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1 Cellular Aut omat a A dynamic syst em I nvent ed by J ohn von - PDF document

Jan 15, 2023 •93 likes •259 views

Last t ime Nonlinear dynamic syst ems The Logist ic map St range at t ract or s The Hnon at t ract or The Lor enz at t r act or Producer-consumer dynamics Equat ion-based modeling I ndividual-based modeling 16/11

  1. Last t ime ❒ Nonlinear dynamic syst ems ❍ The Logist ic map ❒ St range at t ract or s ❍ The Hénon at t ract or ❍ The Lor enz at t r act or ❒ Producer-consumer dynamics ❍ Equat ion-based modeling ❍ I ndividual-based modeling 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 1 Out line f or t oday ❒ Cellular aut omat a ❍ One-dimensional ❍ Wolf r am’s classif icat ion ❍ Langt on’s lambda par amet er ❍ Two-dimensional • Conway’s Game of Lif e ❒ Pat t ern f ormat ion in slime molds ❍ Dict yost elium discoideum ❍ Modeling of pat t er n 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 2 Complex Syst em ❒ Things t hat consist of many similar and simple par t s ❍ Of t en easy t o under st and t he part s ❍ The global behavior much harder t o explain ❍ On many levels ❍ Some ar e capable of univer sal comput at ion 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 3 1

  2. Cellular Aut omat a ❒ A dynamic syst em ❒ I nvent ed by J ohn von Neumann ❍ Wit h help f r om St anislaw Ulam ❍ 1940s ❍ Want ed t o under st and t he pr ocess of r eproduct ion ❍ The essence 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 4 One-Dimensional CA ❒ Linear gr id of cells ❒ Each cell can be in one of k dif f erent st at es ❒ Next st at e is comput ed as an f unct ion of t he st at es of neighbors (and own st at e) ❒ Neighborhood ❍ r = radius � neighborhood = 2 r + 1 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 5 1-Dim CA - Example ❒ k = 2, r = 1 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 6 2

  3. 1-Dim CA - Gener al ❒ Neighbor hood size = 2 r + 1 ❒ k dif f er ent st at es ❒ � a rule t able wit h k 2r + 1 ent r ies ❒ � number of legal r ule t ables, k ^ k 2r + 1 ❒ Usally wr ap-ar ound of t he linear gr id ❒ I nit ial populat ion? ❍ Random or a f ew ”on” 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 7 Wolf ram’s CA Classif icat ion ❒ St ephen Wolf ram ❒ 1980s ❒ Resurrect ed cellular aut omat a research ❒ A ring of n cells wit h k possible st at es � k n dif f erent conf igurat ions of a row ❒ Four dif f erent CA classes 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 8 Wolf ram’s CA – Class 1 ❒ St at ic ❒ Compared t o f ract als – f ixed point 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 9 3

  4. Wolf ram’s CA – Class 2 ❒ Per iodic ❒ Compared t o f ract als – limit cycles 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 10 Wolf ram’s CA – Class 3 ❒ Random-like ❒ Compared t o f ract als – chaos, inst able limit cycles 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 11 Wolf ram’s CA – Class 4 ❒ Complex pat t erns wit h local st r uct ures ❒ Can perf or m comput at ion, some even universal comput at ion ❒ Not regular, periodic or r andom ❍ Bet ween chaos and periodicit y 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 12 4

  5. Langt on’s Lambda Par amet er ❒ Chr is Langt on – ”Founder ” of Ar t if icial Lif e ❒ Was sear ching f or a vir t ual knob t o cont rol t he behavior of a CA ❒ Quiescent st at e – inact ive, of f ❒ Number of ent r ies in a r ule t able, N = k 2r + 1 λ = ( N – n q )/ N ❒ λ = 0 � t he most homogeneous r ule t able ❒ λ = 1 � all r ules map t o non-quiescent st at es ❒ λ = 1 – 1/ k � t he most het erogeneous 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 13 Langt on’s λ - Example ❒ Comparing wit h t he examples of Wolf ram’s classes ❒ The most het erogeneous: ❍ λ = 1 - 1/ k = 1 – 1/ 5 = 0.8 ❒ Class 1: λ = 0.22823267 (average) ❒ Class 2: λ = 0.43941967 (average, biased) ❒ Class 3: λ = 0.8164867 (average) ❒ Class 4: λ = 0.501841 (aver age) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 14 Langt on’s λ - Paramet er 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 15 5

  6. Langt on’s λ - Problems One rule set can have a high λ but st ill 1. produce very simple behavior 2. Lit t le inf ormat ion in a singular value 3. λ says not hing cert ain about t he long- t erm behavior Dangerous t o map t o a single scalar ❒ number 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 16 2-Dim CA – Conway’s Game of Lif e ❒ J ohn Conway, 1960s ❒ Want ed t o f ind t he simplest CA t hat could support univer sal comput at ion ❒ k = 2, very simple rules ❒ 1970, Mart in Gander described Conway’s work in his Scient if ic American column ❒ A global collaborat ive ef f ort succeeded 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 17 Conway’s Game of Lif e ❒ 2-dimensional ❒ 8 neighbor s ❒ Rules: ❍ Loneliness: Less t han t wo neighbors, die ❍ Overcrowding: More t han t hree neighbors, die ❍ Reproduct ion: Empt y cell wit h t hree neighboors, live ❍ St asis: Exact t wo neighboors, st ay t he same 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 18 6

  7. Conway’s Game of Lif e – Univer sal comput at ion ❒ St at ic obj ect s � memor y 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 19 Conway’s Game of Lif e – Univer sal comput at ion ❒ Per iodic obj ect s � count ers 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 20 Conway’s Game of Lif e – Univer sal comput at ion ❒ Moving obj ect s � moving inf ormat ion 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 21 7

  8. Conway’s Game of Lif e – Univer sal comput at ion ❒ Br eeder s, glider guns � collide t o make new moving obj ect s 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 22 Conway’s Game of Lif e – Univer sal comput at ion ❒ To implement univer sal comput at ion ❍ NOT and (AND or OR) ❒ Comput ing science t heorist : The rest ar e bor ing det ails 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 23 Nat ur al CA-like Phenomena ❒ Describes phenomena t hat occur on radically dif f erent t ime and space scales ❒ St at ist ical mechanical syst ems ❍ Lat t ice-gas aut omat on ❒ Aut ocat alyt ic chemical set s ❍ The Belousov-Zhabot insky react ion ❒ Gene regulat ion ❒ Mult icellular organisms ❒ Colonies and ”super-organisms” ❒ Flocks and herds ❒ Economics and Societ y 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 24 8

  9. P at t ern Format ion in Slime Molds ❒ Self -organisat ion r esult ing in pat t er ns ❒ The Belousov-Zhabot insky react ion ❒ Honeybees ❒ Pat t erns gener at ed by or ganisms midway in complexit y ❒ Dict yost elium discoideum 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 25 D. Discoideum – Lif e Cycle 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 26 D. Discoideum – Amoebae St age ❒ Growt h phase ❒ Fr ee moving single cell ❒ Lives in soil engulf s bact er ia ❒ Divides asexually ❍ Doubling t ime ~ 3h (Picture from dictybase.org) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 27 9

  10. D. Discoideum – Amoebaes (Video from dictybase.org) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 28 D. Discoideum – Aggr egat ion St age ❒ When st ar ving � development al phase ❒ Aggregat e by chemot axis ❒ Mult iple concent r ic cir cles and spirals ❒ Up t o 100000 individuals ❒ 1 f r ame/ 36 sec ❒ Wave pr opagat ion 60 – 120 µm/ min ❒ Spiral accelerat e cell (Video from Zool. Inst. Univ. München) aggregat ion (18 vs 3 µm/ min) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 29 D. Discoideum – Spir al Waves ❒ Spir al f or mat ion unclear, involves symmet r y br eaking ❒ 1 f r ame/ 10 sec (Video from Zool. Inst. Univ. München) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 30 10

  11. D. Discoideum – St ream Format ion St age ❒ St r eams depends on movement and symmet r y br eaking ❒ Begin t o f orm slug (Picture from R. Firtel, UCSD (dictybase.org)) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 31 D. Discoideum – Mound St age ❒ 10000 – 100000 cells ❒ Cells begin t o dif f erent iat e ❒ 1 f r ame/ 20 sec (Video from Zool. Inst. Univ. München) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 32 D. Discoideum – Mult i-ar med spir als ❒ Up t o 10 spirals have been observed ❒ This mound has 5 spirals (Video from Zool. Inst. Univ. München) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 33 11

  12. D. Discoideum – Slug St age ❒ Behaves as single organism ❒ Migrat es; seeks light , seeks or avoids heat ❒ No brain, no nervous syst em ❒ 1 f rame/ 10sec (Video from Zool. Inst. Univ. München) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 34 D. Discoideum – Culminat ion St age ❒ Cells dif f erant iat e int o base, st alk and spor es ❒ 1 f r ame/ 5 sec (Video from Zool. Inst. Univ. München) 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 35 D. Discoideum – Fr uit ing Body St age ❒ Spor es ar e disper sed, wind or animal ❒ I f suf f icient moist ure, spor es germinat e, release amoebaes ❒ Cycle begins again 16/11 - 04 Emergent Systems, Jonny Pettersson, UmU 36 12

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