CA to SM
La perspective du signal:
des automates cellulaires aux machines à signaux
Jérôme Durand-Lose
Laboratoire d’Informatique Fondamentale d’Orléans,
Université d’Orléans, Orléans, FRANCE
Journée Graphes et Algorithmes 2008
1e juillet 2008 — LIFO, Orléans
CA to SM
1
Introduction
2
Implicit use of signals
3
Discrete signals
4
Signal Machines
5
Conclusion
CA to SM
Introduction
1
Introduction
2
Implicit use of signals
3
Discrete signals
4
Signal Machines
5
Conclusion
CA to SM
Introduction
Cellular Automata
1 0 0 1202020010000202 1 0 0 0 2 0 0 2 1 Definition
0 1 2 3303030322222202 2 3 2 1 2 1 2 3 0
Q: finite set of states
0 0 1 0032023200220020 0 2 3 0 0 2 1 0 0
0 0 0 1231230222200002 0 2 2 1 2 3 0 0 0
f : Q k → Q local function
0 0 0 0100100022000000 2 0 0 2 1 0 0 0 0
0 0 0 0012332121012321 2 1 2 3 0 0 0 0 0
0 0 0 0001003002120030 0 2 1 0 0 0 0 0 0 Dynamical system
0 0 0 0000123101022221 2 3 0 0 0 0 0 0 0
Global function, G : Q Z → Q Z
0 0 0 0000010012022002 1 0 0 0 0 0 0 0 0
0 0 0 0000001233030123 0 0 0 0 0 0 0 0 0
Orbit and space-time diagram
0 0 0 0000000100320210 0 0 0 0 0 0 0 0 0
0 0 0 0000000012312300 0 0 0 0 0 0 0 0 0 Value in Q Z×N
Image with big pixels
Q = {0, 1, 2, 3}
f (x, y , z) = 3x + 2y + z + xy mod 4
CA to SM
Introduction
Background and Signals
Background
(2-d) Pattern that may form
a valid space-time diagram by bi-periodic repetition.
Signal
Pattern that (legally) repeats 1-periodically on a background
Pattern repeating 1-periodically and separating two
backgrounds
Illustration by examples
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Implicit use of signals
1
Introduction
2
Implicit use of signals
3
Discrete signals
4
Signal Machines
5
Conclusion
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Implicit use of signals
Understanding the dynamics
[Hordijk et al., 2001, Fig. 7]
[Boccara et al., 1991, Fig. 7]
[Siwak, 2001, Fig. 5]
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Implicit use of signals
Generating prime numbers
[Fischer, 1965, Fig. 2]
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Implicit use of signals
Computing by simulating a Turing machine
[Lindgren and Nordahl, 1990, Fig. 4]
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Implicit use of signals
Firing Squad Synchronization
[Goto, 1966, Fig. 3+6]
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Discrete signals
1
Introduction
2
Implicit use of signals
3
Discrete signals
4
Signal Machines
5
Conclusion
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Discrete signals
Firing Squad Synchronization (again)
[Varshavsky et al., 1970, Fig 1 and 3]
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Discrete signals
Multiplication
strongest
digit
* multiplier
1 1 0
0
multiplicand
0 0
1st0 partial
0 0 0
sum
0
0
end of the
words
i-1 th digit of
weakest
the j th partial
digit
sum
1 1
* 0 01 11 0
0
0
0
0
12nd1partial
0 0 1 1
1 sum
1 0 0 1 1
0
1 3rd
1 partial
0 0 1 1
sum
0 1 0 0 1 01 0
0
0
iCell
2
at time
2- j4
i--1 th carry
over of thethj
partial sum
i th digit of the
multiplier
i th carry over
of thethj
partial sum
β
D
α
C
A
A
AAA
AAA
A
AAA
AAA
A
AAA
AAA
AAAA
AAA
A
AA
AAAA
AAA
A
AAAA
A
A
α
j th digit of the
multiplicand
0 0 0 0
0 1 0 0 10 11 0
B
A
β
i th digit of
the j-1 th
partial sum
i th digit of the
multiplier
0
1
0
00
*
0
1
0
0
Bits 0 in transit throught
the network
0
Bits 1 in transit throught
the network
1
of the
0,1,* Bits
result
1
Bits of the
0, 1 multiplier
0
0, 1
Bits of the
multiplicand
* * End of words
0
Bit 0 of the
multiplier
0
*
*
Bit 1 of the
multiplier
1
0
1
1 1 0 0 1 1
0 1 0 0 0 1 0010
0
Time
Bit 0 of the
multiplicand
1
1
0
One cell out of two computes one time
out of two :
( =
C
α ∧ )β A⊕
(
)⊕
B
( =
D
α ∧ β A)
∧ (∨ α ∧ β )B
∧ A(∨
0 0 0
0 1 0010
6
Bit 1 of the
multiplicand
0
1
1
1
B).
∧
1
0
The last partial sum is the result
Figure 1: A human multiplication.
0
0
cells
Figure 3: Computations done on one cell out of two, one unit of time out of
two.
9
[Mazoyer, 1996, Fig. 1, 3 andx 4]
Figure 4: Multiplying 110011 by 10110.
10
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Discrete signals
A whole programming system
1
Bits 1 of the
multiplier
0
Bits 0 of the
multiplier
1
Bit "end " of
the multiplier
*
Bits 1 of the
multiplicand
0
0
<1
Bits 0 of the
multiplicand
0
0
0
Bit "end" of
the multiplicand
0
Limits of a
computation
0
0
Bits 0 in transit
throught the
network
1
Bits 1 in transit
throught the
network
0
1
1
0
*
0
0
*
1
0
0
1
0
1
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(2n, 2t(n))
E1
D1
T
E
Time f(n)
Time f(5)
Signal T
Time f(4)
Signal D
Signal D
Time f(3)
0
1
AAA
AAA
1
1
1
Right space moves
Nodes
Time f(2)
Left space moves
Impact sites
Time f (1)
Signals T
0
Figure 8: Computing (ab)2.
Signals setting up the (1,1)-th node
Figure 9: Setting up an innite family of regular safe grids (the darkness of the
grid indicates its rank).
19
k/2
k
Signal E
Signal E - 1
0
0
D2
E-1
0
1
AAA
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AAA AAA
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AA AAA
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A
A
AA
Time 0
Signal E + 1
Cell 0
Figure 18: Characterization of the sites (n; f (n)).
[Mazoyer, 1996, Fig. 8 and 19] and [Mazoyer and Terrier, 1999, Fig. 18]
CA to SM
Signal Machines
1
Introduction
2
Implicit use of signals
3
Discrete signals
4
Signal Machines
5
Conclusion
CA to SM
Signal Machines
Moving to the continuum
Forget about discreteness
continuous
Time (N)
Time (R + )
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Signal Machines
Space (Z)
Vocabulary
Signal (meta-signal)
Collision (rule)
Space (R)
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Signal Machines
New kinds of monsters
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Signal Machines
Computability and undecidability [Durand-Lose, 2005]
Two-counter simulation
Turing-machine can also
be simulated directly
Undecidable
total erasing
finite number of signal
signal/collision apparition
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Signal Machines
Scaling down and bounding the duration
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Signal Machines
Computing inside bounded room
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Signal Machines
Accumulation forecasting is Σ20 -complete
[Durand-Lose, 2006b]
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Signal Machines
Link with the Black hole model [Durand-Lose, 2006a]
Principe
Two different timelike half-curves such that
they have a point in common (used to set things and start)
one is upward-infinite and fully contained in the casual past of
a point of the other
Solving recursively enumerable problems
Accept
calcul
Refuse
calcul
Does not stop
calcul
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Signal Machines
Links with the Blum, Shub and Smale model
Classical BSS model
Variables holds real numbers in exact precision
input / output
test 0 < x
shift (to access other variables)
compute a polynomial function
Linear BSS [Durand-Lose, 2007]
Restriction
only linear function
i.e. no inner multiplication
CA to SM
Signal Machines
Encoding real numbers
Scale + distance
1
scale scale
Common scale for all variables
Sign test trivial
1
ba
val
CA to SM
Signal Machines
Encoding real numbers
Scale + distance
1
scale scale
Common scale for all variables
Sign test trivial
2.71
ba
val
CA to SM
Signal Machines
Encoding real numbers
Scale + distance
1
scale scale
-3.14
val
Common scale for all variables
Sign test trivial
ba
CA to SM
Signal Machines
val
Copy and Addition
set
−
t
se
set
set
set
−
t
se
−
to
ns 3
ns
to 5
lin
e
sca n val
ns
accum
−
to 5
line
n+1
ns
ns
base{2,7}
−
to 2
−
to 0
base∅
CA to SM
Signal Machines
n+1
line
val
line
val
External multiplication
n+1
+
mul a
mul
accum
+
mul a
mul
val
accum
val
m
ul
b
+
+
ul b
m
+
mul c
+
lc
mu
CA to SM
Signal Machines
Internal multiplication [Durand-Lose, 2008]
Computation
Pre-treatment to ensure
Binary extension of y :
Computation
0<y <1
y = y0 .y1 y2 y3 . . .
xy =
X
0≤i
yi
x Principe
Computation on the margin
the margin is scaling down geometrically
Square rooting is also possible!
2i
CA to SM
Conclusion
1
Introduction
2
Implicit use of signals
3
Discrete signals
4
Signal Machines
5
Conclusion
CA to SM
Conclusion
Natural filiation with CA
Continuous time
Zeno effect
Links with other models
Black hole model
Blum, Shub and Smale model
Future work
Relate with CA
Characterize the analog computing power