Simulation between signal machinesmsablik/Slide...Cellular Automata: signal use Firing Squad...
Transcript of Simulation between signal machinesmsablik/Slide...Cellular Automata: signal use Firing Squad...
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Simulation between signal machines
Simulation between signal machines
Jérôme Durand-Losejoint work with Florent Becker, Tom Besson,
Hadi Foroughmand and Sama Goliaei
Partially funded by PHC Gundichapur n◦909536E
LABORATOIRED'INFORMATIQUE
FONDAMENTALED'ORLEANS
Laboratoire d’Informatique Fondamentale d’Orléans,Université d’Orléans, Orléans, FRANCE
Laboratoire d’Informatique de l’École polytechnique
Algorithmic Questions in Dynamical SystemsMarch 2018 — Toulouse
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Simulation between signal machines
Introduction
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
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Simulation between signal machines
Introduction
Outline
One model/dynamical system
signal machines
Relations to “usual” computational models
Turing machines
Linear Blum, Shub and Smale model
Intrinsic universality
a family only
Non determinism
same power
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Simulation between signal machines
Introduction
Outline
One model/dynamical system
signal machines
Relations to “usual” computational models
Turing machines
Linear Blum, Shub and Smale model
Intrinsic universality
a family only
Non determinism
same power
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Simulation between signal machines
Introduction
Outline
One model/dynamical system
signal machines
Relations to “usual” computational models
Turing machines
Linear Blum, Shub and Smale model
Intrinsic universality
a family only
Non determinism
same power
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Simulation between signal machines
Introduction
Outline
One model/dynamical system
signal machines
Relations to “usual” computational models
Turing machines
Linear Blum, Shub and Smale model
Intrinsic universality
a family only
Non determinismsame power
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Simulation between signal machines
Signal Machines (Introduction and Definition)
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Cellular Automata: signal use
Firing Squad Synchronization (Goto, 1966)
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Simulation between signal machines
Signal Machines (Introduction and Definition)
CA: Conception with signals
Fischer (1965)
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Simulation between signal machines
Signal Machines (Introduction and Definition)
CA: Analyzing with Signals
Das et al. (1995)
in a chromosome. This de�nes one generation of the GA; it is repeated G times for one GA run.FI(�) is a random variable since its value depends on the particular set of I ICs selected toevaluate �. Thus, a CA's �tness varies stochastically from generation to generation. For thisreason, we choose a new set of ICs at each generationFor our experiments we set P = 100, E = 20; I = 100, m = 2; and G = 50. M was chosenfrom a Poisson distribution with mean 320 (slightly greater than 2N). Varying M preventsselecting CAs that are adapted to a particular M . A justi�cation of these parameter settings isgiven in [9].We performed a total of 65 GA runs. Since F100(�) is only a rough estimate of performance,we more stringently measured the quality of the GA's solutions by calculating PN104(�) withN 2 f149; 599; 999g for the best CAs in the �nal generation of each run. In 20% of the runsthe GA discovered successful CAs (PN104 = 1:0). More detailed analysis of these successful CAsshowed that although they were distinct in detail, they used similar strategies for performing thesynchronization task. Interestingly, when the GA was restricted to evolve CAs with r = 1 andr = 2, all the evolved CAs had PN104 � 0 for N 2 f149; 599; 999g. (Better performing CAs withr = 2 can be designed by hand.) Thus r = 3 appears to be the minimal radius for which the GAcan successfully solve this problem.β γ
δν
β γ
(a) Space-time diagram. (b) Filtered space-time diagram.Site0 74Site0 74
0
74
Tim
e
α
γδ
µ
Figure 1: (a) Space-time diagram of �sync starting with a random initial condition. (b) The same space-time diagram after �ltering with a spatial transducer that maps all domains to white and all defects toblack. Greek letters label particles described in the text.Figure 1a gives a space-time diagram for one of the GA-discovered CAs with 100% perfor-mance, here called �sync. This diagram plots 75 successive con�gurations on a lattice of sizeN = 75 (with time going down the page) starting from a randomly chosen IC, with 1-sites col-ored black and 0-sites colored white. In this example, global synchronization occurs at time step58. How are we to understand the strategy employed by �sync to reach global synchronization?Notice that, under the GA, while crossover and mutation act on the local mappings comprising a47 / 58
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Simulation between signal machines
Signal Machines (Introduction and Definition)
SignalsT
ime
(N)
Space (Z)
Tim
e(R
+)
Space (R)
Signal (meta-signal)
Collision (rule)
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Vocabulary and Example: Find the Middle
M M
Meta-signals (speed)
M (0)
div (3)hi (1)lo (3)
back (-3)
Collision rules
{ div, M }→{ M, hi, lo }{ lo, M }→{ back, M }
{ hi, back }→{ M }
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Vocabulary and Example: Find the Middle
div M M
Meta-signals (speed)
M (0)div (3)
hi (1)lo (3)
back (-3)
Collision rules
{ div, M }→{ M, hi, lo }{ lo, M }→{ back, M }
{ hi, back }→{ M }
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Vocabulary and Example: Find the Middle
div M
M hi lo
M
Meta-signals (speed)
M (0)div (3)hi (1)lo (3)
back (-3)
Collision rules
{ div, M }→{ M, hi, lo }
{ lo, M }→{ back, M }{ hi, back }→{ M }
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Vocabulary and Example: Find the Middle
div M
lo
M
M hi
backM
Meta-signals (speed)
M (0)div (3)hi (1)lo (3)
back (-3)
Collision rules
{ div, M }→{ M, hi, lo }{ lo, M }→{ back, M }
{ hi, back }→{ M }
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Vocabulary and Example: Find the Middle
div M
lo
M
hi
backM
MM
Meta-signals (speed)
M (0)div (3)hi (1)lo (3)
back (-3)
Collision rules
{ div, M }→{ M, hi, lo }{ lo, M }→{ back, M }
{ hi, back }→{ M }
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Stack Implantation
Name Speed
Add, Rem 1/3
A, E 1
U, M 0−→R 3←−R −3
Collision rules
{Add,M } → {M,A,−→R }
{−→R ,M } → {
←−R ,M }
{A,←−R } → {U }
{−→R ,U } → {
←−R ,U }
{Rem,M } → {M,E }{E,U } → {} Space R
Tim
e
R+
M
M
M
M
M
M
M
M
Add
A
Add
A−→R
←−R
UU
Rem
E
−→R
←−R
U
U
U
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Stack Implantation
Name Speed
Add, Rem 1/3
A, E 1
U, M 0−→R 3←−R −3
Collision rules
{Add,M } → {M,A,−→R }
{−→R ,M } → {
←−R ,M }
{A,←−R } → {U }
{−→R ,U } → {
←−R ,U }
{Rem,M } → {M,E }{E,U } → {} Space R
Tim
e
R+
M
M
M
M
M
M
M
M
Add
AAdd
A−→R
←−R
U
U
Rem
E
−→R
←−R
U
U
U
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Stack Implantation
Name Speed
Add, Rem 1/3
A, E 1
U, M 0−→R 3←−R −3
Collision rules
{Add,M } → {M,A,−→R }
{−→R ,M } → {
←−R ,M }
{A,←−R } → {U }
{−→R ,U } → {
←−R ,U }
{Rem,M } → {M,E }{E,U } → {} Space R
Tim
e
R+
M
M
M
M
M
M
M
M
Add
AAdd
A−→R
←−R
U
U
Rem
E
−→R
←−R
U
U
U
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Fractal Generation
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Complex Dynamics
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Complex Dynamics
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Complex Dynamics
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Simulation between signal machines
Signal Machines (Introduction and Definition)
Complex Dynamics
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Simulation between signal machines
Relations to Models of Computation
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
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Simulation between signal machines
Relations to Models of Computation
Discrete computation: Turing Machines
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of ComputationDiscrete computation: Turing MachinesAnalog Computation: linear Blum, Shub and Smale
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
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Simulation between signal machines
Relations to Models of Computation
Discrete computation: Turing Machines
Turing-computation
Turing Machine
^
qfb b a b #
^
qfb b a b #
^
qfb b a b #
^
qfb b a b #
^
q2b b a #
^
q1b b #
^
q1b a #
^
q1a a #
^
qia b #
^
qia b #
^
qia b #
Simulation
^
^
^
a
a
b
b
b
a
b
b
#
a
a
b
−→qi
−→qi
−→qi
←−q1
−→q1
−→q1
−→q2←−#
−→#
←−qf
←−#
−→#
←−qf
←−#
−→# −→
#
#←−qf
←−qf
←−qf
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Simulation between signal machines
Relations to Models of Computation
Discrete computation: Turing Machines
Turing-computation
Also with restrictions
all different speed
only 2→ 2 rules(conservative)
one-to-one rules(reversible)
Any above and
rational (Q)
Rational machines
speeds ∈ Qinitial positions ∈ Q⇒ collision coordinates ∈ Qexact simulation on computer/TM
Undecidabilityfinite number de collisionsmeta-signal appereanceuse of a ruledisappearing of all signalsinvolvement of a signal in any collisionextension on the side, etc.
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of ComputationDiscrete computation: Turing MachinesAnalog Computation: linear Blum, Shub and Smale
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Computing with Real Numbers
Encoding
base val
x
Zero
Sign test
base val
sign?
pos
Zero
sign?
zer
val base
sign?
neg
Addition
base
base
base
base
val
val
val
val
up
side
side
downs1down0
add
end
15 −8
7
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Multiplication by a constant
By −.9
base
base
val
val
x
−.9x
By − 12
base
base
val
val
x
− 12x
By 23
base
base
val
val
x
23x
By√
2
base
base
val
val
x
√2x
By π
base
base
val
val
x
πx
Signal speeds are constants of the machine
If x ≤ 0 then val is meet before base
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Multiplication by a constant
By −.9
base
base
val
val
x
−.9x
By − 12
base
base
val
val
x
− 12x
By 23
base
base
val
val
x
23x
By√
2
base
base
val
val
x
√2x
By π
base
base
val
val
x
πx
Signal speeds are constants of the machine
If x ≤ 0 then val is meet before base
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Multiplication by a constant
By −.9
base
base
val
val
x
−.9x
By − 12
base
base
val
val
x
− 12x
By 23
base
base
val
val
x
23x
By√
2
base
base
val
val
x
√2x
By π
base
base
val
val
x
πx
Signal speeds are constants of the machine
If x ≤ 0 then val is meet before base19 / 58
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Zooming out
x−2 x−1 x0 x1 x2 x3 x4[ ]
q1q1
q2
q3
q3
q2
Finite sequence of real numbers
+ Dynamics
finite state automata
sign test
addition, multiplication by constant
(set constant value)
(enlarge the array)
Like a Turing machine with real numbers on the tape
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Zooming out
x−2 x−1 x0 x1 x2 x3 x4[ ]
q1q1
q2
q3
q3
q2
Finite sequence of real numbers + Dynamics
finite state automata
sign test
addition, multiplication by constant
(set constant value)
(enlarge the array)
Like a Turing machine with real numbers on the tape
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Linear Blum, Shub and Smale with shift
Encoding a configuration
b a a c
d1 d2 d3
-1
-1 ...
b
d1 ...
a
d2 ...
a
d3 ...
c
-3 ...
-2
-2 ...
Simulating a signal machine: loop
1 Compute the minimum time to a collision, δ
2 Advance time by δ (update all distances)
3 Process collision(s)
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Linear Blum, Shub and Smale with shift
Encoding a configuration
b a a c
d1 d2 d3
-1 -1
...
b d1
...
a d2
...
a d3
...
c -3
...
-2 -2
...
Simulating a signal machine: loop
1 Compute the minimum time to a collision, δ
2 Advance time by δ (update all distances)
3 Process collision(s)
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Linear Blum, Shub and Smale with shift
Encoding a configuration
b a a c
d1 d2 d3
-1 -1 ... b d1 ... a d2 ... a d3 ... c -3 ... -2 -2 ...
Simulating a signal machine: loop
1 Compute the minimum time to a collision, δ
2 Advance time by δ (update all distances)
3 Process collision(s)
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Simulation between signal machines
Relations to Models of Computation
Analog Computation: linear Blum, Shub and Smale
Linear Blum, Shub and Smale with shift
Encoding a configuration
b a a c
d1 d2 d3
-1 -1 ... b d1 ... a d2 ... a d3 ... c -3 ... -2 -2 ...
Simulating a signal machine: loop
1 Compute the minimum time to a collision, δ
2 Advance time by δ (update all distances)
3 Process collision(s)
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Concept and Definition
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal MachinesConcept and DefinitionGlobal SchemeMacro-CollisionPreparation (shrink, test and check)
5 Non-determinism (work in progress)
6 conclusion23 / 58
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Concept and Definition
Intrinsic Universality
Being able to simulate any other dynamical system of the its class.
Cellular Automata
regular (Albert and Čulik II, 1987; Mazoyer and Rapaport,1998; Ollinger, 2001)
reversible (Durand-Lose, 1997)
freezing [Theyssier et Al.]
Tile Assembly Systems
possible at T=2 and above (Woods, 2013)
impossible at T=1 (Meunier et al., 2014)
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Concept and Definition
Simulation for Signal Machines
Space-Time Diagram Mimickingm
1
m2
m3
m2
m4
space
time
space
time
Signal Machine Simulation
US simulates A if there is function from the configurations of A tothe ones of US s.t. the space-time issued from the image alwaysmimics the original one.
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Concept and Definition
Theorem
For any finite set of real numbers S, there is a signal machineUS , that can simulate any machine whose speeds belong to S.The set of US where S ranges over finite sets of real numbersis an intrinsically universal family of signal machines.
Rest of this section
Let S be any finite set of real numbers,let A be any signal machine whose speeds belongs to S,
US is progressively constructed as simulation is presented.
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Global Scheme
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal MachinesConcept and DefinitionGlobal SchemeMacro-CollisionPreparation (shrink, test and check)
5 Non-determinism (work in progress)
6 conclusion27 / 58
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Global Scheme
Macro-Signal
Meta-signal of A identified with numbersUnary encoding of numbers
Structure
iµσ: σth signal, ith speed
iµσ
ileft-bou
nd
iidi
imainσ times
iright-bou
nd
CollisionRules
encoding
support zone
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Global Scheme
Global scheme
When Support Zones Meet
1 provide a delay
2 test if macro-collision is appropriate and what macro-signalsare involved
3 if OK
start the macro-collision
Hypotheses for macro-collision
no other macro-signal nor macro-collision will interfere
speed of involved macro-signals ranged [j , . . . , i ] (included)
their main∅ signals intersect at a unique point
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Macro-Collision
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal MachinesConcept and DefinitionGlobal SchemeMacro-CollisionPreparation (shrink, test and check)
5 Non-determinism (work in progress)
6 conclusion30 / 58
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Macro-Collision
Removing Unused Tables and Sending ids to Table
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Macro-Collision
Collision Rules Encoding
One rule after the other
Encoding of { 3µ1, 7µ4, 8µ5 } → { 2µ3, 4µ1 } in the direction i .
iru
le-b
ou
nd
ith
en-i
d 2it
hen
-id 2
ith
en-i
d 2
ith
en-i
d 4
iru
le-m
idd
le
iif-
id3
iif-
id7
iif-
id7
iif-
id7
iif-
id7
iif-
id8
iif-
id8
iif-
id8
iif-
id8
iif-
id8
iru
le-b
ou
nd
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Macro-Collision
Comparison of id’s in the if-part of a Rule
iru
le-mi
ddle
iru
le-mi
ddle-
fail
iif 6
iif 6
iif 6
iif-
ok 6
iif 6
iif-
ok 6
speed 6id n◦3
iif 5
iif-
ok 5
iif 5
iif-
ok 5
speed 5id n◦2
iif 4
iif-
ok 4
iif 4
iif-
ok 4
speed 4id n◦2
iru
le-bo
und
iru
le-bo
und
cross-ok4
cross4
cross-ok4
cross-ok4
cross4
cross-ok4
speed4
idn ◦
2
cross-ok6
cross6
cross-ok6
cross-ok6
cross6
cross-ok6
speed6
idn ◦
2
cross-ok5
cross5
cross-ok5
cross-ok5
cross5
cross-ok5
cross-ok5
cross5
cross-ok5
speed5
idn ◦
3
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Macro-Collision
Rule Selection
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Macro-Collision
Generating the Output
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Macro-Collision
Whole resolution
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal MachinesConcept and DefinitionGlobal SchemeMacro-CollisionPreparation (shrink, test and check)
5 Non-determinism (work in progress)
6 conclusion37 / 58
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Good Cases
main∅m
ain∅
space
tim
e
main∅m
ain∅
mai
n∅ ma
in∅m
ain∅
mai
n∅
spaceti
me
Bad Case
main ∅
mai
n∅
mai
n∅ m
ain ∅
mai
n∅
mai
n∅
space
tim
e
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Shrinking Unit
a0
a1
a2
a3
b0
b1
b2
b
b’
b
c
c’
c
a
a’
a
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Shrink
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Testing for Other main∅ Signals
(0, 0)
(si , 1)
im
ain∅
km
ain∅
j main ∅ (
si+(2−ε)smaxε
)(si+(ε−2)smax
ε
)U
(1.5εsi1.5ε
)(2εsi , 2ε)test-lefti
test-rightii
test-rightki
test-rightjitest-upi
test-right-upkitest-left-upki
test-left-okii main
oktest
test-right-okji
test-starti
check-?ji
check-
upji
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Checking the right positioning of Other main∅ Signalsim
ain∅
km
ain∅
j main ∅
im
ain
ok testtest-right-ok j
i
check-?ji
checkji
check-u
pji
check-intersec
tk,ji
check-ok ji
A
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Detecting Potential Overlaps
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Whole Preparation
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Exact 3-signal collision
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Some examplesm
1
m2
m 3 m2
m4
space
time
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Some examples
46 / 58
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Some examples
46 / 58
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Simulation between signal machines
Intrinsically Universal Family of Signal Machines
Preparation (shrink, test and check)
Some examples
46 / 58
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Simulation between signal machines
Non-determinism (work in progress)
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
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Simulation between signal machines
Non-determinism (work in progress)
Definition and example
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)Definition and exampleStrategyImplantation
6 conclusion
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Simulation between signal machines
Non-determinism (work in progress)
Definition and example
Non-determinism in rule output
Meta-signalsa 0b -1c 1
Collision rules{ a, b }→{ a, c }{ a, b }→{ b }{ c, a }→{ b }{ c, a }→{ a }
a b a
a
a c
ba
a b
ba
ac
a
ba
ca
a
ab
a c
ab
b
Shifted superposition
no collisionpossible
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-
Simulation between signal machines
Non-determinism (work in progress)
Definition and example
Non-determinism in rule output
Meta-signalsa 0b -1c 1
Collision rules{ a, b }→{ a, c }{ a, b }→{ b }{ c, a }→{ b }{ c, a }→{ a }
a b a
a
a c
ba
a b
ba
ac
a
ba
ca
a
ab
a c
ab
b
Shifted superposition
no collisionpossible
49 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Definition and example
Non-determinism in rule output
Meta-signalsa 0b -1c 1
Collision rules{ a, b }→{ a, c }{ a, b }→{ b }{ c, a }→{ b }{ c, a }→{ a }
a b a
a
a c
ba
a b
ba
ac
a
ba
ca
a
ab
a c
ab
b
Shifted superposition
no collisionpossible
49 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Definition and example
Non-determinism in rule output
Meta-signalsa 0b -1c 1
Collision rules{ a, b }→{ a, c }{ a, b }→{ b }{ c, a }→{ b }{ c, a }→{ a }
a b a
a
a c
ba
a b
ba
ac
a
ba
ca
a
ab
a c
ab
b
Shifted superposition
no collisionpossible
49 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Definition and example
Non-determinism in rule output
Meta-signalsa 0b -1c 1
Collision rules{ a, b }→{ a, c }{ a, b }→{ b }{ c, a }→{ b }{ c, a }→{ a }
a b a
a
a c
ba
a b
ba
ac
a
ba
ca
a
ab
a c
ab
b
Shifted superposition
no collisionpossible
49 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Definition and example
Non-determinism in rule output
Meta-signalsa 0b -1c 1
Collision rules{ a, b }→{ a, c }{ a, b }→{ b }{ c, a }→{ b }{ c, a }→{ a }
a b a
a
a c
ba
a b
ba
ac
a
ba
ca
a
ab
a c
ab
b
Shifted superposition
no collisionpossible
49 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Strategy
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)Definition and exampleStrategyImplantation
6 conclusion
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Simulation between signal machines
Non-determinism (work in progress)
Strategy
All possible universes
U ={
, , ,
}
Unbounded signals
Information held:
{(vα, uα)}αsuch that:
vα ∈ {�, a, b, c}⊎α uα =U
Future unknown
Distinguish
(
c,
{ })(�,{
, ,
})
(a,
{,
})(�,{
,
})
(c,{})(�,{ }){(c,{ })}{(c.1,{ 1 })}{(
c.3,{
3
})}
(
1
,
{ })(�,{
, ,
})
51 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Strategy
All possible universes
U ={
, , ,
}
Unbounded signals
Information held:
{(vα, uα)}αsuch that:
vα ∈ {�, a, b, c}⊎α uα =U
Future unknown
Distinguish
(
c,
{ })(�,{
, ,
})
(a,
{,
})(�,{
,
})
(c,{})(�,{ }){(c,{ })}{(c.1,{ 1 })}{(
c.3,{
3
})}
(
1
,
{ })(�,{
, ,
})
51 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Strategy
All possible universes
U ={
, , ,
}
Unbounded signals
Information held:
{(vα, uα)}αsuch that:
vα ∈ {�, a, b, c}⊎α uα =U
Future unknown
Distinguish
(
c,
{ })(�,{
, ,
})
(a,
{,
})(�,{
,
})
(
c,
{, ,
})(�,{ })
{(c,{ })}{(
c.1,{
1
})}{(
c.3,{
3
})}
(
1
,
{ })(�,{
, ,
})
51 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Strategy
All possible universes
U ={
, , ,
}
Unbounded signals
Information held:
{(vα, uα)}αsuch that:
vα ∈ {�, a, b, c}⊎α uα =U
Future unknown
Distinguish
(
c,
{ })(�,{
, ,
})
(a,
{,
})(�,{
,
})
(c,{ })(�,{ })
{(c,{ })}{(
c.1,{
1
})}{(
c.3,{
3
})}
(
1
,
{ })(�,{
, ,
})
51 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Strategy
All possible universes
U ={
, , ,
}
Unbounded signals
Information held:
{(vα, uα)}αsuch that:
vα ∈ {�, a, b, c}⊎α uα =U
Future unknown
Distinguish
(
c,
{ })(�,{
, ,
})
(a,
{,
})(�,{
,
})
(c,{})(�,{ })
{(c,{ })}
{(c.1,
{1
})}{(
c.3,{
3
})}
(
1
,
{ })(�,{
, ,
})
51 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Strategy
All possible universes
U ={
, , ,
}
Unbounded signals
Information held:
{(vα, uα)}αsuch that:
vα ∈ {�, a, b, c}⊎α uα =U
Future unknown
Distinguish
(
c,
{ })(�,{
, ,
})
(a,
{,
})(�,{
,
})
(c,{})(�,{ }){(c,{ })}
{(c.1,
{1
})}{(
c.3,{
3
})}
(
1
,
{ })(�,{
, ,
})
51 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Strategy
All possible universes
U ={
, , ,
}
Unbounded signals
Information held:
{(vα, uα)}αsuch that:
vα ∈ {�, a, b, c}⊎α uα =U
Future unknown
Distinguish
(
c,
{ })(�,{
, ,
})
(a,
{,
})(�,{
,
})
(c,{})(�,{ }){(c,{ })}
{(c.1,
{1
})}{(
c.3,{
3
})}
(
1
,
{ })(�,{
, ,
})
51 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Implantation
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)Definition and exampleStrategyImplantation
6 conclusion
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Simulation between signal machines
Non-determinism (work in progress)
Implantation
Macro-collision
{ (vα, uα) }α meets{(
v ′β, u′β
)}β
1 E1 ={(
(vα, v′β), uα ∧ u′β
)}α,β
U =⊎α,β uα ∧ u′β
2 E2 = { ((v , v ′), u ∧ u′) ∈ E1 | u ∧ u′ 6= ∅ }3 E3 = { (∅,w) | ((�,�),w) ∈ E2 }
∪ { ({µ},w) | ((µ,�),w) ∈ E2 ∨ ((�, µ),w) ∈ E2 }∪ { (ρ+.µ, ρ(µ, ν) ∧ w) | ((µ, ν),w) ∈ E2 ∧ ρ− = {µ, ν} }
4 Out Speed = { Speed(µ) | ∃µ, (F ,w) ∈ E3, µ ∈ F}5 ∀s ∈ out,
Outs = { (µ,w) | ∃F , (F ,w) ∈ E3 ∧ µ ∈ F , Speed(µ) = s} }∪ { (�,w) | ∃F , (F ,w) ∈ E3 ∧ ∀µ ∈ F ,Speed(µ) 6= s} }
Compatible string encodings53 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Implantation
Displaying the operations
Preparation
same as before
start
im
ain∅ j m
ain ∅
jm
ain∅
j main ∅
E1
E2
E3
54 / 58
-
Simulation between signal machines
Non-determinism (work in progress)
Implantation
Displaying the operations
Preparation
same as before
start
im
ain∅ j m
ain ∅
jm
ain∅
j main ∅
E1
E2
E3
54 / 58
-
Simulation between signal machines
conclusion
1 Introduction
2 Signal Machines (Introduction and Definition)
3 Relations to Models of Computation
4 Intrinsically Universal Family of Signal Machines
5 Non-determinism (work in progress)
6 conclusion
55 / 58
-
Simulation between signal machines
conclusion
Very rich setting
Intrinsically universal family of signal machines
What is the complexity?
Non-deterministic signal machine are not “more powerful”
How to extract “result”?
What is the complexity?
Augmented signal machines
56 / 58
-
Simulation between signal machines
conclusion
Very rich setting
Intrinsically universal family of signal machines
What is the complexity?
Non-deterministic signal machine are not “more powerful”
How to extract “result”?
What is the complexity?
Augmented signal machines
56 / 58
-
Simulation between signal machines
conclusion
Very rich setting
Intrinsically universal family of signal machines
What is the complexity?
Non-deterministic signal machine are not “more powerful”
How to extract “result”?
What is the complexity?
Augmented signal machines
56 / 58
-
Simulation between signal machines
conclusion
That’s all folks!
Thank you for your attention
57 / 58
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Simulation between signal machines
References
Albert, J. and Čulik II, K. (1987). A Simple Universal Cellular Automatonand its One-Way and Totalistic Version. Complex Systems, 1:1–16.
Das, R., Crutchfield, J. P., Mitchell, M., and Hanson, J. E. (1995).Evolving globally synchronized cellular automata. In Eshelman, L. J.,editor, International Conference on Genetic Algorithms ’95, pages336–343. Morgan Kaufmann.
Durand-Lose, J. (1997). Intrinsic Universality of a 1-DimensionalReversible Cellular Automaton. In STACS 1997, number 1200 inLNCS, pages 439–450. Springer.
Fischer, P. C. (1965). Generation of primes by a one-dimensionalreal-time iterative array. J ACM, 12(3):388–394.
Goto, E. (1966). Ōtomaton ni kansuru pazuru [Puzzles on automata]. InKitagawa, T., editor, Jōhōkagaku eno michi [The Road to informationscience], pages 67–92. Kyoristu Shuppan Publishing Co., Tokyo.
Mazoyer, J. and Rapaport, I. (1998). Inducing an Order on CellularAutomata by a Grouping Operation. In 15th Annual Symposium onTheoretical Aspects of Computer Science (STACS 1998), volume 1373of LNCS, pages 116–127. Springer.
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Simulation between signal machines
conclusion
Meunier, P., Patitz, M. J., Summers, S. M., Theyssier, G., Winslow, A.,and Woods, D. (2014). Intrinsic Universality in Tile Self-AssemblyRequires Cooperation. In Chekuri, C., editor, 25th Annual ACM-SIAMSymposium on Discrete Algorithms, SODA 2014, Portland, Oregon,USA, pages 752–771. SIAM.
Ollinger, N. (2001). Two-States Bilinear Intrinsically Universal CellularAutomata. In FCT ’01, number 2138 in LNCS, pages 369–399.Springer.
Woods, D. (2013). Intrinsic Universality and the Computational Power ofSelf-Assembly. In Neary, T. and Cook, M., editors, ProceedingsMachines, Computations and Universality 2013, MCU 2013, Zürich,Switzerland, volume 128 of EPTCS, pages 16–22.
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IntroductionSignal Machines (Introduction and Definition)Relations to Models of ComputationDiscrete computation: Turing MachinesAnalog Computation: linear Blum, Shub and Smale
Intrinsically Universal Family of Signal MachinesConcept and DefinitionGlobal SchemeMacro-CollisionPreparation (shrink, test and check)
Non-determinism (work in progress)Definition and exampleStrategyImplantation
conclusion