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 1

Introduction 1950s : von Neumann and Ulam, “non-trivial self-reproduction”. ✳ ✳ ✴ ✴ ✳ ✳ ✴ ✴ ✳ ✳ ✷ ✷ ✴ ✴ ✳ ✳ ✻ ✻ ✷ ✷ ✴ ✴ ✳ ✳ ✷ ✷ ✴ ✴ ❅ ❅ ❃ ❃ ✹ ✹ ✼ ❘ ✼ ❘ ✱ ✵ ✱ ✵ ❆ ❆ ❄ ❄ ✺ ✺ ❙ ✻ ✻ ❙ ✶ ✲ ✶ ✲ ✷ ✸ ✷ ✸ ❅ ❅ ❃ ❃ ✹ ✹ ❘ ❘ ✱ ✵ ✱ ✵ ❆ ❆ ❄ ❄ ✺ ✺ ❙ ❙ ✲ ✶ ✲ ✶ ✸ ✸ ❅ ❅ ❃ ❃ ✹ ✹ ❘ ✼ ✼ ❘ ✵ ✱ ✵ ✱ ❆ ❆ ❄ ❄ ✺ ✺ ❙ ❙ ✲ ✶ ✶ ✲ ✸ ✸ ❅ ❅ ❃ ❃ ✹ ✹ ❘ ❘ ✵ ✱ ✵ ✱ ❆ ❆ ❄ ❄ ✺ ✺ ❙ ❙ ✲ ✂ ✶ ✲ ✂ ✶ ✸ ✸ ❇ ❇ ❂ ✄ ✿ ❁ ✄ ❁ ❂ ✿ ✜ ✜ ✫ ✫ ❈ ❈ ✞ ✝ ✝ ✞ ❀ ❀ ✢ ✢ ✘ ✙ ✙ ✘ ✬ ✬ ❇ ❇ ✿ ✿ ✜ ✜ ✫ ✫ ✝ ✝ ✘ ✘ ❈ ❈ ✞ ✞ ❀ ❀ ✢ ✢ ✙ ✙ ✬ ✬ ◆ ◆ ▲ ▲ � ❏ ❏ � ❇ ❇ ❖ ❖ ▼ ▼ ❑ ✁ ❑ ✁ ✿ ✂ ❁ ✿ ✂ ❁ ✜ ✜ ✫ ✫ ✝ ✝ ❂ ✄ ✄ ❂ ✘ ✘ ❈ ❈ ✞ ✞ ❀ ❀ ✢ ✢ ✙ ✙ ✬ ✬ ❉ ❉ ☎ ☎ ✚ ✚ ✽ ✽ ❇ ❇ ✿ ✿ ✜ ✜ ✫ ✫ ❊ ❊ ✆ ✆ ✝ ✝ ✗ ✗ ✘ ✛ ✛ ✘ ✾ ✾ ❈ ❈ ✞ ✞ ❀ ❀ ✢ ✢ ✙ ✙ ✬ ✬ ❉ ❉ ❖ ◆ ◆ ❖ ☎ ▲ ▼ ☎ ▲ ▼ ❑ ❏ � ✁ ❑ � ❏ ✁ ✚ ✚ ✽ ✽ ✖ ✖ ❊ ❊ ✆ ✆ ✗ ✗ ✛ ✛ ✾ ✾ ❉ ❉ ☎ ☎ ✚ ✚ ✽ ✽ ✖ ✖ ❊ ❊ ✆ ✆ ✗ ✗ ✛ ✛ ✾ ✾ ❉ ❉ ❖ ◆ ❖ ◆ ▲ ☎ ▼ ▲ ☎ ▼ ✁ � ❏ ❑ � ✁ ❑ ❏ ✚ ✚ ✽ ✽ ✖ ✖ ❊ ❊ ✆ ✆ ✗ ✗ ✛ ✛ ✾ ✾ ❉ ❉ ☎ ☎ ☞ ☞ ✚ ✚ ✽ ✽ ✖ ✖ ❊ ❊ ✆ ✆ ✗ ✗ ✛ ✛ ✾ ✾ ✒ ✓ ✓ ✒ ✔ P ◗ ◗ ✔ P ✟ ❍ ■ ✟ ❍ ■ ✥ ✥ ✣ ✣ ✕ ✕ ✠ ✠ ✖ ✖ ✦ ✦ ✤ ✤ ✔ ✔ ✟ ✟ ☞ ☞ ✥ ✥ ✣ ✣ ✕ ✕ ✠ ✠ ✦ ✦ ✤ ✤ ✏ ✏ ✍ ✍ ✯ ✯ ❵ ❫ ✭ ✭ ❫ ❵ ✒ ✑ ✒ ✑ ✎ ✔ P P ✔ ✎ ✟ ✰ ❍ ❍ ✰ ✟ ❛ ✌ ❴ ✮ ✌ ❛ ❴ ✮ ✥ ✥ ✣ ✣ ✓ ✓ ◗ ◗ ■ ■ ✕ ✕ ✠ ✠ ✦ ✦ ✤ ✤ ✡ ✡ ❱ ✧ ❱ ✧ ✩ ✩ ✔ ✔ ✟ ✟ ✥ ✥ ✣ ✣ ☛ ☛ ☞ ☞ ★ ❲ ★ ❲ ✪ ✪ ✕ ✕ ✠ ✠ ✦ ✦ ✤ ✤ ✑ ✏ ✏ ✑ ✍ ✎ ✍ ✎ ✡ ✰ ✯ ✰ ✯ ✡ ✭ ❴ ✮ ❵ ❛ ✌ ❫ ❛ ❵ ❴ ✮ ✭ ✌ ❫ ✧ ❱ ✧ ❱ ✩ ✩ ☛ ☛ ❲ ★ ★ ❲ ✪ ✪ ✡ ✡ ✧ ❱ ❱ ✧ ✩ ✩ ☛ ☛ ★ ❲ ❲ ★ ✪ ✪ ❋ ● ❋ ● ✏ ✑ ✏ ✑ ✍ ✎ ✎ ✍ ✯ ✡ ✰ ✡ ✰ ✯ ✮ ❴ ✭ ❛ ❵ ❫ ✌ ❛ ❫ ❵ ✌ ✭ ✮ ❴ ❱ ✧ ✧ ❱ ✩ ✩ ☛ ☛ ❲ ★ ★ ❲ ✪ ✪ ❳ ❳ ✡ ✡ ❱ ✧ ❱ ✧ ✩ ✩ ❨ ❨ ☛ ☛ ★ ❲ ❲ ★ ✪ ✪ ❋ ● ❋ ● ❳ ❳ ❨ ❨ ❳ ❳ ❨ ❨ ❋ ● ● ❋ ❳ ❳ ❨ ❨ ❳ ❳ ❨ ❨ ❯ ❚ ❪ ❭ ❭ ❪ ❯ ❚ ❩ ❩ ❬ ❬ ❩ ❩ ❬ ❬ ❚ ❯ ❪ ❭ ❪ ❯ ❭ ❚ ❩ ❩ ❬ ❬ ❩ ❩ ❬ ❬ 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 2

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) 3

Framework : ECA We restrict our study to “elementary cellular automata”. ... ... a b c a ... ... f(a,b,c) set of states : Q = { 0 , 1 } , size of the ring : n , configuration : x ∈ Q Z /n Z ECA : f : Q 3 → Q double quiescent : f (0 , 0 , 0) = 0 & f (1 , 1 , 1) = 1 4

Asynchronous ECA (1) Def. partial asynchronous behaviour : at each time step t , each cell has a probability α to be updated  f ( x t i − 1 , x t i , x t i +1 ) if p ( α ) = 1  ∀ x ∈ Z /n Z , x t +1 = i x t otherwise  i � ✁ ✁ � ✞ ✝ ✞ ✝ ✆ ✎ ✆ ✎ ✠ ✟ ✠ ✟ ✡ ☛ ✡ ☛ ☎ ✍ ☎ ✍ � � ✝ ✝ ✟ ✟ ✡ ✡ ✁ ✁ ✞ ✞ ✆ ✎ ✆ ✎ ✠ ✠ ☛ ☛ ✍ ☎ ☎ ✍ ✁ � � ✁ ✝ ✞ ✞ ✝ ✆ ✎ ✆ ✎ ✠ ✟ ✠ ✟ ☛ ✡ ☛ ✡ ✍ ☎ ✍ ☎ � � ✝ ✝ ✟ ✟ ✡ ✡ ✁ ✁ ✞ ✞ ✎ ✆ ✆ ✎ ✠ ✠ ☛ ☛ ☎ ✍ ✍ ☎ � ✁ ✁ � ✝ ✞ ✝ ✞ ✎ ✆ ✎ ✆ ✠ ✟ ✠ ✟ ✡ ☛ ✡ ☛ ✂ ✂ ☞ ☞ ✏ ✏ ✄ ✄ ✌ ✌ ✍ ✑ ☎ ☎ ✑ ✍ ✂ ✄ ✄ ✂ ☞ ✌ ☞ ✌ ✏ ✑ ✏ ✑ ✂ ✂ ☞ ☞ ✏ ✏ ✄ ✄ ✌ ✌ ✑ ✑ ✒ ✓ ✓ ✒ ✗ ✖ ✖ ✗ ✼ ✼ ✔ ✕ ✔ ✕ ✂ ✄ ✂ ✄ ☞ ✌ ☞ ✌ ✏ ✑ ✑ ✏ ✻ ✻ ✒ ✒ ✖ ✖ ✔ ✔ ✓ ✓ ✗ ✗ ✼ ✼ ✕ ✕ ✻ ✻ ✓ ✒ ✒ ✓ ✖ ✗ ✖ ✗ ✼ ✼ ✕ ✔ ✕ ✔ ✒ ✒ ✖ ✖ ✻ ✻ ✔ ✔ ✓ ✓ ✗ ✗ ✼ ✼ ✕ ✕ ✻ ✻ ✓ ✒ ✓ ✒ ✗ ✖ ✖ ✗ ✼ ✼ ✔ ✕ ✔ ✕ ✵ ✵ ✷ ✷ ✹ ✹ ✿ ✿ ❁ ❁ ✽ ✽ ✶ ✶ ✸ ✸ ✺ ✺ ❀ ❀ ❂ ❂ ✻ ✻ ✾ ✾ ... ✶ ✵ ✶ ✵ ✷ ✸ ✷ ✸ ✺ ✹ ✺ ✹ ❀ ✿ ❀ ✿ ❁ ❂ ❂ ❁ ✽ ✾ ✽ ✾ ✵ ✵ ✷ ✷ ✹ ✹ ✿ ✿ ❁ ❁ ✽ ✽ ✶ ✶ ✸ ✸ ✺ ✺ ❀ ❀ ❂ ❂ ✾ ✾ . ✘ ✙ ✙ ✘ ✢ ✜ ✢ ✜ ✤ ✣ ✤ ✣ ✛ ✚ ✛ ✚ ✵ ✶ ✵ ✶ ✷ ✸ ✷ ✸ ✹ ✺ ✹ ✺ ✿ ❀ ✿ ❀ ❂ ❁ ❂ ❁ ✽ ✾ ✾ ✽ ✘ ✘ ✜ ✜ ✣ ✣ ✚ ✚ ✙ ✙ ✢ ✢ ✤ ✤ ✛ ✛ ✘ ✙ ✙ ✘ ✢ ✜ ✜ ✢ ✣ ✤ ✣ ✤ ✛ ✚ ✛ ✚ x^2 ✙ ✘ ✘ ✙ ✜ ✢ ✢ ✜ ✤ ✣ ✣ ✤ ✛ ✚ ✚ ✛ ✘ ✙ ✘ ✙ ✜ ✢ ✢ ✜ ✤ ✣ ✤ ✣ ✚ ✛ ✚ ✛ TIME ✥ ✥ ❃ ❃ ✧ ✧ ✩ ✩ ✫ ✫ ✦ ✦ ❄ ❄ ★ ★ ✪ ✪ ✬ ✬ ✦ ✥ ✦ ✥ ❄ ❃ ❃ ❄ ✧ ★ ★ ✧ ✪ ✩ ✩ ✪ ✫ ✬ ✬ ✫ x^1 ✥ ✥ ❃ ❃ ✧ ✧ ✩ ✩ ✫ ✫ ✦ ✦ ❄ ❄ ★ ★ ✪ ✪ ✬ ✬ ✮ ✭ ✭ ✮ ✰ ✯ ✰ ✯ ✱ ✲ ✱ ✲ ✴ ✳ ✳ ✴ ✦ ✥ ✥ ✦ ❄ ❃ ❃ ❄ ✧ ★ ✧ ★ ✩ ✪ ✩ ✪ ✫ ✬ ✬ ✫ ✭ ✮ ✭ ✮ ✰ ✯ ✰ ✯ ✲ ✱ ✲ ✱ ✴ ✳ ✳ ✴ ✮ ✭ ✭ ✮ ✰ ✯ ✰ ✯ ✲ ✱ ✱ ✲ ✳ ✴ ✴ ✳ x^0 ✮ ✭ ✭ ✮ ✯ ✰ ✰ ✯ ✱ ✲ ✲ ✱ ✴ ✳ ✴ ✳ ✭ ✭ ✯ ✯ ✱ ✱ ✳ ✳ ✮ ✮ ✰ ✰ ✲ ✲ ✴ ✴ α : synchrony rate , α → 1 classical case, ACA ⊂ PCA 5