Geometric computation on the plane
– signal machines –
Jérôme D URAND -L OSE
jerome.durand-lose@ens-lyon.fr
MC2 LIP
UMR CNRS - ÉNS Lyon - UCB Lyon - INRIA 5668
Geometric computation on the plane – p.1/43
Methodology
Cellular automata word
(Discrete) Space-time diagrams
Dynamics
Conception
Geometric computation on the plane – p.2/43
Methodology
Cellular automata word
(Discrete) Space-time diagrams
Observation
Discrete lines
Interpretation
Lines on the plane
Dynamics
Conception
Geometric computation on the plane – p.2/43
Methodology
Cellular automata word
(Discrete) Space-time diagrams
Implementation
Observation
Discrete lines
Interpretation
Discretization
Lines on the plane
Dynamics
Conception
Geometric computation on the plane – p.2/43
Methodology
Cellular automata word
(Discrete) Space-time diagrams
Implementation
Observation
Discrete lines
Interpretation
Discretization
Lines on the plane
Dynamics
Model on its own
Conception
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Outline
Cellular automata, particles and signals
Signal machines
Computational universality
Geometrical modifications
Accumulation
Undecidability of appearance
Accumulation
Conclusion and perspectives
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Origins
— Cellular Automata —
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Cellular Automata
Modeling tools in biology, physics. . . parallelism
Dynamical systems
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Cellular Automata
Modeling tools in biology, physics. . . parallelism
Dynamical systems
Cells
...
...
t=0
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Cellular Automata
Modeling tools in biology, physics. . . parallelism
Dynamical systems
Cells
|States|< ∞ . . .
...
t=0
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Cellular Automata
Modeling tools in biology, physics. . . parallelism
Dynamical systems
Cells
|States|< ∞ . . .
t=1
...
t=0
Updating rules
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Cellular Automata
Modeling tools in biology, physics. . . parallelism
Dynamical systems
t=4
t=3
t=2
Iteration
Cells
|States|< ∞ . . .
t=1
...
t=0
Updating rules
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Cellular Automata
local
Space-time diagram
uniform
t=4
synchronous
t=3
t=2
Iteration
Cells
|States|< ∞ . . .
t=1
...
t=0
Updating rules
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Example of particles
[Hordijk et al., 2001, Fig. 7]
[Boccara et al., 1991, Fig. 7]
[Siwak, 2001, Fig. 5]
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To build an Turing-universal CA
[Lindgren and Nordahl, 1990, Fig. 4]
[Lindgren and Nordahl, 1990, Fig. 3]
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Signal algorithmic
1
A AAA
A
AA AAA
AA
A AAA
AA AA
AAA
AA
AAA AA
AA
AA AA
A AAA
AA
AAAAA
AAA
AA
AAA
AA
AA
AAAA
AA
AA
AA
AAA
AAA
AA AA
AA
AAA
AA
A
A
AAA
A
AAA
AA
AA
AAA
AA
AA
AA
A AA
AAAA
AA
A AA
AA
AA
AAA
AA
AAA
AA
A
A
AAA
AAAA
A
AA
A
AA
AAAAA
AA
AAAAA
AA
A
AA
A
(2n, 2t(n))
Bits 1 of the
multiplier
E1
0
Bits 0 of the
multiplier
1
D1
Bit "end " of
the multiplier
*
T
Bits 1 of the
multiplicand
0
0
<1
Bits 0 of the
multiplicand
0
0
Bit "end" of
the multiplicand
E
0
0
Limits of a
computation
0
0
Bits 0 in transit
throught the
network
Time f(n)
1
Bits 1 in transit
throught the
network
0
1
Time f(5)
1
0
Signal T
Time f(4)
0
*
0
Signal D
*
1
0
Signal D
Time f(3)
0
1
0
1
1
Time f(2)
k
Signal E - 1
0
0
k/2
Signal E
0
0
1
Time f (1)
1
1
Time 0
1
[Fischer, 1965, Fig. 2]
D2
E-1
0
Signal E + 1
Cell 0
Figure 8: Computing (ab) .
Figure 18: Characterization of the sites (n; f (n)).
[Mazoyer, 1996, Fig. 8]
[Mazoyer and Terrier, 1999, Fig. 18]
2
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Geometric algorithmic
[Goto, 1966, Fig. 3]
[Goto, 1966, Fig. 6]
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Signal machines
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Signal Machines
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Time
Model analyze
Space
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Model analyze
Signal
(Meta-signal, position)
Position
(x, t)
Time
Meta-signal
µ = (ι, ν)
Space
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Model analyze
Signal
(Meta-signal, position)
Time
Collision
(Rule, position)
Position
(x, t)
Meta-signal
µ = (ι, ν)
Rule
+
ρ = {µ−
}
→
{µ
i
i
j }j
Space
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Model definition
Machine
Space-time diagram
M = ({µi }i , {ρj }j )
Finite description
Deterministic
Configuration (at t)
Positions of
signals and collisions
Computation
collisions +
dependence ordering
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Properties
Uniform in space and time
Local
Light cone
Finite number of values & rules
Continuous space & time
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Turing-universal
2-counter automata simulation
A and B two non-negative integer counters
beg: B++
A-A != 0
B != 0
beg1: A-A != 0
pair: B-A++
B != 0
A != 0
imp: B-A++
A++
B != 0
A != 0
imp1: B-A++
A++
A++
B != 0
A != 0
beg1
imp
beg
pair
beg
imp1
beg
imp1
beg
Operations
A++
B++
A-A-A != 0 <label > B != 0 <label >
Encoding
bord a0 = 6 instruction b0 = 2
Unbounded place for signals
bord
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Transcription of the instructions
A++
a=0
a 6= 0
After A++
B != 0 m
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Geometric computation on the plane – p.17/43
Turing-universal
Any recursive computation can be performed
Highly unpredictable
e.g. all these are undecidable
Finite number of collisions
Appearance of a meta-signal
Collision with a signal
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Geometrical constructions
Modify the space-time diagram
but preserve the computation (i.e. ordering of collisions)
Dynamic
Freezing
Translations
Homotecy
Constructions with these “operators”
basic
iterated
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Freezing the computation
Freeze then restore
Rest
of computation
Rest
of computation
n
Tra
Beginning
tio
a
l
s
n
Beginning
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Freezing the computation
Freeze then restore
Only modifications: new meta-signals and rules
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Contraction
Rest
of computation
Rest
of computation
Beginning
Contraction
Beginning
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Contraction
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Contraction to a ribbon
3
Contraction
3
2
2
Contraction
1
Contraction
1
Beginning
Beginning
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Contraction to a ribbon
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Distortion
Rest
Rest
of computation
of computation
×2
Beginning
Beginning
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Distortion
×2
×2
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Continuous contraction
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Contraction to a triangle
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Any computation can be embedded in a triangle
Great malleableness of space-time
Problem generation of an accumulation point
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Accumulations
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Undecidability of accumulation prevision
Instance
M: signal machine,
c0 : finite initial configuration,
(all values in Q)
Question
Will there be any accumulation?
∃(x, t) ∈ Z×N, ∀n ∈ N,
There is at least n collisions
in the light cone ending in (x, t)
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Undecidability of accumulation prevision
Instance
M: signal machine,
c0 : finite initial configuration,
(all values in Q)
Question
Will there be any accumulation?
∃(x, t) ∈ Z×N, ∀n ∈ N,
recursive
total predicate
There is at least n collisions
in the light cone ending in (x, t)
in Σ02 (arithmetical hierarchy)
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Reduction for
0
Σ1-hardness
Accumulation ⇔ the 2 counter automaton does not stop
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Placing all possible computations
Σ02 -complete
Instance
A: 2-counter automata,
Question
Does the computation finished for all initials values?
(0, 3)
(1, 2)
(0, 2)
(2, 1)
(1, 1)
(0, 1)
(3, 0)
(2, 0)
(1, 0)
(0, 0)
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0
Σ2-hardness
Σ02 -complete
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Topological reformulation
Space-time diagram
D
R × R>0 7−→ {⊘} ∪ {µi }i ∪ {ρj }j ∪ S
void
singularities
Conditions
D (p) = ⊘
⇐⇒ ∃r > 0, D(ball(p, r)) =
D (p) =
⇐⇒ ∃r > 0, D(ball(p, r)) =
D (p) =
⇐⇒ ∃r > 0, D(ball(p, r)) =
|S| < ∞
and
♦∈S
Matches machine approach definition
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Accumulated values
∀p ∈ R × R>0
AccVal(p) set of values accumulating around p
Singularity
⇐⇒
AccVal(p) ∩ {{ρj }j ∪ S} =
6 ∅
Isolated
⇐⇒
AccVal(p) ∩ S = ∅
Closer look at isolated singularities
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Meta-singularities
,
, ∅
{∅, ∅}
(
(
,
,
))
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Non deterministic meta-singularities
(
∅,
(
,
))
{∅, ∅}
(
∅,
(
,
))
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Non deterministic meta-singularities
(
∅,
(
,
))
{∅, ∅}
(
∅,
(
,
))
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Non deterministic meta-singularities
(
∅,
(
,
))
{∅, ∅}
(
∅,
(
,
))
Geometric computation on the plane – p.35/43
Non deterministic meta-singularities
(
∅,
(
,
(
∅,
(
,
))
, ∅
))
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Non deterministic meta-singularities
(
∅,
(
,
(
∅,
(
,
))
, ∅
))
Geometric computation on the plane – p.36/43
Non deterministic meta-singularities
(
∅,
(
,
(
∅,
(
,
))
, ∅
))
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Second order
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Non isolated singularities
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Non isolated singularities
Cantor
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Conclusion
Geometrical computation model
Turing-computability
Continuous space and time aspects
Great malleableness of space-time
Accumulations
Decidability
Partial treatment
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Perspectives – on the model
Singularities
Comprehension
Utilization (super-Turing capability)
Algorithmic principles
Complexity
Intrinsic universality
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Perspectives – relating the model
Link to other model
Understand continuous models
Continuous classes of complexity and decidability
Cellular automata
Automatic discretization
Transfer theorem
Validate proofs
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References
[Boccara et al., 1991] Boccara, N., Nasser, J., and Roger, M. (1991). Particle-like structures
and interactions in spatio-temporal patterns generated by one-dimensional deterministic
cellular automaton rules. Phys. Rev. A, 44(2):866–875.
[Fischer, 1965] Fischer, P. C. (1965). Generation of primes by a one-dimensional real-time
iterative array. Journal of the ACM, 12(3):388–394.
[Goto, 1966] Goto, E. (1966). Ōtomaton ni kansuru pazuru [Puzzles on automata]. In
Kitagawa, T., editor, Jōhōkagaku eno michi [The Road to Information Science], pages
67–92. Kyoristu Shuppan Publishing Co., Tokyo.
[Hordijk et al., 2001] Hordijk, W., Shalizi, C. R., and Crutchfield, J. P. (2001). An upper
bound on the products of particle interactions in cellular automata. Physica D,
154:240–258.
[Lindgren and Nordahl, 1990] Lindgren, K. and Nordahl, M. G. (1990). Universal
computation in simple one-dimensional cellular automata. Complex Systems, 4:299–318.
[Mazoyer, 1996] Mazoyer, J. (1996). Computations on one dimensional cellular automata.
Annals of Mathematics and Artificial Intelligence, 16:285–309.
[Mazoyer and Terrier, 1999] Mazoyer, J. and Terrier, V. (1999). Signals in one-dimensional
cellular automata. Theoretical Computer Science, 217(1):53–80.
[Siwak, 2001] Siwak, P. (2001). Soliton-like dynamics of filtrons of cycle automata. Inverse
Geometric computation on the plane – p.43/43
Problems, 17:897–918.