Simulation between signal machinesmsablik/Slide...Cellular Automata: signal use Firing Squad...

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

1 / 58


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

2 / 58


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

3 / 58


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

3 / 58


Signal Machines (Introduction and Definition)

1 Introduction





6 conclusion

4 / 58



Cellular Automata: signal use

Firing Squad Synchronization (Goto, 1966)

5 / 58



CA: Conception with signals

Fischer (1965)

6 / 58



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



SignalsT

ime

(N)

Space (Z)

Tim

e(R

+)

Space (R)

Signal (meta-signal)

Collision (rule)

8 / 58



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 }

9 / 58




div M M


M (0)div (3)

hi (1)lo (3)

back (-3)

Collision rules


{ hi, back }→{ M }

9 / 58




div M

M hi lo

M


M (0)div (3)hi (1)lo (3)

back (-3)

Collision rules

{ div, M }→{ M, hi, lo }

{ lo, M }→{ back, M }{ hi, back }→{ M }

9 / 58




div M

lo

M

M hi

backM


M (0)div (3)hi (1)lo (3)

back (-3)

Collision rules


{ hi, back }→{ M }

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div M

lo

M

hi

backM

MM


M (0)div (3)hi (1)lo (3)

back (-3)

Collision rules


{ hi, back }→{ M }

9 / 58



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

10 / 58



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

10 / 58



Fractal Generation

11 / 58



Complex Dynamics

12 / 58


Relations to Models of Computation

1 Introduction





6 conclusion

13 / 58



Discrete computation: Turing Machines

1 Introduction


3 Relations to Models of ComputationDiscrete computation: Turing MachinesAnalog Computation: linear Blum, Shub and Smale



6 conclusion

14 / 58




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

15 / 58




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.

16 / 58



Analog Computation: linear Blum, Shub and Smale

1 Introduction


3 Relations to Models of ComputationDiscrete computation: Turing MachinesAnalog Computation: linear Blum, Shub and Smale



6 conclusion

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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

18 / 58




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

19 / 58




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




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

20 / 58




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

20 / 58




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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b a a c

d1 d2 d3

-1 -1

...

b d1

...

a d2

...

a d3

...

c -3

...

-2 -2

...





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b a a c

d1 d2 d3

-1 -1 ... b d1 ... a d2 ... a d3 ... c -3 ... -2 -2 ...





21 / 58


Intrinsically Universal Family of Signal Machines

1 Introduction





6 conclusion

22 / 58



Concept and Definition

1 Introduction



4 Intrinsically Universal Family of Signal MachinesConcept and DefinitionGlobal SchemeMacro-CollisionPreparation (shrink, test and check)


6 conclusion23 / 58




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)

24 / 58




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.

25 / 58




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.

26 / 58



Global Scheme

1 Introduction





6 conclusion27 / 58



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

28 / 58



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

29 / 58



Macro-Collision

1 Introduction





6 conclusion30 / 58



Macro-Collision

Removing Unused Tables and Sending ids to Table

31 / 58



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

32 / 58



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

33 / 58



Macro-Collision

Rule Selection

34 / 58



Macro-Collision

Generating the Output

35 / 58



Macro-Collision

Whole resolution

36 / 58



Preparation (shrink, test and check)

1 Introduction





6 conclusion37 / 58




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

38 / 58




Shrinking Unit

a0

a1

a2

a3

b0

b1

b2

b

b’

b

c

c’

c

a

a’

a

39 / 58




Shrink

40 / 58




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

41 / 58




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

42 / 58




Detecting Potential Overlaps

43 / 58




Whole Preparation

44 / 58




Exact 3-signal collision

45 / 58




Some examplesm

1

m2

m 3 m2

m4

space

time

46 / 58




Some examples

46 / 58


Non-determinism (work in progress)

1 Introduction





6 conclusion

47 / 58



Definition and example

1 Introduction




5 Non-determinism (work in progress)Definition and exampleStrategyImplantation

6 conclusion

48 / 58



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



Strategy

1 Introduction





6 conclusion

50 / 58



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



Strategy


U ={

, , ,

}

Unbounded signals

Information held:


vα ∈ {�, a, b, c}⊎α uα =U

Future unknown

Distinguish

(

c,

{ })(�,{

, ,

})

(a,

{,

})(�,{

,

})

(

c,

{, ,

})(�,{ })

{(c,{ })}{(

c.1,{

1

})}{(

c.3,{

3

})}

(

1

,

{ })(�,{

, ,

})

51 / 58



Strategy


U ={

, , ,

}

Unbounded signals

Information held:


vα ∈ {�, a, b, c}⊎α uα =U

Future unknown

Distinguish

(

c,

{ })(�,{

, ,

})

(a,

{,

})(�,{

,

})

(c,{ })(�,{ })

{(c,{ })}{(

c.1,{

1

})}{(

c.3,{

3

})}

(

1

,

{ })(�,{

, ,

})

51 / 58



Strategy


U ={

, , ,

}

Unbounded signals

Information held:


vα ∈ {�, a, b, c}⊎α uα =U

Future unknown

Distinguish

(

c,

{ })(�,{

, ,

})

(a,

{,

})(�,{

,

})

(c,{})(�,{ })

{(c,{ })}

{(c.1,

{1

})}{(

c.3,{

3

})}

(

1

,

{ })(�,{

, ,

})

51 / 58



Strategy


U ={

, , ,

}

Unbounded signals

Information held:


vα ∈ {�, a, b, c}⊎α uα =U

Future unknown

Distinguish

(

c,

{ })(�,{

, ,

})

(a,

{,

})(�,{

,

})

(c,{})(�,{ }){(c,{ })}

{(c.1,

{1

})}{(

c.3,{

3

})}

(

1

,

{ })(�,{

, ,

})

51 / 58



Implantation

1 Introduction





6 conclusion

52 / 58



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



Implantation

Displaying the operations

Preparation

same as before

start

im

ain∅ j m

ain ∅

jm

ain∅

j main ∅

E1

E2

E3

54 / 58


conclusion

1 Introduction





6 conclusion

55 / 58


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


conclusion

That’s all folks!

Thank you for your attention

57 / 58


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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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.

57 / 58

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

Simulation between signal machinesmsablik/Slide...Cellular Automata: signal use Firing Squad...

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