CDMTCS
Research
Rep ort
Series
Randomness on Full Shift
Spaces
Cristian S. Calude
Peter P. Hertling
Department of Computer Science
University of Auckland
Helmut Jurgensen
Department of Computer Science
University of Western Ontario, Canada
Klaus Weihrauch
Theoretische Informatik I, Fern-Universitat
Hagen, Germany
CDMTCS-100
April 1999
Centre for Discrete Mathematics and
Theoretical Computer Science
Randomness on Full Shift Spaces
Cristian S. Calude,
y
Peter P. Hertling,
z
Helmut J urgensen,
x
Klaus Weihrauch
{
Abstract
We givevarious characterizations for algorithmically random congurations on full
shift spaces, based on randomness tests. We show that all nonsurjective cellular au-
tomata destroy randomness and surjective cellular automata preserve randomness. Fur-
thermore all one-dimensional cellular automata preserve nonrandomness. The last three
assertions are also true if one replaces randomness by richness,|a form of pseudoran-
domness, which is compatible with computability, the last assertion even for an arbitrary
dimension.
1 Intro duction
Cellular automata were originally introduced by Ulam and von Neumann [28] as mo dels for
natural complex systems, esp ecially self{repro ducing biological systems. Since then they
have b een analyzed in many other contexts, e.g. for the simulation of physical phenomena,
for computability questions (cellular automata are capable of universal computation), for
random number generation, in the framework of formal language theory, in symbolic dy-
namics, and many more; compare e.g. Wolfram [30] and other pap ers in the same volume,
Tooli, Margolus [27], Culik, Hurd, Yu [11], and Lind, Marcus [17].
Cellular automata show a uniform b ehavior over a certain region of the space. They
op erate on congurations which consist of a discrete lattice of cells each of which is in
one of nitely many states. Time is discrete; at each time step the value of each cell is
up dated uniformly according to a nite set of rules. The new value of a cell dep ends only
on the currentvalues of nitely many cells in its neighborhood. Although cellular automata
can be describ ed easily by a nite set of rules (the lo cal function) they exhibit a rich
and complicated global b ehavior which often seems chaotic or random. In [31] Wolfram
discussed some asp ects of cellular automata with resp ect to randomness in the sense of
Calude was supported in part byAURC Grants, A18/XXXXX/62090/F3414044-50. Hertling was sup-
ported by the DFG Research Grant No. HE 2489/2-1. Jurgensen was supp orted by National Sciences and
Engineering Council of Canada Grant OGP0000243.
y
Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New
Zealand, email:
cristian@cs.auckland.ac.nz
.
z
Department of Computer Science, The University of Auckland, Private Bag 92019, Auckland, New
Zealand, email:
hertling@cs.auckland.ac.nz
.
x
Department of Computer Science, The UniversityofWestern Ontario, London, Ontario, Canada N6A
5B7, and Institut f ur Informatik, Universiat Potsdam, Am Neunen Palais 10, D-14469, Potsdam, Germany,
email:
helmut@uwo.ca
.
{
Theoretische Informatik I, Fern-Universitat Hagen, D-58084 Hagen, Germany, email:
klaus.weihrauch@fernuni-hagen.de
.
algorithmic information theory; compare Chaitin [7], Calude [4]. In this pap er we give
several rigorous mathematical characterizations of random congurations and analyze the
b ehavior of cellular automata on random and nonrandom congurations.
The characterizations of random congurations are based on Martin{Lof 's [19 ] idea
to use randomness tests and the generalization of his ideas carried out by Hertling and
Weihrauch [13, 14 ]. We show that a cellular automaton is surjective if and only if it preserves
randomness of congurations. This gives a new characterization of the class of surjective
cellular automata. Note that the analysis and comparison of the classes of injective (or
reversible) cellular automata and surjective cellular automata have received great attention
in the past, starting with Mo ore's
Garden of Eden Theorem
[22]; compare Myhill [23],
Richardson [24], Maruoka and Kimura [20, 21], and others. It follows directly from known
results that nonsurjective cellular automata destroy randomness. Furthermore, we show
that every cellular automaton of dimension 1 preserves nonrandomness, i.e., if started on a
nonrandom conguration then the following conguration is nonrandom as well. The same
statements are shown to be true also if randomness is replaced by the simpler \richness"
prop erty (following Compton's [10 ] terminology for one way innite sequences we call a
conguration
rich
if it contains every nite pattern). In fact, cellular automata of arbitrary
dimension preserve nonrichness. At present it seems to b e op en whether arbitrary cellular
automata of dimension 2 or greater preserve nonrandomness. These denitions and results
may serve as a rst step towards a b etter understanding of the \random" b ehavior of cellular
automata. Further p ossible questions in this context are formulated in the conclusions
section.
We give a short overview over the pap er. In the next section weintroduce and describ e
full shift spaces and basic notions connected with them. We also introduce the notion of
an algorithmically random conguration. In Section 3 more characterizations (based on
randomness tests) and prop erties of random congurations are discussed; the construction
of new randomness spaces in terms of products and quotients of randomness spaces is also
analyzed. In Section 4, we dene cellular automata and analyze their b ehavior with respect
to randomness and nonrandomness of congurations. Finally, in Section 5, we indicate some
p ossible further questions for study.
2 Full Shift Spaces
We intro duce full shift spaces and several elementary notions connected with them, esp e-
cially richness of congurations.
By
N
;
Z
we denote the sets
f
0
;
1
;
2
;:::
g
(of nonnegative integers)
and
f
:::;
,
2
;
,
1
;
0
;
1
;
2
;:::
g
integers, resp ectively. Let be a nite set with at least 2
elements, and let
d
1 be a p ositive integer. Then
Z
d
is the
d
{dimensional lattice over
the integers
Z
. The space
Z
d
is called a
ful l shift space
. We call the elements of the
states
, the number
d
the
dimension
, and the elements
c
2
Z
d
the
congurations
of the full
shift space. On such spaces we use the pro duct topology induced by innitely many copies
of the discrete top ology on the nite space . For a conguration
c
2
Z
d
and
a
2
Z
d
we
write
c
a
instead of
c
(
a
); elements of
Z
d
will b e sometimes called cells and
c
a
will then be the
content
s
of cell
a
. For
r
2
N
, let [
,
r;r
] denote the set
f,
r;:::;
0
;::: ;r
g
. By Tychono 's
Theorem the space
Z
d
is compact b ecause it is a countable pro duct of compact spaces.
2
This space is in fact a metric space. One can, for example, use the metric
dist
dened by
dist
(
c; c
0
)=2
,
m
(
c;c
0
)
where
m
(
c; c
0
) = min
f
r
2
N
j 9
a
2
[
,
r;r
]
d
:
c
a
6
=
c
0
a
g
;
for
c; c
0
2
Z
d
; here min
;
=
1
. The sets
f
c
2
Z
d
j
c
z
=
s
g
; s
2
; z
2
Z
d
form a subbase of the top ology on
Z
d
. Cellular automata op erate on full shift spaces.
Related questions will be discussed in Section 4.
The name
shift spaces
comes from the fact that the
shift mappings
on the space
Z
d
play an important role. Eachinteger vector
a
=(
1
;:::;
d
)
2
Z
d
induces a bijection
(
d
)
a
:
Z
d
!
Z
d
dened by
(
d
)
a
(
c
)
b
=
c
b
+
a
, for every
b
2
Z
d
; it is called the
shift map associated
with
a
. In the sequel the sup erscript (
d
) will be omitted when the dimension is clear from the
context. The shift map
e
i
asso ciated with the unit vector
e
i
=(0
;::: ;
0
;
1
;
0
;::: ;
0)
2
Z
d
having a 1 in p osition
i
and zero es on all other p ositions is also written
i
. The shift
mapping
1
is the usual left shift in the one-dimensional case.
We wish to dene a random conguration of a full shift space. First let us lo ok at
the simplest case, when the dimension
d
is equal to 1. For one-way innite sequences
1
in
N
=
f
p
j
p
:
N
!
S
g
one obtains the well{known randomness notion from algorithmic
information theory; see Calude [4 ], Li, Vitanyi [16 ]. Random one-way sequences can be
dened via Martin{Lof 's [19] randomness tests or Chaitin [6, 7, 9] program{size complexity.
This \notion of randomness" will be dened precisely b elow. The simplest way to dene
randomness for two-way innite sequences over , that is, for elements of
Z
, is to use a
standard bijection from
Z
to
N
, e.g. the bijection
hi
:
Z
!
N
dened by
h
z
i
=
(
2
z;
if
z
0
;
2(
,
z
)
,
1
;
if
z<
0
:
This bijection induces a bijection from
N
to
Z
in the obvious way: one maps an element
p
=(
p
i
)
i
2
Z
to the one-way sequence
q
=(
q
z
)
z
2
N
dened by
q
z
=
p
h
z
i
, for all
z
2
Z
.
Now it seems natural to call a two-way innite sequence
q
2
Z
random
if and only if the
corresp onding one-way innite sequence
p
2
N
is random.
This pro cedure can also be carried out in the case of a dimension
d
1. For this
aim we use a bijection from
Z
d
onto
N
. The mapping
:
N
2
!
N
dened by
(
i; j
) =
(
i
+
j
)(
i
+
j
+1)+
i
is a bijection. For
d
2we dene
hi
:
Z
d
!
N
recursively by
h
z
1
;:::;z
d
i
=
(
h
z
1
i
;
h
z
2
;:::;z
d
i
)
:
This is a bijection for each
d
1.
If
L
1
and
L
2
are countable sets, then a total mapping
f
:
L
1
!
L
2
induces a mapping
f
:
L
2
!
L
1
via
f
(
p
)
l
1
=
p
f
(
l
1
)
;
for all
p
2
L
2
and
l
1
2
L
1
. If
f
is a bijection, then also
f
is a bijection. Hence, for
each
d
1, the induced mapping
hi
:
N
!
Z
d
is a bijection. It is clear that it is
1
In formal language theory one writes
!
instead of
N
and the elements of
N
are called
!
-words.
3
even a homeomorphism, and induces a bijection of the following subbases of the resp ective
top ologies: the pre-image under
hi
of the cylinder
f
c
2
Z
d
j
c
z
=
s
g
Z
d
for
s
2
and
z
2
Z
d
is the cylinder
f
c
2
N
j
c
h
z
i
=
s
g
, and these sets form a subbase of the pro duct
top ology on
N
. Furthermore, if we consider the product measure ~
on
N
and ~
on
Z
d
of
the uniform measure
on , given by
(
f
s
g
)=1
=
j
j
, then
hi
is also measure preserving,
i.e., ~
(
hi
,
1
(
U
)) = ~
(
U
) for all op en
U
Z
d
. Thus, the mapping
hi
really shows that
the spaces
N
and
Z
d
are identical with resp ect to top ology and measure. Using these
considerations we shall see later that it makes sense to call a conguration
c
2
Z
d
random
if and only if the one-way innite sequence
hi
(
c
)
2
N
is random.
There is just one more p oint which should be discussed: do es the construction ab ove
dep end up on the bijection
hi
:
Z
d
!
N
? Do es the choice of the bijection inuence the
denition? Certainly it do es, b ecause the notion of randomness for elements of
N
is not
invariant under an arbitrary p ermutation of its entries.
Example 2.1
For every sequence
c
0
c
1
c
2
:::
2
N
, there exists a bijection
:
N
!
N
such that the sequence
c
(0)
c
(1)
c
(2)
:::
2
N
is nonrandom. Such a
can be obtained
for example as follows. If the
c
0
c
1
c
2
:::
is not random we can take
to be the identity.
Otherwise we can assume, without loss of generality, that =
f
0
;
1
;::: ;q
,
1
g
, for some
q
2. Some element of app ears in the sequence innitely many times, say
c
i
= 0, for
innitely many
i
. Let
f
:
N
!
N
b e the unique and increasing function such that
c
f
(
i
)
is
the (
i
+ 1){st zero in
c
0
c
1
c
2
:::
for all
i
. We dene
by
(
i
)=
8
>
<
>
:
f
(2
j
+1)
;
if
i
=
f
(2
j
)+1
;
f
(2
j
)+1
;
if
i
=
f
(2
j
+1)
;
i;
if
i
62
S
j
2
N
f
f
(2
j
)+1
;f
(2
j
+1)
g
:
Then the sequence
c
(
i
)
c
(1)
c
(2)
:::
do es not contain an isolated zero, hence it do es not
contain the word 101, hence it is nonrandom.
2
But if
:
N
!
N
is a computable bijection, then a sequence
c
0
c
1
c
2
:::
2
N
is random
if and only if the sequence
c
(0)
c
(1)
c
(2)
:::
2
N
is random (see Bo ok, Lutz, Martin [2,
Lemma 3.4] or Hertling, Weihrauch [14 , Corollary 4.9]). Hence,
if a bijection
b
:
Z
d
!
N
is
chosen such that
hi
b
,
1
is computable we obtain via
b
the same randomness notion on
Z
d
as via the bijection
hi
.
We would like to consider also a very weak form of randomness for which this is not
true: richness. Following Compton [10], we call a one-way innite sequence
c
2
N
rich
if
and only if every word
w
2
o ccurs in
c
.
3
This can be transferred to congurations as
follows.
Two elements
v
2
A
and
w
2
B
for nite sets
A; B
Z
d
are called
equivalent
if
and only if there exists an integer vector
a
2
Z
d
such that
A
=
a
+
B
, and
v
a
+
b
=
w
b
for
all
b
2
B
. The equivalence classes of elements of
A
for nite subsets
A
Z
d
are called
patterns (over
and of dimension
d
)
. The equivalence classes of elements of
f
1
;
2
;:::;n
g
d
for
any p ositive integer
n
are called
cube patterns
. The number
n
is called the
side length
of
such a cub e pattern. We say that a pattern, given by a representative
w
2
A
for some
2
A random sequence in
N
contains every word in
, see Calude [4 ].
3
Richness is called disjunctiveness in formal language theory, cf. Jurgensen and Thierrin [15].
4