Abstract geometrical computation: small Turing universal signal - - PowerPoint PPT Presentation

SLIDE 1

Abstract geometrical computation: small Turing universal signal machines

Abstract geometrical computation: small Turing universal signal machines

J´ erˆ

• me Durand-Lose

Laboratoire d’Informatique Fondamentale d’Orl´ eans, Universit´ e d’Orl´ eans, Orl´ eans, FRANCE

IW Complexity of Simple Programs December 6-7, 2008 – Cork, Irland

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Abstract geometrical computation: small Turing universal signal machines

1

Introduction

2

Turing machines

3

Cellular automata

4

Cyclic tag systems

5

Conclusion

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Abstract geometrical computation: small Turing universal signal machines Introduction

1

Introduction

2

Turing machines

3

Cellular automata

4

Cyclic tag systems

5

Conclusion

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Abstract geometrical computation: small Turing universal signal machines Introduction

Context

Collision based computing Idealization continuous space continuous time dimensionless particles/signals

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Abstract geometrical computation: small Turing universal signal machines Introduction

Abstract geometrical computation

Signal machines meta-signals (finitely many) their speed/velocity collision rules Signals, e.g. red (with speed 1) at position xx blue (with speed -1) at position yy Collision, e.g. rule {green, red} → {blue} application

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Abstract geometrical computation: small Turing universal signal machines Introduction

An example

Time Space

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Abstract geometrical computation: small Turing universal signal machines Introduction

More examples

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Abstract geometrical computation: small Turing universal signal machines Introduction

More complex examples

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Abstract geometrical computation: small Turing universal signal machines Turing machines

1

Introduction

2

Turing machines

3

Cellular automata

4

Cyclic tag systems

5

Conclusion

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Abstract geometrical computation: small Turing universal signal machines Turing machines

Turing machines?

Finite automata Read/write head Tape q ^ input # # . . .

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Abstract geometrical computation: small Turing universal signal machines Turing machines

Simulation

Iterations of a Turing machine

^ qf b b a b # ^ qf b b a b # ^ q2 b b a # # ^ q1 b b # # # ^ q1 b a # # # ^ q1 a a # # # ^ qi a b # # # ^ qi a b # # # ^ qi a b # # #

Corresponding signal machine

^ ^ ^ 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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Abstract geometrical computation: small Turing universal signal machines Turing machines

How many meta-signals?

Meta-signal 1 symbol

• 1

1 state

• 2

for finitness #, ← − # , − → # , − → # 4 |Γ| + 2|Q| + 4 Results universal semi-universal 18 (Woods and Neary, 2007) 7 (Smith, 2007)

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Abstract geometrical computation: small Turing universal signal machines Cellular automata

1

Introduction

2

Turing machines

3

Cellular automata

4

Cyclic tag systems

5

Conclusion

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Abstract geometrical computation: small Turing universal signal machines Cellular automata

Cellular automata

Rule 110 and one transition implementation

Output 1 1 1 1 1 Input 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0

z e r

• R
• ne

z e r

• L
• n

e

L

• ne
• n

e

R

Evolution and simulation on 11 framed by ω(10) and (011)ω

0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 0 1

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Abstract geometrical computation: small Turing universal signal machines Cellular automata

How many meta-signals?

Meta-signal 1 state

• 3

for finitness Regular pattern on both side expensive 3|Q|+??? Results universal semi-universal not interesting 6 (Cook, 2004)

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Abstract geometrical computation: small Turing universal signal machines Cellular automata

Link CA-ACG

in a c hromosome. This denes

• ne

generation

• f

the GA; it is rep eated G times for

• ne

GA run. F I () is a random v ariable since its v alue dep ends

• n

the particular set

• f

I ICs selected to ev aluate . Th us, a CA's tness v aries sto c hastically from generation to generation. F

• r

this reason, w e c ho

• se

a new set

• f

ICs at eac h generation F

• r
• ur

exp erimen ts w e set P = 100, E = 20; I = 100, m = 2; and G = 50. M w as c hosen from a P

• isson

distribution with mean 320 (sligh tly greater than 2N ). V arying M prev en ts selecting CAs that are adapted to a particular M . A justication

• f

these parameter settings is giv en in [9]. W e p erformed a total

• f

65 GA runs. Since F 100 () is

• nly

a rough estimate

• f

p erformance, w e more stringen tly measured the qualit y

• f

the GA's solutions b y calculating P N 10 4 () with N 2 f149; 599; 999g for the b est CAs in the nal generation

• f

eac h run. In 20%

• f

the runs the GA disco v ered successful CAs (P N 10 4 = 1:0). More detailed analysis

• f

these successful CAs sho w ed that although they w ere distinct in detail, they used similar strategies for p erforming the sync hronization task. In terestingly , when the GA w as restricted to ev

• lv

e CAs with r = 1 and r = 2, all the ev

• lv

ed CAs had P N 10 4

• for

N 2 f149; 599; 999 g. (Better p erforming CAs with r = 2 can b e designed b y hand.) Th us r = 3 app ears to b e the minimal radius for whic h the GA can successfully solv e this problem.

β γ δ ν β γ (a) Space-time diagram. (b) Filtered space-time diagram.

Site 74 Site 74 74 Time

α γ δ µ

Figure 1: (a) Space-time diagram

• f
• sy

nc 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 to blac k. Greek letters lab el particles describ ed in the text. Figure 1a giv es a space-time diagram for

• ne
• f

the GA-disco v ered CAs with 100% p erfor- mance, here called

• sy

nc . This diagram plots 75 successiv e congurations

• n

a lattice

• f

size N = 75 (with time going do wn the page) starting from a randomly c hosen IC, with 1-sites col-

• red

blac k and 0-sites colored white. In this example, global sync hronization

• ccurs

at time step 58. Ho w are w e to understand the strategy emplo y ed b y

• sy

nc to reac h global sync hronization? Notice that, under the GA, while crosso v er and m utation act

• n

the lo cal mappings comprising a 4

c Das-Crutchfield-Mitchell-Hanson95

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

1

Introduction

2

Turing machines

3

Cellular automata

4

Cyclic tag systems

5

Conclusion

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Cyclic tag system?

Definition a binary word a circular list Dynamics 101

011 :: h :: 0110 :: 01011

Halt empty word halt appendant (here h) cycle (too expensive to test)

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Cyclic tag system?

Definition a binary word a circular list Dynamics 101011

011 :: h :: 0110 :: 01011 :: 011

Halt empty word halt appendant (here h) cycle (too expensive to test)

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Cyclic tag system?

Definition a binary word a circular list Dynamics 101011

011 :: h :: 0110 :: 01011 :: 011 :: h

Halt empty word halt appendant (here h) cycle (too expensive to test)

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Cyclic tag system?

Definition a binary word a circular list Dynamics 1010110110

011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110

Halt empty word halt appendant (here h) cycle (too expensive to test)

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Cyclic tag system?

Definition a binary word a circular list Dynamics 1010110110

011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110 :: 01011

Halt empty word halt appendant (here h) cycle (too expensive to test)

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Cyclic tag system?

Definition a binary word a circular list Dynamics 1010110110011

011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110 :: 01011:: 011

Halt empty word halt appendant (here h) cycle (too expensive to test)

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Cyclic tag system?

Definition a binary word a circular list Dynamics 1010110110011

011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110 :: 01011:: 011 :: h

Halt empty word halt appendant (here h) cycle (too expensive to test)

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

Simulation

101 and 011 :: 10 :: 10 :: 01 initial configuration

last goLL go

R R

last

• n

eR

• ne

zero

• ne

first zero

• ne
• ne

sep

• ne

zero sep

• ne

zero sep zero

• ne

last

• ne iteration

101 & 011 :: h :: 0110 :: 01011

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Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems

How many meta-signals?

Universality 13 meta-signals 21 non-blank rules

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Abstract geometrical computation: small Turing universal signal machines Conclusion

1

Introduction

2

Turing machines

3

Cellular automata

4

Cyclic tag systems

5

Conclusion

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Abstract geometrical computation: small Turing universal signal machines Conclusion

Conclusion

Bounds universal semi-universal 13 6 Future work lower this bound bounds for

accumulation (3 or 4) black hole implementation (13 ? 14 ? more) analog computation (BSS or computable analysis)