Machines à signaux :origine, puissance, retour au raisonnable Machines à signaux :
origine, puissance, retour au raisonnable Jérôme Durand-Lose, Vincent Levorato et Maxime Senot
Laboratoire d'Informatique Fondamentale d'Orléans,
Université d'Orléans, Orléans, FRANCE
LRI, Orsay 3 mai 2011
Machines à signaux :origine, puissance, retour au raisonnable Motivation
Discrete space and time
Cellular automata
Continuous reasoning
Continuous space and time
Machines à signaux :origine, puissance, retour au raisonnable Motivation
Discrete space and time
Cellular automata
Continuous reasoning
abstraction
Signal machines
(Abstract geometrical computation)
Continuous space and time
Machines à signaux :origine, puissance, retour au raisonnable Motivation
Discrete space and time
Cellular automata
Continuous reasoning
abstraction
Autonomous
model of computation
dynamical system
Signal machines
(Abstract geometrical computation)
Continuous space and time
Machines à signaux :origine, puissance, retour au raisonnable Motivation
Discrete space and time
Cellular automata
discretization
Autonomous
model of computation
dynamical system
Continuous reasoning
abstraction
Signal machines
(Abstract geometrical computation)
Continuous space and time
Machines à signaux :origine, puissance, retour au raisonnable 1
Cellular automata to signal machines
2
Generic QSat solving in constant space and duration (with D.
Duchier)
3
Discretization into CA
4
Conclusion
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
1
Cellular automata to signal machines
2
Generic QSat solving in constant space and duration (with D.
Duchier)
3
Discretization into CA
4
Conclusion
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Analyzing CA with signals
[Boccara et al., 1991, Fig. 7]
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Analyzing CA with signals
[Das, Crutcheld, Mitchell 95]
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Designing CA with signals
Goto's solution to the Firing Squad Synchronization Problem
[Goto66]
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Designing CA with signals
Generating primes [Fischer, 1965, Fig. 2]
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Designing CA with signals
Computing by simulating a Turing machine
[Lindgren and Nordahl, 1990, Fig. 4]
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
A whole programming system with discrete signals
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
0
E
Time f(n)
Time f(5)
Signal T
Time f(4)
Signal D
Signal D
Time f(3)
1
AAA
AAA
1
1
1
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
Right space moves
Signal E - 1
0
0
D2
E-1
0
1
AAA
AAA
AAA
AAA AAA
AAA
AA
AAA
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AA
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AAA AA
AA
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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]
Time (N)
Time (R + )
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Space (Z)
Vocabulary
Signal (meta-signal)
Collision (rule)
Space (R)
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Example: nding the middle
Meta-signal, speed
M,
S (M) = 0
Collision rules
M
M
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Example: nding the middle
Meta-signal, speed
S (M) = 0
div, S (div) = 3
M,
Collision rules
div
M
M
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Example: nding the middle
Meta-signal, speed
S (M) = 0
div, S (div) = 3
hi, S (hi) = 1
lo, S (lo) = 3
M,
M
div
hi
lo
M
M
Collision rules
{ div, M }
→
{ M, hi, lo }
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Example: nding the middle
Meta-signal, speed
S (M) = 0
div, S (div) = 3
hi, S (hi) = 1
lo, S (lo) = 3
back, S (back) = −3
M,
back
M
hi
M
lo
M
div
M
Collision rules
{ div, M }
{ lo, M }
→ { M, hi, lo }
→ { back, M }
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Example: nding the middle
Meta-signal, speed
S (M) = 0
div, S (div) = 3
hi, S (hi) = 1
lo, S (lo) = 3
back, S (back) = −3
M,
M
M
back
hi
M
lo
M
div
M
Collision rules
{ div, M }
{ lo, M }
{ hi, back
→ { M, hi, lo }
→ { back, M }
} → { M }
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
New kinds of
monsters
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Scaling down and bounding the duration
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Machines à signaux :origine, puissance, retour au raisonnable Cellular automata to signal machines
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
1
Cellular automata to signal machines
2
Generic QSat solving in constant space and duration (with D.
Duchier)
3
Discretization into CA
4
Conclusion
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
QSat: quantied satisfaction problem
Quantied boolean formula
(without free variable)
Find its logical value
PSPACE-complete problem
Running example
φ = ∃x1 ∀x2 ∀x3 (x1 ∧ ¬x2 ) ∨ x3
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
QSat: quantied satisfaction problem
Quantied boolean formula
(without free variable)
Find its logical value
PSPACE-complete problem
Running example
φ = ∃x1 ∀x2 ∀x3 (x1 ∧ ¬x2 ) ∨ x3
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
Building the tree / combinatorial comb
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
Lens and variables assignment
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
Formula evaluation
−t
←
t
=c
⇒
−
←
γ+
=t
⇒
γ←−
+
a
a
a
a
=
⇒
f
←
T−
=
⇒
¬
←
T−
=t
⇒
=t
⇒
←
T−
=t
⇒
γ←−
+
=t
⇒
−F
←
⇒◦
=
¬
−
←
γ+
⇒◦
=
¬
γ←−
+
=t
⇒
=t
⇒
a
=t
⇒
−
←
γ+
γ←−
+
−
←
γ+
=γ
⇒
a
=γ
⇒
=γ
⇒
a
=
⇒
¬
−
←
T
=t
⇒
−
←
T
γ←−
+
=t
⇒
⇒
=
f+
←
−
F
=t
⇒
−
←
T
⇒
=
∧◦
=∧
⇒
⇒
=
t+
=∧
⇒
−
←
T
=∨
⇒
a
=s
⇒
Case here
(true ∧ ¬true) ∨ true
=
⇒
T←
T−
=t
⇒
←
−
F
=
⇒
f
a
φ = ∃x1 ∀x2 ∀x3 (x1 ∧ ¬x2 ) ∨ x3
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
Complexities
Space
constant (as a width)
exponential (as max number of signals)
Time
constant (as a duration)
cubic (as max length of collision chain)
Machines à signaux :origine, puissance, retour au raisonnable Generic QSat solving in constant space and duration (with D. Duchier)
Complexities
Space
constant (as a width)
exponential (as max number of signals)
Time
constant (as a duration)
cubic (as max length of collision chain)
NB: Super-Turing Model with accumulations
Decide Halt in nite duration and width. . .
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
1
Cellular automata to signal machines
2
Generic QSat solving in constant space and duration (with D.
Duchier)
3
Discretization into CA
4
Conclusion
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Returning to a reasonnable model of computation
Natural candidate
Cellular automata
Limits
No more than a
crude approximation
No unbounded computation in nite space and time
No accumulation
Goal
Keep as much as possible the signal geometry
Transfer of properties
(nding the middle is already a problem)
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Considering pretopology
Based on pseudo-closure
A ⊆ a(A) and no idempotency requirement)
(
Use to relate geometrical properties between
continuous and discrete images
A signal is considered as the trace of the dilatation of a set
(orle:
Goal
Generate a
o (A) = a(A) − A)
discrete signal machine
which is nothing but a special case of CA
Experimental prototype
Simulation sotware in JAVA with the
PretopoLib [LevoratoBui08] library
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Illustration of meta-signals
Meta-Signal
Base of
neighborhood
B(x)=
(blue, 2)
or
B(x)={-2, -1, 0}
B(x)=
(red, -1)
or
B(x)={0,1}
B(x)=
(green, 3)
or
B(x)={-3, -2, -1, 0}
Meta-Signal movement (successive orles)
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Example
t=10
t=5
t=0
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Example
t=10
t=5
t=0
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Example
t=10
t=5
t=0
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Example
t=10
t=5
t=0
Machines à signaux :origine, puissance, retour au raisonnable Discretization into CA
Test: middle geometrical algorithm
(a) Continuous Signal Machine
(b) Discrete Signal Machine
Machines à signaux :origine, puissance, retour au raisonnable Conclusion
1
Cellular automata to signal machines
2
Generic QSat solving in constant space and duration (with D.
Duchier)
3
Discretization into CA
4
Conclusion
Machines à signaux :origine, puissance, retour au raisonnable Conclusion
Future work
Fractal computation
modularity
fractal characterization
higher classes of complexity
Automatic discretization
simple translation of discrete signal machine as CA
accuracy / approximation characterization
robustness
develop a logic to express preserved properties