Abstract geometrical computation: small Turing universal signal - PowerPoint PPT Presentation

Abstract geometrical computation: small Turing universal signal machines Abstract geometrical computation: small Turing universal signal machines J´ erˆ ome Durand-Lose Laboratoire d’Informatique Fondamentale d’Orl´ eans, Universit´ e d’Orl´ eans, Orl´ eans , FRANCE IW Complexity of Simple Programs December 6-7, 2008 – Cork, Irland

Abstract geometrical computation: small Turing universal signal machines Introduction 1 Turing machines 2 Cellular automata 3 Cyclic tag systems 4 Conclusion 5

Abstract geometrical computation: small Turing universal signal machines Introduction Introduction 1 Turing machines 2 Cellular automata 3 Cyclic tag systems 4 Conclusion 5

Abstract geometrical computation: small Turing universal signal machines Introduction Context Collision based computing Idealization continuous space continuous time dimensionless particles/signals

Abstract geometrical computation: small Turing universal signal machines Introduction Abstract geometrical computation Signal machines meta-signals (finitely many) their speed/velocity collision rules Signals, e.g. red (with speed 1) at position xx blue (with speed -1) at position yy Collision, e.g. rule { green , red } → { blue } application

Abstract geometrical computation: small Turing universal signal machines Introduction An example Time Space

Abstract geometrical computation: small Turing universal signal machines Introduction More examples

Abstract geometrical computation: small Turing universal signal machines Introduction More complex examples

Abstract geometrical computation: small Turing universal signal machines Turing machines Introduction 1 Turing machines 2 Cellular automata 3 Cyclic tag systems 4 Conclusion 5

Abstract geometrical computation: small Turing universal signal machines Turing machines Turing machines? q Finite automata Read/write head . . . input Tape ^ # #

Abstract geometrical computation: small Turing universal signal machines Turing machines Simulation Iterations of a Turing machine Corresponding signal machine q f ← − q f ^ ^ b b a b # b ← − q f b q f a # ^ b b a b # ← − q f b q 2 ← − q f ^ b b a # # q 1 − → → − ← − q f # # ← − ^ b b # # # a # b b − → − → − → # # q 2 q 1 ← ← − − # # ^ ^ b a # # # − → q 1 q 1 → − q 1 ^ a a # # # a q i ← − q 1 ^ a b # # # a # → − q i q i ^ a b # # # → − q i b q i a − → q i ^ ^ a b # # #

Abstract geometrical computation: small Turing universal signal machines Turing machines How many meta-signals? Meta-signal 1 symbol 1 � 1 state 2 � for finitness − → # , ← # , − − → # , 4 # � | Γ | + 2 | Q | + 4 Results universal semi-universal 18 (Woods and Neary, 2007) 7 (Smith, 2007)

Abstract geometrical computation: small Turing universal signal machines Cellular automata Introduction 1 Turing machines 2 Cellular automata 3 Cyclic tag systems 4 Conclusion 5

Abstract geometrical computation: small Turing universal signal machines Cellular automata Cellular automata Rule 110 and one transition implementation o one n e R e n o Output 0 1 1 0 1 1 1 0 L z Input 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 0 0 0 R e o r r one o e z L Evolution and simulation on 11 framed by ω (10) and (011) ω 0 1 1 1 0 0 1 1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 1 0 1 1 0 1

Abstract geometrical computation: small Turing universal signal machines Cellular automata How many meta-signals? Meta-signal 1 state 3 � for finitness Regular pattern on both side � expensive 3 | Q | +??? Results universal semi-universal not interesting 6 (Cook, 2004)

in a c hromosome. This de�nes one generation of the GA; it is rep eated G times for one GA run. F ( � ) is a random v ariable since its v alue dep ends on the particular set of I ICs selected to I ev aluate � . Th us, a CA's �tness v aries sto c hastically from generation to generation. F or this reason, w e c ho ose a new set of ICs at eac h generation F or our exp erimen ts w e set P = 100, E = 20; I = 100, m = 2; and G = 50. M w as c hosen from a P oisson distribution with mean 320 (sligh tly greater than 2 N ). V arying M prev en ts selecting CAs that are adapted to a particular M . A justi�cation of these parameter settings is giv en in [9]. W e p erformed a total of 65 GA runs. Since F ( � ) is only a rough estimate of p erformance, 100 N w e more stringen tly measured the qualit y of the GA's solutions b y calculating P ( � ) with 4 10 N 2 f 149 ; 599 ; 999 g for the b est CAs in the �nal generation of eac h run. In 20% of the runs N the GA disco v ered successful CAs ( P = 1 : 0). More detailed analysis of these successful CAs 4 10 sho w ed that although they w ere distinct in detail, they used similar strategies for p erforming the sync hronization task. In terestingly , when the GA w as restricted to ev olv e CAs with r = 1 and N r = 2, all the ev olv ed CAs had P � 0 for N 2 f 149 ; 599 ; 999 g . (Better p erforming CAs with 4 10 r = 2 can b e designed b y hand.) Th us r = 3 app ears to b e the minimal radius for whic h the GA can successfully solv e this problem. Abstract geometrical computation: small Turing universal signal machines Cellular automata Link CA-ACG 0 Figure 1: (a) Space-time diagram of � starting with a random initial condition. (b) The same space- sy nc time diagram after �ltering with a spatial transducer that maps all domains to white and all defects to blac k. Greek letters lab el particles describ ed in the text. Figure 1a giv es a space-time diagram for one of the GA-disco α v ered CAs with 100% p erfor- γ β β γ mance, here called � . This diagram plots 75 successiv e con�gurations on a lattice of size sy nc N = 75 (with time going do wn the page) starting from a randomly c hosen IC, with 1-sites col- Time ν ored blac k and 0-sites colored white. In this example, global sync hronization o ccurs at time step µ δ 58. γ Ho w are w e to understand the strategy emplo y ed b y � to reac h global sync hronization? sy nc δ Notice that, under the GA, while crosso v er and m utation act on the lo cal mappings comprising a 4 74 0 Site 74 0 Site 74 (a) Space-time diagram. (b) Filtered space-time diagram. � Das-Crutchfield-Mitchell-Hanson95 c

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Introduction 1 Turing machines 2 Cellular automata 3 Cyclic tag systems 4 Conclusion 5

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Cyclic tag system? Definition a binary word a circular list Dynamics 101 011 :: h :: 0110 :: 01011 Halt empty word halt appendant (here h) cycle (too expensive to test)

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Cyclic tag system? Definition a binary word a circular list Dynamics 101 011 011 :: h :: 0110 :: 01011 :: 011 Halt empty word halt appendant (here h) cycle (too expensive to test)

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Cyclic tag system? Definition a binary word a circular list Dynamics 101011 011 :: h :: 0110 :: 01011 :: 011 :: h Halt empty word halt appendant (here h) cycle (too expensive to test)

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Cyclic tag system? Definition a binary word a circular list Dynamics 101011 0110 011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110 Halt empty word halt appendant (here h) cycle (too expensive to test)

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Cyclic tag system? Definition a binary word a circular list Dynamics 1010110110 011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110 :: 01011 Halt empty word halt appendant (here h) cycle (too expensive to test)

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Cyclic tag system? Definition a binary word a circular list Dynamics 1010110110 011 011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110 :: 01011:: 011 Halt empty word halt appendant (here h) cycle (too expensive to test)

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Cyclic tag system? Definition a binary word a circular list Dynamics 101011 0110011 011 :: h :: 0110 :: 01011 :: 011 :: h :: 0110 :: 01011:: 011 :: h Halt empty word halt appendant (here h) cycle (too expensive to test)

Abstract geometrical computation: small Turing universal signal machines Cyclic tag systems Simulation 101 and 011 :: 10 :: 10 :: 01 101 & 011 :: h :: 0110 :: 01011 initial configuration e R n last o zero zero zero zero zero one first one one one one one last sep sep sep R go R one last go LL one iteration