The Discrete Baker
Transformation
Burton Voorhees
Center for Science
Athabasca University
References
Bulitko, V., Voorhees, B., & Bulitko, V. (2006) Discrete baker
transformation for linear cellular automata analysis. Journal of
Cellular Automata 1(1) 40 - 70.
Voorhees, B. (2007) Discrete baker transformation for binary valued
cylindrical cellular automata. Lecture Notes in Computer Science
(4173) 182 - 191.
Bulitko, V. & Voorhees, B. (2008) Index permutations and classes of
additive cellular automata rules with isomorphic STD. Journal of
Cellular Automata (to appear)
Bulitko, V. & Voorhees, B. (2008) Cycle structure of baker
transformation in one dimension. Journal of Cellular Automata
(submitted, in revision).
The Baker Transformation
The baker transformation (or Bernoulli shift) in dynamical
systems theory is the transformation of the interval [0,1]
by
x --> 2x mod(1)
Where mod(1) means that only digits to the right of the
decimal are retained. E.g., .75 --> .5 --> 0; 1/3 <--> 2/3.
All rationals with denominator a power of 2 iterate to 0,
other rationals to cycles. 1/7 --> 2/7 --> 4/7 --> 1/7 is a
period 3 cycle, and period 3 implies chaos.
Additive Cellular Automata in
One Dimension
An additive cellular automaton rule in one dimension, with
periodic boundary conditions and alphabet {0,…,p-1}, is
defined in terms of the left shift operator  by:
n1
X   as  s
0  as  p  1
s0
Equivalently, by the string (a0,…,an-1). The set of pn such
strings defines the rule space of additive CA, and also the
state space on which these CA operate.
Discrete Baker Transformation
For a rule T = (a0,…,an-1) with alphabet {0,…,p-1} (p
prime), the discrete baker transformation is defined by:
  a j if {k | pk  i mod(n)}  

 B p T    j:pji mod(n)
i
 0
otherwise
Discrete Baker Transformation
Examples for p = 2
n=6:
B2T = (a0+a3,0,a1+a4,0,a2+a5,0)
n=7:
B2T = (a0,a4,a1,a5,a2,a6,a3)
n=9:
B2T = (a0,a5,a1,a6,a2,a7,a3,a8,a4)
Examples for p = 3
n=6:
B2T = (a0+a2+a4,0,0,a1+a3+a5,0,0)
n=7:
B2T = (a0,a5,a3,a1,a6,a4,a2)
n=9:
B2T = (a0+a3+a6,0,0,a1+a4+a7,0,0a2+a5+a8,0,0)
Discrete Baker Transformation
If T is an additive rule acting on a d-dimensional torus with

T  ai1
id
0  ai1
id

 p  1, 0  is  ns  1
Then
 B p T 
i1
id

a
j :pj imod(n ) j1
s
 s s s
 0
jd
if {ks | pks  is mod(n)}  
otherwise
Discrete Baker Transformation
T = (a0,…an-1) defines a right circulant matrix C(T) that gives
an expression of the update rule on a state µ by µ --> C(T)µ.
The baker transformation exponentially speeds up rule
evolution:
C(T )
pr

 C Bpr T
For p = 2 this is just B2T = T2.

Properties of Baker
Transformation
1. If p and n are relatively prime then Bp is a permutation
(analogous results hold for all dimensions).
2. Let k be the largest integer such that pk|n. For all r > k,
r1
Bp is a permutation on the indices of Bp T
Properties of Baker
Transformation
Example: p = 2, n = 6 = 2x3, 12 = 22x3
(a0,a1,a2,a3,a4,a5) -->
(a0+a3,0,a1+a4,0,a2+a5,0) <--> (a0+a3,0,a2+a5,0,a1+a4,0)
(a0,a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11) -->
(a0+a6,0,a1+a7,0,a2+a8,0,a3+a9,0,a4+a10,0,a5+a11,0) -->
(a0+a3+a6+a9,0,0,0,a1+a4+a7+a10,0,0,0,a2+a5+a8+a11,0,0,0 ) <-->
(a0+a3+a6+a9,0,0,0,a2+a5+a8+a11,0,0,0,a1+a4+a7+a10,0,0,0 )
Properties of Baker
Transformation
Let n = pkm with p and m relatively prime and c = ordmp.
Then
Bpkc T  Bpk T
In addition, all cycle periods in STD(T) divide pk(pc - 1)
and the height of trees in STD(T) does not exceed pk.
Baker Diagram
The baker transformation is a mapping on the space of
additive CA rules. The state transition diagram for this
mapping is the Baker Diagram. Suppose that X and Y are
two additive rules belonging to the same cycle in the Baker
Diagram, with cycle length L and let (c1,…,cr) be the set of
cycle lengths in STD(X). Then:
1. STD(X) and STD(Y) are isomorphic as graphs
2. Tree heights in STD(X) do not exceed 1


3. For all s, cs|(pL - 1), L  lcm ordc1 p, ,ordcr p and L|c
Baker Diagram
Let T be an additive rule, h(T) the height of T in the baker
diagram (i.e., its distance from an attractor) and c(T) the
period of the attractor (cycle or fixed point). Then the length
of all cycles in STD(T) divides ph(T)(pc(T) - 1).
Example: p = 2, n = 6, T = (1,1,0,0,0,0) = I + (rule 102)
Under the baker transformation
(1,1,0,0,0,0) --> (1,0,1,0,0,0) <--> (1,0,0,0,1,0)
h(T) = 1, c(T) = 2 so ph(T)(pc(T) - 1) = 6 and the cycle periods
in STD(I + ) are 1, 2, 3, and 6.
Baker Diagram
For n = 5 the baker diagram consists of four fixed points, two
period 2 cycles, and six period 4 cycles. with X = (a0a1a2a3a4)
Fixed Points: (00000), (11111), (01111), and (10000)
Period 2: {(01001), (00110)}, {(11001), (10110)}
Period4: {(00001), (00010), (01000), (00100)},
{(01100), (00101), (00011), (01010)},
{(01110), (01101), (00111), (01011)},
{(10001), (10010), (11000), (10100)},
{(11100), (10101), (10011), (11010)},
{(11110), (11101), (10111), (11011)}
Baker Diagram
Fixed Points: 0, 1, I, and I + 1 mod(2)
Period 2: , }, {I+, I+}
Period4: {,,,},
{,,,},
{,,,},
{I+,,,},
{I+,,,},
{I+,,,}
Baker Diagram
For n = 6 the baker diagram consists of four fixed points and
two period 2 cycles with all states on cycles the root of height
1 trees containing 7 states. with X = (a0a1a2a3a4,a5)
Fixed Points: (000000), (100000), (001010), (101010)
0,
I,
,
Period 2: {(001000), (000010)}, {(101000), (100010)}
{,,
Rules having isomorphic STDs are at the same level in the
baker diagram (but rules at the same level but not on cycles do
not necessarily have isomorphic STDs).
Index Baker Transformation
For any rule T with h(T) its height in the baker diagram, the
*
h(T )
rule T  Bp T is on a cycle so Bp acts as a permutation
on T*. Thus, Bp can be represented by a permutation on the
index space: bp:{0,…,n-1} --> {0,…,n-1}. This is called the
index baker transformation. It has the same cycle structure
as the baker transformation.
Index Baker Transformation
In a recent paper (Bulitko, V. & Voorhees, B., Cycle structure
of baker transformation in one dimension) we show the
complete cycle structure for Bp in terms of the bp cycles and
their multiplicities on the prime factors of n. Thus the index
baker transformation only needs to be considered on values
of n that are relatively prime to p.
In one dimension, all non-singular transformations that map
every additive CA into an additive CA with an isomorphic
STD are compositions of shifts and powers of baker (e.g., if
n = 7 the powers of baker showing up are 1/2, 1, 3/2 and
their inverses).
Brief Number Theoretic
Excursions
If n is prime, all non-trivial cycles of bp on {0,…,n-1} have
the same cycle length.
Let s be a positive integer, p and q prime. Then s is the length
of a non-trivial cycle of bp on {0,…,q-1} if and only if s|(q-1)
and there exist positive integers m, t such that (a) s = mt and
(b) q|(pt - 1)
For any p, s there are only a finite number of primes q with
non-trivial bp cycles of length s.
Brief Number Theoretic
Excursions
For p and q prime, define the height of p modulo q as
 
hq ( p)  max k q k | p
ordq p

1
with ordqp the smallest integer s such that ps = 1 mod(q).
Define:
p (s) 

q
hq ( p)
q prime
sordq p
This represents all primes having non-trivial bp cycles of
length s.
Brief Number Theoretic
Excursions
Some properties of this “universal” number:
1.

2. If s ≠ r
3. gcd
4.

gcd 
s
(s)
p
1
p

p
(s),
p
p
(s),

p

(r)  1
(r)  1  s  r or
p
(s)  1
m p (s)  non-trivial bp cycles on {0,…,m-1} have
length s and p (s) is the largest number for which this is
true.
Brief Number Theoretic
Excursions
For s a positive integer and p prime set i(q) to the largest
integer such that qi(q)|(ps - 1) and define
N ( p, s) 

q|( p1)
q i(q)
ps  1
M ( p, s) 
N( p, s)
M(2,s) = ps - 1 is the s-th Mersenne number.
If s is prime, p (s)  M ( p, s) Thus, for prime s the s-th
Mersenne number is the largest number such that all nontrivial cycles of b2 have length s.
Brief Number Theoretic
Excursions
There are similar results for the Fermat numbers: Define
p1
F( p, s)   p ,
rp s
2s
so F(2, s)  2  1
r0
p  F( p, s)
s1
F(2,s) is the s-th Fermat number and
p
hence F(2,s) is the largest number for which b2 has non-trivial
cycles of length 2s+1.
Extension to General Rules
In one dimension (and perhaps higher dimensions) the baker
transformation can be extended to all rules acting on cylinders.
Doing this requires a messy, but theoretically useful extension
of the CA rule table to maximal neighborhoods; i.e., on a
cylinder of size n, the neighborhoods are also of size n and in
line with the expression of a rule in terms of the left shift, are
left justified. The rule table has size pn.
From here on, p = 2.
Extension to General Rules
For k-site additive rules X = (a0,…,an-1) there is a relation
between the rule coefficients as and the rule table entries xi
(0 ≤ i ≤ 2k-1). For k = n, taking the binary form i = i0…in-1
1
as  
0
x2ns1  1
0  s  n 1
otherwise
In terms of the components xi the condition that X be
additive is
n1
xi   ins1 x2s mod(2)
s0
Extension to General Rules
The baker transformation can be given in matrix form, both
in terms of the rule coefficients as and the rule table
components xi:
a  ba
x  Bx
xi  xi b
The second form of this doesn’t depend on additivity, hence
gives a generalization of the baker transformation to all CA.
The matrix b is size n, the matrix B is size 2n. If n is odd,
both b and B are non-singular permutation matrices.
Extension to General Rules
Example: n = 5
1
0

b  0
0

0
1
0
A
0
 0
0 0 0 0
0 0 1 0

1 0 0 0
0 0 0 1

0 1 0 0
A
0

D
0
B
0
0

0

0
0 0 0 0 0 0 0
0 0 0 1 0 0 0

1 0 0 0 0 0 0
0 0 0 0 1 0 0 
0
0
A
0
0
0
D
0
0
A
0
0
0
D
0
0
0
0
D
0
 0
0
0

0
0

0
A

0

D
0 1 0 0 0 0 0
0 0 0 0 0 1 0

0 0 1 0 0 0 0
0 0 0 0 0 0 1 
Extension to General Rules
Let n be odd and let A(X) be the adjacency matrix for
STD(X): Aij = 1 iff X(i0…in-1) = j0…jn-1.
Then A(B2X) = B2A(X)B2-1.
Since for odd n B2 is a permutation, B2-1 exists and this
defines an isomorphism between STD(X) and STD(B2X).
This is the equivalent non-linear expression for the linear
equation [b2C(T)b2-1](b2µ) = b2ß where C(T) is the circulant
matrix representing T with C(T)µ = ß.
Extension to General Rules
It is no longer true that (B2T) = T2.
For rule 18 with n = 5, for example, B2T equals T2 on states
00000, 00001, 00010, 00100, 01000, 10000, 01111, 10111,
11011, 11101, 11110, 11111. It equals T on states 00011,
00110, 01100, 11000, 10001, 00101, 01010, 10100, 01001,
10010.
With substitutions f:00111 --> 11010, f-1:11010 --> 00111
and all shifts of this (B2T)(fµ) = T2µ.