The Nature of Computation Introduction of Wolfram s NKS Complex systems research center Zhang Jiang
What can we do by computers? Scientific computation Processing data Computer simulations
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
What can we say? Objects: Artificial worlds Computational universe (CU) NKS is studying these Begin from Cellular automata But including all kinds of CUs
A Brief History In 1940 s von Neumann began to study the self-reproducing automata
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
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
What is A New Kind of Science?
What is NKS? Study all kinds of computational universe Cellular Automata Turing Machines
1-D Cellular Automata Space of the Universe
1-D Cellular Automata Physics of the universe Neighborhood Rules
1-D Cellular Automata Time of the universe
Implementation Definition
Game of life Living
Game of life Die
Turing Machine
Turing Machine As a computational universe
Turing Machine Implementation
Substitution systems A AB, B BA A B,B BA
Implementation
Systems based on Numbers Unary representation of n n=n+1
Systems based on Numbers Binary represent of n 100 steps
Standard approach of NKS Implementation: Observation Classification Systematic Searching
Observations and classification 4 classes of CA Class I: Fixed value Class II: Cyclic Class III: Random Class IV: Complex
Information propagation
Self-similar is very common
Self-similar is very common CA225 start with 0,1,0,0, Transform
Complex rules Complex behavior A slice of Game of life
It seems Complexity of behavior A threshold? Complexity of rules
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 0 0 1 1 0 0 1 1 Coding 51 There are 2 8 =256 rules
Searching Searching for conserved number of black cell For all 256 k=2,r=1 rules, And 2 w possible initial conditions
Searching For k=2, r=2 CAs There are 428 in 2 32 = 4294967296 possible rules
Applications Simulating natural phenomena Flake Tree growth Fluid Not only simulating
CA Time Serials Jason Cawley, Wolfram Research
CA and time series Microstate: Black Buy, White Sell 20 Macrostate: Resultant Price Series 0 20 40 60 CA 90 0 10 20 30 40 50 60
ICA: Mix up two CAs Run CA 90 3 steps Run CA 110 7 steps Adjust portions of 3:7 can generate different time serials 1.0 0.8 0.6 0.4 0.2 100 200 300 400
Fitting to the real data 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 50 100 150 200 250 300 350 50 100 150 200 250 300 350
Evolving DNA sequence Dawei Li Ph.D The Rockefeller University
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
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.
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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
Emulation and Universality That s what I really like
What is simulation? But what is simulation on earth? Observation Simulation Decision
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
A Turing Machine 3 states,2 colors
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
Emulation CA TM CA TM No head on it Head state 1 Head state 2 Head state 3 0 1
Rules Mapping Each rule corresponds two adjacent cells CA: r=1 is enough For one rule (1,1) (2,0,r) Don t care
Compare their behavior
CAs can emulate TMs This approach can be generalized for all Turing Machines CA as a class can emulate TM class
Turing machine emulates CAs ECA 90 Conflict: TM is serial, CA is parallel
Basic Idea Using serial to emulate parallel
Emulation by Turing Machine
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
Church - Turing Thesis Any effective computation can be done by TM All of those computational systems are equivalent They are universal
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
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
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
Universal Cellular Automata A specific CA can emulate any other CA
Universal Cellular Automaton
CA 110 CA 110 is universal, it is really a non-trivial discovery!!! Skill: Emulation by emergent behavior not by the rules
The proof of CA110 is universal
Finding Minimum universal machine 1962: TM (7 states, 4 colors) 2002: CA110 2002: Turing machine (2 states, 5 colors) Wolfram prize:
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
Thank you!!!
The Core Question What is Life? In 1944
What is life
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
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
An example: Virtual Machine
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
Something Special
Good Movies Deep thoughts
Example of virtual worlds 读者张三 神雕侠侣 真实世界 小龙女 杨过
13th Floor 读者张三 真实世界 界虚拟世界的虚拟世 虚拟世界
Implication of Universal Machine If a universal system is a universe Then the universal machine builds a virtual universe
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
Some Heuristics in Fundamental Physics Space as Network Causal network
Space as network Suppose space of our universe is a network How can we obtain spatial dimension from a network?
It is easy from space to network
How about the inverse problem? One network has different layouts
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
Layout as r~n(r)
Causal network Every thing is causal Event is node, causal effect is edge
Different ways to view causal network
The metabolism of science Observations Nature Pure nature Science Artificial world Technology
Artificial = inferior? Popper s artificial world H.A. Simon s artificial science Pure nature Pure nature Artificial world Artificial world