)	O
moving	O
in	O
tracks	O
and	O
guided	O
by	O
a	O
fixed	O
set	O
of	O
obstacles	O
.	O
</s>
<s>
As	O
showed	O
,	O
billiard-ball	B-Application
computers	I-Application
may	O
be	O
simulated	O
using	O
a	O
two-state	O
reversible	O
block	O
cellular	O
automaton	O
with	O
the	O
Margolus	O
neighborhood	O
.	O
</s>
<s>
Connected	O
groups	O
of	O
more	O
than	O
one	O
live	O
cell	O
behave	O
instead	O
like	O
the	O
fixed	O
obstacles	O
of	O
the	O
billiard-ball	B-Application
computer	I-Application
.	O
</s>
<s>
In	O
an	O
appendix	O
,	O
Margolus	O
also	O
showed	O
that	O
a	O
three-state	O
second-order	O
cellular	O
automaton	O
using	O
the	O
two-dimensional	O
Moore	O
neighborhood	O
could	O
simulate	O
billiard-ball	B-Application
computers	I-Application
.	O
</s>
<s>
were	O
the	O
first	O
to	O
ask	O
whether	O
every	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
has	O
a	O
locally	O
reversible	O
update	O
rule	O
.	O
</s>
<s>
showed	O
that	O
for	O
one	O
-	O
and	O
two-dimensional	O
automata	O
the	O
answer	O
is	O
positive	O
,	O
and	O
showed	O
that	O
any	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
could	O
be	O
simulated	O
by	O
a	O
(	O
possibly	O
different	O
)	O
locally	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
.	O
</s>
<s>
Kari	O
suggests	O
that	O
a	O
system	O
such	O
as	O
this	O
may	O
be	O
used	O
as	O
a	O
public-key	B-Application
cryptosystem	I-Application
.	O
</s>
<s>
However	O
,	O
Kari	O
did	O
not	O
specify	O
which	O
types	O
of	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
should	O
be	O
used	O
for	O
such	O
a	O
system	O
,	O
or	O
show	O
how	O
a	O
cryptosystem	O
using	O
this	O
approach	O
would	O
be	O
able	O
to	O
generate	O
encryption/decryption	O
key	B-Application
pairs	I-Application
.	O
</s>
<s>
Quantum	O
cellular	O
automata	O
are	O
arrays	O
of	O
automata	O
whose	O
states	B-Application
and	O
state	O
transitions	O
obey	O
the	O
laws	O
of	O
quantum	O
dynamics	O
.	O
</s>
<s>
asked	O
whether	O
it	O
was	O
possible	O
for	O
a	O
cellular	O
automaton	O
to	O
be	O
physically	O
universal	O
,	O
meaning	O
that	O
,	O
for	O
any	O
bounded	O
region	O
of	O
the	O
automaton	O
's	O
cells	O
,	O
it	O
should	O
be	O
possible	O
to	O
surround	O
that	O
region	O
with	O
cells	O
whose	O
states	B-Application
form	O
an	O
appropriate	O
support	O
scaffolding	O
that	O
causes	O
the	O
automaton	O
to	O
implement	O
any	O
arbitrary	O
transformation	O
on	O
sets	O
of	O
states	B-Application
within	O
the	O
region	O
.	O
</s>
<s>
constructed	O
a	O
reversible	B-Application
cellular	I-Application
automaton	I-Application
that	O
is	O
physically	O
universal	O
in	O
this	O
sense	O
.	O
</s>
<s>
Schaeffer	O
's	O
automaton	O
is	O
a	O
block	O
cellular	O
automaton	O
with	O
two	O
states	B-Application
and	O
the	O
Margolis	O
neighborhood	O
,	O
closely	O
related	O
to	O
the	O
automata	O
for	O
the	O
billiard	O
ball	O
model	O
and	O
for	O
the	O
HPP	O
lattice	O
gas	O
.	O
</s>
<s>
Thus	O
,	O
his	O
automaton	O
is	O
not	O
Turing	B-Algorithm
complete	I-Algorithm
.	O
</s>
