The Nature of Computation
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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
61
62
63 Universal Cellular Automaton
64
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
89
90
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
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