The Nature of Computation

Transcription

1 The Nature of Computation Introduction of Wolfram s NKS Complex systems research center Zhang Jiang

2 What can we do by computers? Scientific computation Processing data Computer simulations

3 New field emerging Computer Games World of Warcraft Second life W.S. Bainbridge: The Scientific Research Potential of VIRTUAL WORLDs, Science, vol 317, 2007 Jim Giles, Social Sciences: Life's A Game, Nature 445, 18-20, 2007/01/04

4 What can we say? Objects: Artificial worlds Computational universe (CU) NKS is studying these Begin from Cellular automata But including all kinds of CUs

5 A Brief History In 1940 s von Neumann began to study the self-reproducing automata

6 A Brief history Godel Von Neumann A.Turing Arthur Burk Codd John Conway Wolfram John Holland C. Langton CA NKS GA AL,SA Self-ref D. Hofstader

7 About Stephen Wolfram Published his first paper in 15 years old, the youngest recipient of a MacArthur Prize Fellowship in 22 years old Worked for Princeton, Illinois university Launched Wolfram Research Inc. in 1986 Transferred from physics to complexity, study CA in mid 1980 s Began to write NKS book from 1991 Launched NKS book in May, 2002

8 What is A New Kind of Science?

9 What is NKS? Study all kinds of computational universe Cellular Automata Turing Machines

10 1-D Cellular Automata Space of the Universe

11 1-D Cellular Automata Physics of the universe Neighborhood Rules

12 1-D Cellular Automata Time of the universe

13 Implementation Definition

14 Game of life Living

15 Game of life Die

16 Turing Machine

17 Turing Machine As a computational universe

18 Turing Machine Implementation

19 Substitution systems A AB, B BA A B,B BA

20 Implementation

21 Systems based on Numbers Unary representation of n n=n+1

22 Systems based on Numbers Binary represent of n 100 steps

23 Standard approach of NKS Implementation: Observation Classification Systematic Searching

24 Observations and classification 4 classes of CA Class I: Fixed value Class II: Cyclic Class III: Random Class IV: Complex

25 Information propagation

26 Self-similar is very common

27 Self-similar is very common CA225 start with 0,1,0,0, Transform

28 Complex rules Complex behavior A slice of Game of life

29 It seems Complexity of behavior A threshold? Complexity of rules

30 Systematic searching Enumeration: Coding any CA with a number For any k=2, r=1 CAs, how many rules are there? Possible inputs: Possible output Coding 51 There are 2 8 =256 rules

31 Searching Searching for conserved number of black cell For all 256 k=2,r=1 rules, And 2 w possible initial conditions

32 Searching For k=2, r=2 CAs There are 428 in 2 32 = possible rules

33 Applications Simulating natural phenomena Flake Tree growth Fluid Not only simulating

34 CA Time Serials Jason Cawley, Wolfram Research

35 CA and time series Microstate: Black Buy, White Sell 20 Macrostate: Resultant Price Series CA

36 ICA: Mix up two CAs Run CA 90 3 steps Run CA steps Adjust portions of 3:7 can generate different time serials

37 Fitting to the real data

38 Evolving DNA sequence Dawei Li Ph.D The Rockefeller University

39 Evolving DNA Sequence Consider A,G,C,T sequence in DNA as a binary sequence, So given a sequence, we can evolve it to get a pattern

40 SARS BJ01, partial genome; SARS BJ02, partial genome; SARS BJ03, partial genome; SARS BJ04, partial genome; SARS CUHK-W1, complete genome; SARS GZ01, partial genome; SARS HKU-39849, complete genome; SARS TOR2, complete genome; SARS Urbani, complete genome; SARS coronavirus CUHK-Su10, complete genome; SARS coronavirus isolate SIN2774 complete genome; SARS coronavirus TW1, complete genome; SARS coronavirus, complete genome.

41 复制酶

42 Summary There are many heuristics and ideas in NKS Set bits free!!! Forgetting about the meaning of bits Observation with no purpose Different from artificial life models

43 Emulation and Universality That s what I really like

44 What is simulation? But what is simulation on earth? Observation Simulation Decision

45 Emulation Mapping between different systems Once a program is found mapping A to B, then B can emulate A Emulation is the only rigorous proof in NKS B=f(A) A B Emulate

46 A Turing Machine 3 states,2 colors

47 How can we emulate it using CA? The tape of Turing machine Finite Cells How about the head of the Turing Machine? Head (3+1)*2=8 colors One Cell Of CA One Cell Of TM Color One Cell Of CA

48 Emulation CA TM CA TM No head on it Head state 1 Head state 2 Head state 3 0 1

49 Rules Mapping Each rule corresponds two adjacent cells CA: r=1 is enough For one rule (1,1) (2,0,r) Don t care

50 Compare their behavior

51 CAs can emulate TMs This approach can be generalized for all Turing Machines CA as a class can emulate TM class

52 Turing machine emulates CAs ECA 90 Conflict: TM is serial, CA is parallel

53 Basic Idea Using serial to emulate parallel

54 Emulation by Turing Machine

55 Conclusion Any CA can be emulated by TM CA and TM can emulate each other They are computationally equivalent In NKS book, almost all of computational universes can emulate each other They are equivalent in terms of computation

56 Church - Turing Thesis Any effective computation can be done by TM All of those computational systems are equivalent They are universal

57 Universality Any single or a class of systems can emulate all of TMs, it is universal Universality of a class Universality of a specific machine in a class

58 Universal Machine A universal machine can emulate any other machines by right initial configure x M o M+x y o M Transform M +y Transform Universal Machine z M o M +z Transform

59 Universal Turing Machine The first universal machine is found by Turing in 1936 It is possible because: Any TM x can be emulated by its coding D(x) D(x) can be input to Universal TM U as initial state. U just decomposes D(x) to several single steps of D s computation

60 Universal Cellular Automata A specific CA can emulate any other CA

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63 Universal Cellular Automaton

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65 CA 110 CA 110 is universal, it is really a non-trivial discovery!!! Skill: Emulation by emergent behavior not by the rules

66 The proof of CA110 is universal

67 Finding Minimum universal machine 1962: TM (7 states, 4 colors) 2002: CA : Turing machine (2 states, 5 colors) Wolfram prize:

68 Computational equivalence principle Any class 4 system is universal There is no random class Universality instead of complexity Capability Threshold of universality Complexity of rules

69 Thank you!!!

70 The Core Question What is Life? In 1944

71 What is life

72 A Whole spectrum of theories Model, theory Prigogine s dissipative structure Kauffman s self-catalytic network VN s self-rp Wolfram s NKS John Holland s CAS What is Life? Data, facts Brown & West s Ecology, food webs metabolism ecology System biology Physics (Material energy constraints) Bio-infomics Information, Computation

73 Emulation Hierarchy and Virtual Worlds If universal machine A emulates universal machine B, and B is emulating a machine x, then B x A Emulation Hierarchy

74 An example: Virtual Machine

75 Self-emulation How about Universal Machine A emulate itself? An infinite depth of virtual worlds This is self-reference Godel Theorem Von Neumann s self-reproducing automata

76 Something Special

77 Good Movies Deep thoughts

78 Example of virtual worlds 读者张三 神雕侠侣 真实世界 小龙女 杨过

79 13th Floor 读者张三 真实世界 界虚拟世界的虚拟世 虚拟世界

80 Implication of Universal Machine If a universal system is a universe Then the universal machine builds a virtual universe

81 Enumerating IPD P1\P2 C D C 3,3 0,5 D 5,0 1,1 For two players: 1: CCC,CDDDCD 2: DCD,CDCDCD Strategy: (3 History) (CDC) C, (DDD) D, There are 2 8 =256 strategies There are 2 6 =64 initial conditions

82 Some Heuristics in Fundamental Physics Space as Network Causal network

83 Space as network Suppose space of our universe is a network How can we obtain spatial dimension from a network?

84 It is easy from space to network

85 How about the inverse problem? One network has different layouts

86 Dimension of network Distance r: minimal number of connections between two nodes For given node, number of neighbors of distance r is N(r) There is a power law: N(r)~r d-1 So A~r 2, V~r 3

87 Layout as r~n(r)

88 Causal network Every thing is causal Event is node, causal effect is edge

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91 Different ways to view causal network

92 The metabolism of science Observations Nature Pure nature Science Artificial world Technology

93 Artificial = inferior? Popper s artificial world H.A. Simon s artificial science Pure nature Pure nature Artificial world Artificial world