Estimating the robustness to asynchronism for cellular automata - - PowerPoint PPT Presentation

Cellular Automata Workshop – Gdansk – September 2005

Estimating the robustness to asynchronism for cellular automata models

Nazim Fatès

ENS LYON – Laboratoire de l’informatique du parallélisme – Modèles de calcul et complexité

joint work with Michel Morvan, Nicolas Schabanel, Eric Thierry, Damien Regnault

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Introduction 1950s : von Neumann and Ulam, “non-trivial self-reproduction”.

• ✁

• ❍

But one possible criticism : assumption of perfect synchony What happens to a cellular automaton when the cells are not iterated synchronously ? History : – exp. : Ingerson & Buvel (Physica D 10) – exp. : Schönfish – th : Gacs, Louis

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Methodology A first work has been done by simulation techniques presented in Automata 2003 (Leuven), to appear in Complex Systems → need of better evaluation of the asymptotic behaviour How long do I run simulations ? How long does it take for a perturbed system to re-stabilise ? e.g. computers linked with a ring topology obeying local constraints (tokens)

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Framework : ECA We restrict our study to “elementary cellular automata”.

a c f(a,b,c) a b

... ... ... ...

set of states : Q = {0, 1}, size of the ring : n, configuration : x ∈ QZ/nZ ECA : f : Q3 → Q double quiescent : f(0, 0, 0) = 0 & f(1, 1, 1) = 1

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Asynchronous ECA (1)

• Def. partial asynchronous behaviour :

at each time step t, each cell has a probability α to be updated ∀x ∈ Z/nZ, xt+1

i

=    f(xt

i−1, xt i, xt i+1)

if p(α) = 1 xt

i

• therwise

...

TIME

x^1 x^2 x^0

• ✁

.

α : synchrony rate , α → 1 classical case, ACA ⊂ PCA

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Asynchronous ECA (2)

• Def. fully asynchronous behaviour :

at each time step t,

• ne cell ωt is chosen at random, uniformly in Z/nZ

∀x ∈ Z/nZ, xt+1

i

=    f(xt

i−1, xt i, xt i+1)

if i = ωt xt

i

• therwise

Real time or limiting case for α → 0.

TIME

x^n x^2n

...

x^0

• ✁

.

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Convergence time → Predicting behaviour of CA : difficult problem in with synchronous updating Asynchronous mode ? We no longer have "cycles". We are interested in trying to precict the behaviour of ACA, i.e., : – What are the rechable fixed points ? – How frequent do we reach a fixed point ? – What is the average time of convergence to a fixed point ?

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Main result For fully asynchronous beaviour, among the 64 DQECA, – 55 converge a.s. to a fixed point for all initial configuration, – 9 never reach a fixed point for a non fixed point initial configuration. Among the converging rules, the worst expected convergence times are : 0, Θ(n ln n), Θ(n2), Θ(n3) and Θ(n2n). For partially asynchronous behaviour, the converging times are more diverse and the proofs are more ad hoc. Two rules are not tractable with our analytical methods.

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Behaviour ACE ( # ) Rule 01 01 01 10 10 10 010 010 010 101 101 101 convergence Identity 204 ( 1 ) ∅ · · · · Coupon collector 200 ( 2 ) E · · + · Θ(n ln n) 232 ( 1 ) DE · · + + Monotone 206 ( 4 ) B ← · · · Θ(n2) 132 ( 2 ) BC ← → · · 234 ( 4 ) BDE ← · + + 250 ( 2 ) BCDE ← → + + 202 ( 4 ) BE ← · + · 192 ( 4 ) EF → · + · 218 ( 2 ) BCE ← → + · 128 ( 2 ) EFG → ← + · Biased Random Walk 242 ( 4 ) BCDEF

• →

+ + 130 ( 4 ) BEFG

• ←

+ · Random Walk 226 ( 2 ) BDEF

• ·

+ + Θ(n3) 170 ( 2 ) BDEG ← ← + + 178 ( 1 ) BCDEFG

• +

+ 194 ( 4 ) BEF

• ·

+ · 138 ( 4 ) BEG ← ← + · 146 ( 2 ) BCEFG

• +

· Biased Random Walk 210 ( 4 ) BCEF

• →

+ · Ω(n2n) No convergence to Fixed point 198 ( 2 ) BF

• ·

· · no convergence 142 ( 2 ) BG ← ← · · 214 ( 4 ) BCF

• →

· · 150 ( 1 ) BCFG

• ·

·

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Notation Classical notation (Wolfram) : e.g. majority = ECA 232 Alternative notation : each active transition is labelled with a letter :

A C F H E B G D

Main idea : from code to "regions" – A and H : never present – D and E : fusion of regions – B and F : frontiers 01 – C and G : frontiers 10

D B

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The majority (fully asynchronous case)

A C F H E B G D

i active sites : p = i/n and Ti = n/i majoration of cv. time : T ≤ n

n + n n−1 + · · · + n 1 ∼ n ln n 11

Behaviour ACE ( # ) Rule 01 01 01 10 10 10 010 010 010 101 101 101 convergence Identity 204 ( 1 ) ∅ · · · · Coupon collector 200 ( 2 ) E · · + · Θ(n ln n) 232 ( 1 ) DE · · + + Monotone 206 ( 4 ) B ← · · · Θ(n2) 132 ( 2 ) BC ← → · · 234 ( 4 ) BDE ← · + + 250 ( 2 ) BCDE ← → + + 202 ( 4 ) BE ← · + · 192 ( 4 ) EF → · + · 218 ( 2 ) BCE ← → + · 128 ( 2 ) EFG → ← + · Biased Random Walk 242 ( 4 ) BCDEF

• →

+ + 130 ( 4 ) BEFG

• ←

+ · Random Walk 226 ( 2 ) BDEF

• ·

+ + Θ(n3) 170 ( 2 ) BDEG ← ← + + 178 ( 1 ) BCDEFG

• +

+ 194 ( 4 ) BEF

• ·

+ · 138 ( 4 ) BEG ← ← + · 146 ( 2 ) BCEFG

• +

· Biased Random Walk 210 ( 4 ) BCEF

• →

+ · Ω(n2n) No convergence to Fixed point 198 ( 2 ) BF

• ·

· · no convergence 142 ( 2 ) BG ← ← · · 214 ( 4 ) BCF

• →

· · 150 ( 1 ) BCFG

• ·

·

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Rules with a Lyapunov function DEFG = 160 BEFG = 130 Lemma (simplified version) : If there exists (Xt) with values in {0, . . . , n} with E[∆Xt] ≤ −1/n, then the process converges in time Ω(n2). Finding an "energy function" allows to bound convergence time for many rules with both type of asynchronism.

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Behaviour ACE ( # ) Rule 01 01 01 10 10 10 010 010 010 101 101 101 convergence Identity 204 ( 1 ) ∅ · · · · Coupon collector 200 ( 2 ) E · · + · Θ(n ln n) 232 ( 1 ) DE · · + + Monotone 206 ( 4 ) B ← · · · Θ(n2) 132 ( 2 ) BC ← → · · 234 ( 4 ) BDE ← · + + 250 ( 2 ) BCDE ← → + + 202 ( 4 ) BE ← · + · 192 ( 4 ) EF → · + · 218 ( 2 ) BCE ← → + · 128 ( 2 ) EFG → ← + · Biased Random Walk 242 ( 4 ) BCDEF

• →

+ + 130 ( 4 ) BEFG

• ←

+ · Random Walk 226 ( 2 ) BDEF

• ·

+ + Θ(n3) 170 ( 2 ) BDEG ← ← + + 178 ( 1 ) BCDEFG

• +

+ 194 ( 4 ) BEF

• ·

+ · 138 ( 4 ) BEG ← ← + · 146 ( 2 ) BCEFG

• +

· Biased Random Walk 210 ( 4 ) BCEF

• →

+ · Ω(n2n) No convergence to Fixed point 198 ( 2 ) BF

• ·

· · no convergence 142 ( 2 ) BG ← ← · · 214 ( 4 ) BCF

• →

· · 150 ( 1 ) BCFG

• ·

·

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The shift (1)

G B D E H F C A

BDEG = 170

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The shift (2) easy case : the configuration contains only two regions +

• 1

1 1 1 1 1 1 1 state = number of cells in state 1

ε ε ε ε ε 1 1 1 2 n−1 n 1−2ε 1−2ε 1−2ε ε

ǫ = 1/n. First-step analysis gives : T ≥ X0(n−X0)

2ǫ

∼ n3 and pi = X0/n = d0 X0 is the size of the inital black (or white) region

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The shift (3) More than one region ? +

• +
• +
• +
• 1

1 1 1 1 1 1 Lemma (simple version) : If a process Xt is a martingale on {0, · · · , n} if Pr [|∆Xt| ≥ 1] is higher than 1/n , → the convergence time of the process is in Ω(n3). This lemma allows the generalisation for five other DQECA using different process Xt.

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Behaviour ACE ( # ) Rule 01 01 01 10 10 10 010 010 010 101 101 101 convergence Identity 204 ( 1 ) ∅ · · · · Coupon collector 200 ( 2 ) E · · + · Θ(n ln n) 232 ( 1 ) DE · · + + Monotone 206 ( 4 ) B ← · · · Θ(n2) 132 ( 2 ) BC ← → · · 234 ( 4 ) BDE ← · + + 250 ( 2 ) BCDE ← → + + 202 ( 4 ) BE ← · + · 192 ( 4 ) EF → · + · 218 ( 2 ) BCE ← → + · 128 ( 2 ) EFG → ← + · Biased Random Walk 242 ( 4 ) BCDEF

• →

+ + 130 ( 4 ) BEFG

• ←

+ · Random Walk 226 ( 2 ) BDEF

• ·

+ + Θ(n3) 170 ( 2 ) BDEG ← ← + + 178 ( 1 ) BCDEFG

• +

+ 194 ( 4 ) BEF

• ·

+ · 138 ( 4 ) BEG ← ← + · 146 ( 2 ) BCEFG

• +

· Biased Random Walk 210 ( 4 ) BCEF

• →

+ · Ω(n2n) No convergence to Fixed point 198 ( 2 ) BF

• ·

· · no convergence 142 ( 2 ) BG ← ← · · 214 ( 4 ) BCF

• →

· · 150 ( 1 ) BCFG

• ·

·

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Rule BCEF (1) BCEF = 210

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Rule BCEF (2) 1−3ε 1−3ε

2ε 2ε

1−ε

n n−1 2 1

1 1 ε

ε

ε

Two regions : – there is only one reachable fixed point <0> – but the number of 1s is increasing on average – except when it is equal to the maximum value n − 1

• > biased random walk in the “wrong” direction T ≥ n.2n

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Rule BCEF (3) Multi regions fully asynchronous : coupling technique

Original process Coupled process alignement update

Fully asynchronous mode : T = Θ(n.2n) Partial asynchronous mode : convergence time is polynomial. → Importance of analytical tools but some rules are out of reach.

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Phase transitions ? macroscopic description of the system : f e.g. density or number of active sites control parameter : p e.g. temperature, density of particules or cars 1st order f shows a discontinuity water, percolation 2nd order f is continuous but its de- rivative is not Ising, percolation PT identification : f ∼ [p − pc]λ critical exponents & universality class

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Asynchronous ECA 50

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Synchrony Rate ECA50 50 100 200 1000 10000

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Asynchronous ECA 58

0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Synchrony Rate ECA58 50 100 200 1000 10000

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Asynchronous ECA 6

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Synchrony Rate ECA6 50 100 200 1000 10000

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Directed percolation ? We proceed in two steps : For critical rate αc, theory predicts d ∼ t−δ δ ∼ 0.159 For upper-criticial rate, we are to measure d = K(α − αc)β β = 0.266 irrational number ? → for six ECA good agreement with DP theory

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Synthesis

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Coupon collectors Lyapunov functions (Quasi−)martingales Coupling n^2 n^3 n.ln(n) n.2^n applications: Dictostelium discoidum, robust computer networks "Phase transitions" phenomena Modelling synchronization phenomena

Partial asynchronism Extension of states

Transition Code Two−regions : Markov chains & 1st step analysis Movments and fusion of regions

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