II. Spatial Systems A. Cellular Automata 8/24/08 1
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1 II. Spatial Systems A. Cellular Automata 8/24/08 1
2 Cellular Automata (CAs) Invented by von Neumann in 1940s to study reproduction He succeeded in constructing a self-reproducing CA Have been used as: massively parallel computer architecture model of physical phenomena (Fredkin, Wolfram) Currently being investigated as model of quantum computation (QCAs) 8/24/08 2
3 Structure Discrete space (lattice) of regular cells 1D, 2D, 3D, rectangular, hexagonal, At each unit of time a cell changes state in response to: its own previous state states of neighbors (within some radius ) All cells obey same state update rule an FSA Synchronous updating 8/24/08 3
4 Example: Conway s Game of Life Invented by Conway in late 1960s A simple CA capable of universal computation Structure: 2D space rectangular lattice of cells binary states (alive/dead) neighborhood of 8 surrounding cells (& self) simple population-oriented rule 8/24/08 4
5 State Transition Rule Live cell has 2 or 3 live neighbors! stays as is (stasis) Live cell has < 2 live neighbors! dies (loneliness) Live cell has > 3 live neighbors! dies (overcrowding) Empty cell has 3 live neighbors! comes to life (reproduction) 8/24/08 5
6 Demonstration of Life Run NetLogo Life or < Go to CBN Online Experimentation Center <mitpress.mit.edu/books/flaoh/cbnhtml/java.html> 8/24/08 6
7 Some Observations About Life 1. Long, chaotic-looking initial transient unless initial density too low or high 2. Intermediate phase isolated islands of complex behavior matrix of static structures & blinkers gliders creating long-range interactions 3. Cyclic attractor typically short period 8/24/08 7
8 From Life to CAs in General What gives Life this very rich behavior? Is there some simple, general way of characterizing CAs with rich behavior? It belongs to Wolfram s Class IV 8/24/08 8
9 8/24/08 fig. from Flake via EVALife 9
10 Wolfram s Classification 8/24/08 10
11 Wolfram s Classification Class I: evolve to fixed, homogeneous state 8/24/08 10
12 Wolfram s Classification Class I: evolve to fixed, homogeneous state ~ limit point 8/24/08 10
13 Wolfram s Classification Class I: evolve to fixed, homogeneous state ~ limit point Class II: evolve to simple separated periodic structures 8/24/08 10
14 Wolfram s Classification Class I: evolve to fixed, homogeneous state ~ limit point Class II: evolve to simple separated periodic structures ~ limit cycle 8/24/08 10
15 Wolfram s Classification Class I: evolve to fixed, homogeneous state ~ limit point Class II: evolve to simple separated periodic structures ~ limit cycle Class III: yield chaotic aperiodic patterns 8/24/08 10
16 Wolfram s Classification Class I: evolve to fixed, homogeneous state ~ limit point Class II: evolve to simple separated periodic structures ~ limit cycle Class III: yield chaotic aperiodic patterns ~ strange attractor (chaotic behavior) 8/24/08 10
17 Wolfram s Classification Class I: evolve to fixed, homogeneous state ~ limit point Class II: evolve to simple separated periodic structures ~ limit cycle Class III: yield chaotic aperiodic patterns ~ strange attractor (chaotic behavior) Class IV: complex patterns of localized structure 8/24/08 10
18 Wolfram s Classification Class I: evolve to fixed, homogeneous state ~ limit point Class II: evolve to simple separated periodic structures ~ limit cycle Class III: yield chaotic aperiodic patterns ~ strange attractor (chaotic behavior) Class IV: complex patterns of localized structure ~ long transients, no analog in dynamical systems 8/24/08 10
19 Langton s Investigation Under what conditions can we expect a complex dynamics of information to emerge spontaneously and come to dominate the behavior of a CA? 8/24/08 11
20 Approach Investigate 1D CAs with: random transition rules starting in random initial states Systematically vary a simple parameter characterizing the rule Evaluate qualitative behavior (Wolfram class) 8/24/08 12
21 Why a Random Initial State? How can we characterize typical behavior of CA? Special initial conditions may lead to special (atypical) behavior Random initial condition effectively runs CA in parallel on a sample of initial states Addresses emergence of order from randomness 8/24/08 13
22 Assumptions 8/24/08 14
23 Assumptions Periodic boundary conditions 8/24/08 14
24 Assumptions Periodic boundary conditions no special place 8/24/08 14
25 Assumptions Periodic boundary conditions no special place Strong quiescence: 8/24/08 14
26 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same 8/24/08 14
27 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same persistence of uniformity 8/24/08 14
28 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same persistence of uniformity Spatial isotropy: 8/24/08 14
29 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same persistence of uniformity Spatial isotropy: all rotations of neighborhood state result in same new state 8/24/08 14
30 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same persistence of uniformity Spatial isotropy: all rotations of neighborhood state result in same new state no special direction 8/24/08 14
31 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same persistence of uniformity Spatial isotropy: all rotations of neighborhood state result in same new state no special direction Totalistic [not used by Langton]: 8/24/08 14
32 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same persistence of uniformity Spatial isotropy: all rotations of neighborhood state result in same new state no special direction Totalistic [not used by Langton]: depend only on sum of states in neighborhood 8/24/08 14
33 Assumptions Periodic boundary conditions no special place Strong quiescence: if all the states in the neighborhood are the same, then the new state will be the same persistence of uniformity Spatial isotropy: all rotations of neighborhood state result in same new state no special direction Totalistic [not used by Langton]: depend only on sum of states in neighborhood implies spatial isotropy 8/24/08 14
34 Langton s Lambda Designate one state to be quiescent state Let K = number of states Let N = 2r + 1 = size of neighborhood Let T = K N = number of entries in table Let n q = number mapping to quiescent state Then 8/24/08 15
35 Langton s Lambda Designate one state to be quiescent state Let K = number of states Let N = 2r + 1 = size of neighborhood Let T = K N = number of entries in table Let n q = number mapping to quiescent state Then! = T " n q T 8/24/08 15
36 Range of Lambda Parameter If all configurations map to quiescent state:! = 0 If no configurations map to quiescent state:! = 1 If every state is represented equally:! = 1 1/K A sort of measure of excitability 8/24/08 16
37 Example States: K = 5 Radius: r = 1 Initial state: random Transition function: random (given!) 8/24/08 17
38 Demonstration of 1D Totalistic CA Run NetLogo 1D CA General Totalistic or < CA-1D-General-Totalistic.html> Go to CBN Online Experimentation Center <mitpress.mit.edu/books/flaoh/cbnhtml/java.html> 8/24/08 18
39 Class I (! = 0.2) time 8/24/08 19
40 Class I (! = 0.2) Closeup 8/24/08 20
41 Class II (! = 0.4) 8/24/08 21
42 Class II (! = 0.4) Closeup 8/24/08 22
43 Class II (! = 0.4) Closeup 8/24/08 22
44 Class II (! = 0.31) 8/24/08 23
45 Class II (! = 0.31) Closeup 8/24/08 24
46 Class II (! = 0.37) 8/24/08 25
47 Class II (! = 0.37) Closeup 8/24/08 26
48 Class III (! = 0.5) 8/24/08 27
49 Class III (! = 0.5) Closeup 8/24/08 28
50 Class IV (! = 0.35) 8/24/08 29
51 Class IV (! = 0.35) Closeup 8/24/08 30
52 Class IV (! = 0.34) 8/24/08 31
53 8/24/08 32
54 8/24/08 33
55 8/24/08 33
56 8/24/08 34
57 8/24/08 34
58 8/24/08 34
59 8/24/08 34
60 8/24/08 35
61 8/24/08 35
62 8/24/08 35
63 8/24/08 35
64 Class IV Shows Some of the Characteristics of Computation Persistent, but not perpetual storage Terminating cyclic activity Global transfer of control/information 8/24/08 36
65 ! of Life For Life,!! which is near the critical region for CAs with: K = 2 N = 9 8/24/08 37
66 Transient Length (I, II) 8/24/08 38
67 Transient Length (III) 8/24/08 39
68 Shannon Information (very briefly!) Information varies directly with surprise Information varies inversely with probability Information is additive!the information content of a message is proportional to the negative log of its probability 8/24/08 40
69 Shannon Information (very briefly!) Information varies directly with surprise Information varies inversely with probability Information is additive!the information content of a message is proportional to the negative log of its probability Is {}=!lgpr{} s 8/24/08 40
70 Entropy Suppose have source S of symbols from ensemble {s 1, s 2,, s N } Average information per symbol: N! Pr{ s k }Is { k }= Pr s k k=1 This is the entropy of the source: N " N! k=1 ( ) { } "lgpr{ s } k H{ S}=! Pr{ s k }lgpr s k k=1 { } 8/24/08 41
71 Maximum and Minimum Entropy Maximum entropy is achieved when all signals are equally likely No ability to guess; maximum surprise H max = lg N Minimum entropy occurs when one symbol is certain and the others are impossible No uncertainty; no surprise H min = 0 8/24/08 42
72 Entropy Examples 8/24/08 43
73 Entropy Examples 8/24/08 43
74 Entropy Examples 8/24/08 43
75 Entropy Examples 8/24/08 43
76 Entropy Examples 8/24/08 43
77 Entropy Examples 8/24/08 43
78 Entropy Examples 8/24/08 43
79 Entropy of Transition Rules Among other things, a way to measure the uniformity of a distribution Distinction of quiescent state is arbitrary Let n k = number mapping into state k Then p k = n k / T H =! " i H = lgt! 1 T p i lg p i K " k=1 n k lgn k 8/24/08 44
80 Entropy Range Maximum entropy (! = 1 1/K): uniform as possible all n k = T/K H max = lg K Minimum entropy (! = 0 or! = 1): non-uniform as possible one n s = T all other n r = 0 (r " s) H min = 0 8/24/08 45
81 2-D CAs K = 8 Further Investigations by N = 5 64! 64 lattice Langton periodic boundary conditions 8/24/08 46
82 Avg. Transient Length vs.! (K=4, N=5) 8/24/08 47
83 Avg. Cell Entropy vs.! (K=8, N=5) H(A) = K! k=1 p k lg p k 8/24/08 48
84 Avg. Cell Entropy vs.! (K=8, N=5) 8/24/08 49
85 Avg. Cell Entropy vs.! (K=8, N=5) 8/24/08 50
86 Avg. Cell Entropy vs.!" (K=8, N=5) 8/24/08 51
87 Avg. Cell Entropy vs.! (K=8, N=5) 8/24/08 52
88 Avg. Cell Entropy vs.!" (K=8, N=5) 8/24/08 53
89 Entropy of Independent Systems Suppose sources A and B are independent Let p j = Pr{a j } and q k = Pr{b k } Then Pr{a j, b k } = Pr{a j } Pr{b k } = p j q k 8/24/08 54
90 Entropy of Independent Systems Suppose sources A and B are independent Let p j = Pr{a j } and q k = Pr{b k } Then Pr{a j, b k } = Pr{a j } Pr{b k } = p j q k H(A,B) =! j,k Pr( a j,b k )lgpr( a j,b ) k ( ) =! p j q k lg p j q k = p j q k lg p j + lgq k j,k! j,k ( ) =! p j lg p j +! q k lgq k = H(A) + H(B) j k 8/24/08 54
91 Mutual Information Mutual information measures the degree to which two sources are not independent A measure of their correlation 8/24/08 55
92 Mutual Information Mutual information measures the degree to which two sources are not independent A measure of their correlation IA,B ( )= HA ( )+ HB ( )! HA,B ( ) 8/24/08 55
93 Mutual Information Mutual information measures the degree to which two sources are not independent A measure of their correlation IA,B ( )= HA ( )+ HB ( )! HA,B ( ) I(A,B) = 0 for completely independent sources 8/24/08 55
94 Mutual Information Mutual information measures the degree to which two sources are not independent A measure of their correlation IA,B ( )= HA ( )+ HB ( )! HA,B ( ) I(A,B) = 0 for completely independent sources I(A,B) = H(A) = H(B) for completely correlated sources 8/24/08 55
95 Avg. Mutual Info vs.! (K=4, N=5) I(A,B) = H(A) + H(B) H(A,B) 8/24/08 56
96 Avg. Mutual Info vs.!" (K=4, N=5) 8/24/08 57
97 Mutual Information vs. Normalized Cell Entropy 8/24/08 58
98 Critical Entropy Range 8/24/08 59
99 Critical Entropy Range Information storage involves lowering entropy 8/24/08 59
100 Critical Entropy Range Information storage involves lowering entropy Information transmission involves raising entropy 8/24/08 59
101 Critical Entropy Range Information storage involves lowering entropy Information transmission involves raising entropy Computation requires a tradeoff between low and high entropy 8/24/08 59
102 Suitable Media for Computation How can we identify/synthesize novel computational media? especially nanostructured materials for massively parallel computation Seek materials/systems exhibiting Class IV behavior may be identifiable via entropy, mut. info., etc. Find physical properties (such as!) that can be controlled to put into Class IV 8/24/08 60
103 Complexity vs.! 8/24/08 61
104 Schematic of CA Rule Space vs.! Fig. from Langton, Life at Edge of Chaos 8/24/08 62
105 Some of the Work in this Area Wolfram: A New Kind of Science Langton: Computation/life at the edge of chaos Crutchfield: Computational mechanics Mitchell: Evolving CAs and many others 8/24/08 63
106 Some Other Simple Computational Systems Exhibiting the Same Behavioral Classes CAs (1D, 2D, 3D, totalistic, etc.) Mobile Automata Turing Machines Substitution Systems Tag Systems Cyclic Tag Systems Symbolic Systems (combinatory logic, lambda calculus) Continuous CAs (coupled map lattices) PDEs Probabilistic CAs Multiway Systems 8/24/08 64
107 Universality A system is computationally universal if it can compute anything a Turing machine (or digital computer) can compute The Game of Life is universal Several 1D CAs have been proved to be universal Are all complex (Class IV) systems universal? Is universality rare or common? 8/24/08 65
108 Rule 110: A Universal 1D CA K = 2, N = 3 New state = (p!q!r)!(q"r) where p, q, r are neighborhood states Proved by Wolfram 8/24/08 66
109 Fundamental Universality Classes of Dynamical Behavior space time 8/24/08 67
110 Fundamental Universality Classes of Dynamical Behavior Classes I, II space time solid halt 8/24/08 67
111 Fundamental Universality Classes of Dynamical Behavior space Classes I, II Class III time solid halt fluid don t halt 8/24/08 67
112 Fundamental Universality Classes of Dynamical Behavior space Classes I, II Class IV Class III time solid halt phase transition halting problem fluid don t halt 8/24/08 67
113 Wolfram s Principle of Computational Equivalence a fundamental unity exists across a vast range of processes in nature and elsewhere: despite all their detailed differences every process can be viewed as corresponding to a computation that is ultimately equivalent in its sophistication (NKS 719) Conjecture: among all possible systems with behavior that is not obviously simple an overwhelming fraction are universal (NKS 721) 8/24/08 68
114 Computational Irreducibility systems one uses to make predictions cannot be expected to do computations that are any more sophisticated than the computations that occur in all sorts of systems whose behavior we might try to predict (NKS 741) even if in principle one has all the information one needs to work out how some particular system will behave, it can still take an irreducible amount of computational work to do this (NKS 739) That is: for Class IV systems, you can t (in general) do better than simulation. 8/24/08 69
115 Additional Bibliography 1. Langton, Christopher G. Computation at the Edge of Chaos: Phase Transitions and Emergent Computation, in Emergent Computation, ed. Stephanie Forrest. North-Holland, Langton, Christopher G. Life at the Edge of Chaos, in Artificial Life II, ed. Langton et al. Addison-Wesley, Emmeche, Claus. The Garden in the Machine: The Emerging Science of Artificial Life. Princeton, Wolfram, Stephen. A New Kind of Science. Wolfram Media, Part 2B 8/24/08 70
116 Project 1 Investigation of relation between Wolfram classes, Langton s!, and entropy in 1D CAs Due Sept. 12 Information is on course website (scroll down to Projects/Assignments ) Read it over and questions or ask in class Part 2B 8/24/08 71
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