Simulation between signal machines

Transcription

1 Jérôme Durand-Lose joint work with Florent Becker, Tom Besson, Hadi Foroughmand and Sama Goliaei Partially funded by PHC Gundichapur n E LABORATOIRE D'INFORMATIQUE FONDAMENTALE D'ORLEANS Laboratoire d Informatique Fondamentale d Orléans, Université d Orléans, Orléans, FRANCE Laboratoire d Informatique de l École polytechnique Algorithmic Questions in Dynamical Systems March 2018 Toulouse 1 / 58

2 Introduction 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 2 / 58

3 Introduction Outline One model/dynamical system signal machines Relations to usual computational models Intrinsic universality Non determinism 3 / 58

4 Introduction Outline One model/dynamical system signal machines Relations to usual computational models Turing machines Linear Blum, Shub and Smale model Intrinsic universality Non determinism 3 / 58

5 Introduction Outline One model/dynamical system signal machines Relations to usual computational models Turing machines Linear Blum, Shub and Smale model Intrinsic universality a family only Non determinism 3 / 58

6 Introduction Outline One model/dynamical system signal machines Relations to usual computational models Turing machines Linear Blum, Shub and Smale model Intrinsic universality a family only Non determinism same power 3 / 58

7 Signal Machines (Introduction and Definition) 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 4 / 58

8 Signal Machines (Introduction and Definition) Cellular Automata: signal use Firing Squad Synchronization (Goto, 1966) 5 / 58

9 Signal Machines (Introduction and Definition) CA: Conception with signals Fischer (1965) 6 / 58

10 Simulation We between performed signal machines a total of 65 GA runs. Since F 100 () is only a rough estimate of performance, Signal wemachines more (Introduction stringentlyand measured Definition) the quality of the GA's solutions by calculating P N 104() with N 2 f149; 599; 999g for the best CAs in the nal generation of each run. In 20% of the runs the GA discovered successful CAs (P CA: Analyzing with Signals N 10 = 1:0). More detailed analysis of these successful CAs 4 showed that although they were distinct in detail, they used similar strategies for performing the synchronization task. Interestingly, when the GA was restricted to evolve CAs with r = 1 and r = 2, all the evolved CAs had P N for N 2 f149; 599; 999g. (Better performing CAs with r = 2 can be designed by hand.) Thus r = 3 appears to be the minimal radius for which the GA can successfully solve this problem. Das et al. (1995) 0 β α γ β γ Time δ µ γ ν δ 74 0 Site 74 (a) Space-time diagram. 0 Site 74 (b) Filtered space-time diagram. Figure 1: (a) Space-time diagram of sync starting with a random initial condition. (b) The same spacetime diagram after ltering with a spatial transducer that maps all domains to white and all defects to black. Greek letters label particles described in the text. 7 / 58

11 Signal Machines (Introduction and Definition) Signals Time (N) Time (R + ) Space (Z) Space (R) Signal (meta-signal) Collision (rule) 8 / 58

12 Signal Machines (Introduction and Definition) Vocabulary and Example: Find the Middle Meta-signals (speed) M (0) M M Collision rules 9 / 58

13 Signal Machines (Introduction and Definition) Vocabulary and Example: Find the Middle Meta-signals (speed) M (0) div (3) div M M Collision rules 9 / 58

14 Signal Machines (Introduction and Definition) Vocabulary and Example: Find the Middle Meta-signals (speed) M (0) div (3) hi (1) lo (3) div M M hi lo M Collision rules { div, M } { M, hi, lo } 9 / 58

15 Signal Machines (Introduction and Definition) Vocabulary and Example: Find the Middle Meta-signals (speed) M hi lo backm M (0) div (3) hi (1) lo (3) back (-3) div M M Collision rules { div, M } { M, hi, lo } { lo, M } { back, M } 9 / 58

16 Signal Machines (Introduction and Definition) Vocabulary and Example: Find the Middle Meta-signals (speed) M hi M lo back M M (0) div (3) hi (1) lo (3) back (-3) div M M Collision rules { div, M } { M, hi, lo } { lo, M } { back, M } { hi, back } { M } 9 / 58

17 Signal Machines (Introduction and Definition) Stack Implantation R + Name Speed Add, Rem 1/3 A, E 1 U, M 0 R 3 R 3 Collision rules Time M U M { Add, M } { M, A, R } { R, M } { R, M } { A, R } { U } { R, U } { R, U } { Rem, M } { M, E } { E, U } { } Add A M Space R R M R 10 / 58

18 Signal Machines (Introduction and Definition) Stack Implantation R + Name Speed Add, Rem 1/3 A, E 1 U, M 0 R 3 R 3 Collision rules Time M U R A R U U M { Add, M } { M, A, R } { R, M } { R, M } { A, R } { U } { R, U } { R, U } { Rem, M } { M, E } { E, U } { } Add M Add A M Space R R M R 10 / 58

19 Signal Machines (Introduction and Definition) Stack Implantation Name Speed Add, Rem 1/3 A, E 1 U, M 0 R 3 R 3 Collision rules { Add, M } { M, A, R } { R, M } { R, M } { A, R } { U } { R, U } { R, U } { Rem, M } { M, E } { E, U } { } R + Time M M Rem M Add Add E U R A R A M Space R U U R M M R 10 / 58

20 Signal Machines (Introduction and Definition) Fractal Generation 11 / 58

21 Signal Machines (Introduction and Definition) Complex Dynamics 12 / 58

22 Signal Machines (Introduction and Definition) Complex Dynamics 12 / 58

23 Signal Machines (Introduction and Definition) Complex Dynamics 12 / 58

24 Signal Machines (Introduction and Definition) Complex Dynamics 12 / 58

25 Relations to Models of Computation 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 13 / 58

26 Relations to Models of Computation Discrete computation: Turing Machines 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation Discrete computation: Turing Machines Analog Computation: linear Blum, Shub and Smale 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 14 / 58

27 Relations to Models of Computation Discrete computation: Turing Machines Turing-computation Turing Machine Simulation q f ^ b b a b # q f ^ b b a b # q f ^ b b a b # q f ^ b b a b # q 2 ^ b b a # q 1 ^ b b # qf ^ qf ^ b qf b b a # b qf qf # # b a # # # q2 # q1 q 1 ^ b a # q 1 ^ a a # q i ^ a b # q1 q1 a qi a # q i ^ a b # q i ^ a b # qi qi ^ a b 15 / 58

28 Relations to Models of Computation Discrete computation: Turing Machines Turing-computation Also with restrictions all different speed only 2 2 rules (conservative) one-to-one rules (reversible) Any above and rational (Q) Rational machines speeds Q initial positions Q collision coordinates Q exact simulation on computer/tm Undecidability finite number de collisions meta-signal appereance use of a rule disappearing of all signals involvement of a signal in any collision extension on the side, etc. 16 / 58

29 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation Discrete computation: Turing Machines Analog Computation: linear Blum, Shub and Smale 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 17 / 58

30 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Computing with Real Numbers Encoding Sign test x base val Zero sign? pos base val sign? zer Zero sign? neg val base Addition base val end base 7 val up add base side downs 1 down side val val base 18 / 58

31 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Multiplication by a constant By.9 By 1 2 By 2 3 val.9x base val base 1 2 x base val 2 3 x x x x base val base val base val 19 / 58

32 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Multiplication by a constant By.9 By 1 2 By 2 3 By 2 By π val.9x base val base 1 2 x base val 2 3 x base 2x val base πx val x x x x x base val base val base val base val base val 19 / 58

33 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Multiplication by a constant By.9 By 1 2 By 2 3 By 2 By π val.9x base val base 1 2 x base val 2 3 x base 2x val base πx val x x x x x base val base val base val base val base val Signal speeds are constants of the machine If x 0 then val is meet before base 19 / 58

34 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Zooming out [ ] x 2 x 1 x 0 x 1 x 2 x 3 x 4 Finite sequence of real numbers 20 / 58

35 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Zooming out q 1 q 1 q 2 q 3 q 3 q 2 [ ] x 2 x 1 x 0 x 1 x 2 x 3 x 4 Finite sequence of real numbers + Dynamics finite state automata sign test addition, multiplication by constant (set constant value) (enlarge the array) Like a Turing machine with real numbers on the tape 20 / 58

36 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Linear Blum, Shub and Smale with shift Encoding a configuration b a a c -1 b a a c / 58

37 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Linear Blum, Shub and Smale with shift Encoding a configuration b a a c d 1 d 2 d b d 1 a d 2 a d 3 c / 58

38 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Linear Blum, Shub and Smale with shift Encoding a configuration b a a c d 1 d 2 d b d 1... a d 2... a d 3... c / 58

39 Relations to Models of Computation Analog Computation: linear Blum, Shub and Smale Linear Blum, Shub and Smale with shift Encoding a configuration b a a c d 1 d 2 d b d 1... a d 2... a d 3... c Simulating a signal machine: loop 1 Compute the minimum time to a collision, δ 2 Advance time by δ (update all distances) 3 Process collision(s) 21 / 58

40 Intrinsically Universal Family of Signal Machines 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 22 / 58

41 Intrinsically Universal Family of Signal Machines Concept and Definition 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines Concept and Definition Global Scheme Macro-Collision Preparation (shrink, test and check) 5 Non-determinism (work in progress) 6 conclusion 23 / 58

42 Intrinsically Universal Family of Signal Machines Concept and Definition Intrinsic Universality Being able to simulate any other dynamical system of the its class. Cellular Automata regular (Albert and Čulik II, 1987; Mazoyer and Rapaport, 1998; Ollinger, 2001) reversible (Durand-Lose, 1997) freezing [Theyssier et Al.] Tile Assembly Systems possible at T=2 and above (Woods, 2013) impossible at T=1 (Meunier et al., 2014) 24 / 58

43 Intrinsically Universal Family of Signal Machines Concept and Definition Simulation for Signal Machines Space-Time Diagram Mimicking time m 3 m 2 time m4 m 1 m 2 space space Signal Machine Simulation U S simulates A if there is function from the configurations of A to the ones of U S s.t. the space-time issued from the image always mimics the original one. 25 / 58

44 Intrinsically Universal Family of Signal Machines Concept and Definition Theorem For any finite set of real numbers S, there is a signal machine U S, that can simulate any machine whose speeds belong to S. The set of U S where S ranges over finite sets of real numbers is an intrinsically universal family of signal machines. Rest of this section Let S be any finite set of real numbers, let A be any signal machine whose speeds belongs to S, U S is progressively constructed as simulation is presented. 26 / 58

45 Intrinsically Universal Family of Signal Machines Global Scheme 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines Concept and Definition Global Scheme Macro-Collision Preparation (shrink, test and check) 5 Non-determinism (work in progress) 6 conclusion 27 / 58

46 Intrinsically Universal Family of Signal Machines Global Scheme Macro-Signal Meta-signal of A identified with numbers Unary encoding of numbers Structure iµ σ : σth signal, ith speed iµ σ ileft-bound iidi σ times imain Collision Rules encoding iright-bound support zone 28 / 58

47 Intrinsically Universal Family of Signal Machines Global Scheme Global scheme When Support Zones Meet 1 provide a delay 2 test if macro-collision is appropriate and what macro-signals are involved 3 if OK start the macro-collision Hypotheses for macro-collision no other macro-signal nor macro-collision will interfere speed of involved macro-signals ranged [j,..., i] (included) their main signals intersect at a unique point 29 / 58

48 Intrinsically Universal Family of Signal Machines Macro-Collision 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines Concept and Definition Global Scheme Macro-Collision Preparation (shrink, test and check) 5 Non-determinism (work in progress) 6 conclusion 30 / 58

49 Intrinsically Universal Family of Signal Machines Macro-Collision Removing Unused Tables and Sending ids to Table 31 / 58

50 Intrinsically Universal Family of Signal Machines Macro-Collision Collision Rules Encoding One rule after the other Encoding of { 3 µ 1, 7 µ 4, 8 µ 5 } { 2 µ 3, 4 µ 1 } in the direction i. i rule-bound i then-id 2 i then-id 2 i then-id 2 i then-id 4 i rule-middle i if-id 3 i if-id 7 i if-id 7 i if-id 7 i if-id 7 i if-id 8 i if-id 8 i if-id 8 i if-id 8 i if-id 8 i rule-bound 32 / 58

51 Intrinsically Universal Family of Signal Machines Macro-Collision Comparison of id s in the if-part of a Rule i rule-middle i rule-middle-fail cross-ok 5 cross-ok 5 cross-ok 5 cross-ok 5 speed 5 cross-ok 5 cross 5 6 cross 5 cross-ok cross-ok 6 cross 4 cross-ok 6 speed 6 cross 6 cross-ok 4 cross-ok 4 cross 4 cross-ok 4 i if 6 i if 6 i if 6 i if-ok 6 i if 6 i if-ok 6 i if 5 i if-ok 5 i if 5 i if-ok 5 i if 4 i if-ok 4 i if 4 i if-ok 4 i rule-bound i rule-bound speed 6 id n 3 speed 5 id n 2 cross-ok 4 speed 4 cross 6 cross 5 cross-ok 6 cross-ok 5 id n 2 id n 3 speed 4 id n 2 id n 2 33 / 58

52 Intrinsically Universal Family of Signal Machines Macro-Collision Rule Selection 34 / 58

53 Intrinsically Universal Family of Signal Machines Macro-Collision Generating the Output 35 / 58

54 Intrinsically Universal Family of Signal Machines Macro-Collision Whole resolution 36 / 58

55 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines Concept and Definition Global Scheme Macro-Collision Preparation (shrink, test and check) 5 Non-determinism (work in progress) 6 conclusion 37 / 58

56 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Good Cases time time main space main main main main space main main main Bad Case time main main main space main main main 38 / 58

57 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Shrinking Unit a 3 b 2 a 2 b c b c b 1 a 1 a a b a c a 0 b 0 39 / 58

58 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Shrink 40 / 58

59 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Testing for Other main Signals ( si+(ε 2)s max ) ε test-left i test-left-ok i test-left-up k i (0, 0) i main i mainok test (2εsi, 2ε) ( ) 1.5εsi 1.5ε U test-up i k main j main test-right-up k i test-right i i test-right k i test-right j i test-start i check-up j i check-? j i test-right-ok j i ( si+(2 ε)s max ) ε 41 / 58

60 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Checking the right positioning of Other main Signals i mainok test i main check-up j i test-right-ok j i k main check-intersect k,j i A check-? j i check j i check-ok j i j main 42 / 58

61 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Detecting Potential Overlaps 43 / 58

62 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Whole Preparation 44 / 58

63 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Exact 3-signal collision 45 / 58

64 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Some examples m 2 time m4 m 3 m 1 m 2 space 46 / 58

65 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Some examples 46 / 58

66 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Some examples 46 / 58

67 Intrinsically Universal Family of Signal Machines Preparation (shrink, test and check) Some examples 46 / 58

68 Non-determinism (work in progress) 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 47 / 58

69 Non-determinism (work in progress) Definition and example 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) Definition and example Strategy Implantation 6 conclusion 48 / 58

70 Non-determinism (work in progress) Definition and example Non-determinism in rule output Meta-signals a 0 b -1 c 1 Collision rules { a, b } { a, c } { a, b } { b } { c, a } { b } { c, a } { a } a b a 49 / 58

71 Non-determinism (work in progress) Definition and example Non-determinism in rule output Meta-signals a 0 b -1 c 1 Collision rules { a, b } { a, c } { a, b } { b } { c, a } { b } { c, a } { a } a c a b a b a b a a b a 49 / 58

72 Non-determinism (work in progress) Definition and example Non-determinism in rule output Meta-signals a 0 b -1 c 1 Collision rules { a, b } { a, c } { a, b } { b } { c, a } { b } { c, a } { a } a c a c a b a a c b b a a b a a a a b a 49 / 58

73 Non-determinism (work in progress) Definition and example Non-determinism in rule output a c b a b a b Meta-signals a 0 b -1 c 1 Collision rules { a, b } { a, c } { a, b } { b } { c, a } { b } { c, a } { a } b a a a c c a a a c a b a b a a b a b a 49 / 58

74 Non-determinism (work in progress) Definition and example Non-determinism in rule output a c b a b a b Shifted superposition Meta-signals a 0 b -1 c 1 Collision rules { a, b } { a, c } { a, b } { b } { c, a } { b } { c, a } { a } b a a a c c a a a c a b a b a a b a b a 49 / 58

75 Non-determinism (work in progress) Definition and example Non-determinism in rule output a c b a b a b Shifted superposition Meta-signals a 0 b -1 c 1 Collision rules { a, b } { a, c } { a, b } { b } { c, a } { b } { c, a } { a } b a a a c c a a a c a b a b a a b no collision possible a b a 49 / 58

76 Non-determinism (work in progress) Strategy 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) Definition and example Strategy Implantation 6 conclusion 50 / 58

77 Non-determinism (work in progress) Strategy All possible universes { } U =,,, Unbounded signals Information held: {(v α, u α )} α such that: v α {, a, b, c} α u α = U ( { }) c, ( { }),,, ( { }) a,, ( { }),, 51 / 58

78 Non-determinism (work in progress) Strategy All possible universes { } U =,,, Unbounded signals Information held: {(v α, u α )} α such that: v α {, a, b, c} α u α = U ( { }) c, ( { }),,, ( { }) a,, ( { }),, 51 / 58

79 Non-determinism (work in progress) Strategy All possible universes { } U =,,, Unbounded signals Information held: {(v α, u α )} α such that: v α {, a, b, c} α u α = U Future unknown ( { c,,, ( { }), }) 51 / 58

80 Non-determinism (work in progress) Strategy All possible universes { } U =,,, Unbounded signals Information held: {(v α, u α )} α such that: v α {, a, b, c} α u α = U Future unknown ( { }) ( c, { }), 51 / 58

81 Non-determinism (work in progress) Strategy All possible universes { } U =,,, Unbounded signals Information held: {(v α, u α )} α such that: v α {, a, b, c} α u α = U {( c, { })} Future unknown Distinguish 51 / 58

82 Non-determinism (work in progress) Strategy All possible universes { } U =,,, Unbounded signals Information held: {(v α, u α )} α such that: v α {, a, b, c} α u α = U Future unknown {( c.1, { {( c.3, { })} 1 })} 3 Distinguish 51 / 58

83 Non-determinism (work in progress) Strategy All possible universes { } U =,,, Unbounded signals Information held: {(v α, u α )} α such that: v α {, a, b, c} α u α = U Future unknown ( { }), 1 ( { }),,, {( c.1, { {( c.3, { })} 1 })} 3 Distinguish 51 / 58

84 Non-determinism (work in progress) Implantation 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) Definition and example Strategy Implantation 6 conclusion 52 / 58

85 Non-determinism (work in progress) Implantation Macro-collision { ( ) } { (v α, u α ) } α meets v β, u β β { ( ) } 1 E 1 = (v α, v β ), u α u β α,β 2 E 2 = { ((v, v ), u u ) E 1 u u } U = α,β u α u β 3 E 3 = { (, w) ((, ), w) E 2 } { ({µ}, w) ((µ, ), w) E 2 ((, µ), w) E 2 } { (ρ +.µ, ρ(µ, ν) w) ((µ, ν), w) E 2 ρ = {µ, ν} } 4 Out Speed = { Speed(µ) µ, (F, w) E 3, µ F } 5 s out, Out s = { (µ, w) F, (F, w) E 3 µ F, Speed(µ) = s} } { (, w) F, (F, w) E 3 µ F, Speed(µ) s} } Compatible string encodings 53 / 58

86 Non-determinism (work in progress) Implantation Displaying the operations j main j main E 3 E 2 i main E 1 j main start 54 / 58

87 Non-determinism (work in progress) Implantation Displaying the operations j main j main E 2 E 3 Preparation same as before i main E 1 j main start 54 / 58

88 conclusion 1 Introduction 2 Signal Machines (Introduction and Definition) 3 Relations to Models of Computation 4 Intrinsically Universal Family of Signal Machines 5 Non-determinism (work in progress) 6 conclusion 55 / 58

89 conclusion Very rich setting Intrinsically universal family of signal machines Non-deterministic signal machine are not more powerful 56 / 58

90 conclusion Very rich setting Intrinsically universal family of signal machines What is the complexity? Non-deterministic signal machine are not more powerful How to extract result? What is the complexity? 56 / 58

91 conclusion Very rich setting Intrinsically universal family of signal machines What is the complexity? Non-deterministic signal machine are not more powerful How to extract result? What is the complexity? Augmented signal machines 56 / 58

92 conclusion That s all folks! Thank you for your attention 57 / 58

93 References Albert, J. and Čulik II, K. (1987). A Simple Universal Cellular Automaton and its One-Way and Totalistic Version. Complex Systems, 1:1 16. Das, R., Crutchfield, J. P., Mitchell, M., and Hanson, J. E. (1995). Evolving globally synchronized cellular automata. In Eshelman, L. J., editor, International Conference on Genetic Algorithms 95, pages Morgan Kaufmann. Durand-Lose, J. (1997). Intrinsic Universality of a 1-Dimensional Reversible Cellular Automaton. In STACS 1997, number 1200 in LNCS, pages Springer. Fischer, P. C. (1965). Generation of primes by a one-dimensional real-time iterative array. J ACM, 12(3): Goto, E. (1966). Ōtomaton ni kansuru pazuru [Puzzles on automata]. In Kitagawa, T., editor, Jōhōkagaku eno michi [The Road to information science], pages Kyoristu Shuppan Publishing Co., Tokyo. Mazoyer, J. and Rapaport, I. (1998). Inducing an Order on Cellular Automata by a Grouping Operation. In 15th Annual Symposium on Theoretical Aspects of Computer Science (STACS 1998), volume 1373 of LNCS, pages Springer. 57 / 58

94 conclusion Meunier, P., Patitz, M. J., Summers, S. M., Theyssier, G., Winslow, A., and Woods, D. (2014). Intrinsic Universality in Tile Self-Assembly Requires Cooperation. In Chekuri, C., editor, 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, pages SIAM. Ollinger, N. (2001). Two-States Bilinear Intrinsically Universal Cellular Automata. In FCT 01, number 2138 in LNCS, pages Springer. Woods, D. (2013). Intrinsic Universality and the Computational Power of Self-Assembly. In Neary, T. and Cook, M., editors, Proceedings Machines, Computations and Universality 2013, MCU 2013, Zürich, Switzerland, volume 128 of EPTCS, pages / 58