Computation by Asynchronously Updating Cellular Automata

A known method to compute on an asynchronously updating cellular automaton is the simulation of a synchronous computing model on it. Such a scheme requires not only an increased number of cell states, but also the simulation of a global synchronization mechanism. Asynchronous systems tend to use synchronization only on a local scale—if they use it at all. Research on cellular automata that are truly asynchronous has been limited mostly to trivial phenomena, leaving issues such as computation unexplored. This paper presents an asynchronously updating cellular automaton that conducts computation without relying on a simulated global synchronization mechanism. The two-dimensional cellular automaton employs a Moore neighborhood and 85 totalistic transition rules describing the asynchronous interactions between the cells. Despite the probabilistic nature of asynchronous updating, the outcome of the dynamics is deterministic. This is achieved by simulating delay-insensitive circuits on it, a type of asynchronous circuit that is known for its robustness to variations in the timing of signals. We implement three primitive operators on the cellular automaton from which any arbitrary delay-insensitive circuit can be constructed and show how to connect the operators such that collisions of crossing signals are avoided.

[1]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[2]  John von Neumann,et al.  Theory Of Self Reproducing Automata , 1967 .

[3]  E. F. Codd,et al.  Cellular automata , 1968 .

[4]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[5]  P. H. Kourtz,et al.  A Model a Small Forest Fire ... to Simulate Burned and Burning Areas for Use in a Detection Model , 1971 .

[6]  Robert M. Keller,et al.  Towards a Theory of Universal Speed-Independent Modules , 1974, IEEE Transactions on Computers.

[7]  K. Adkins.  Theory of spin glasses , 1974 .

[8]  Tommaso Toffoli.  Integration of the Phase-Difference Relations in Asynchronous Sequential Networks , 1978, ICALP.

[9]  Michael Creutz,et al.  Confinement and the critical dimensionality of space-time , 1979 .

[10]  E. Berlekamp,et al.  Winning Ways for Your Mathematical Plays , 1983 .

[11]  S. Wolfram.  Statistical mechanics of cellular automata , 1983 .

[12]  D. G. Green.  Shapes of simulated fires in discrete fuels , 1983 .

[13]  K. Kaneko.  Period-Doubling of Kink-Antikink Patterns, Quasiperiodicity in Antiferro-Like Structures and Spatial Intermittency in Coupled Logistic Lattice*) -- Towards a Prelude of a "Field Theory of Chaos"-- , 1984 .

[14]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  T. E. Ingerson,et al.  Structure in asynchronous cellular automata , 1984 .

[16]  C. Langton.  Self-reproduction in cellular automata , 1984 .

[17]  N. Margolus.  Physics-like models of computation☆ , 1984 .

[18]  Kaski,et al.  Growth of unstable domains in the two-dimensional Ising model. , 1985, Physical review. B, Condensed matter.

[19]  Effective discrete-time dynamics in Monte Carlo simulations. , 1986 .

[20]  Effective discrete-time dynamics in Monte Carlo simulations. , 1986, Physical review. B, Condensed matter.

[21]  Teruo Serizawa,et al.  Three-state neumann neighbor cellular automata capable of constructing self-reproducing machines , 1987, Systems and Computers in Japan.

[22]  Daniel H. Rothman,et al.  Immiscible cellular-automaton fluids , 1988 .

[23]  Alain J. Martin.  The limitations to delay-insensitivity in asynchronous circuits , 1990 .

[24]  Tommaso Toffoli,et al.  Cellular Automaton Simulation of Polymers , 1991 .

[25]  Janusz A. Brzozowski,et al.  On the Delay-Sensitivity of Gate Networks , 1992, IEEE Trans. Computers.

[26]  Walter J. Freeman,et al.  TUTORIAL ON NEUROBIOLOGY: FROM SINGLE NEURONS TO BRAIN CHAOS , 1992 .

[27]  Paul Manneville,et al.  Collective Behaviors in Spatially Extended Systems with Local Interactions and Synchronous Updating , 1992 .

[28]  M. Nowak,et al.  Evolutionary games and spatial chaos , 1992, Nature.

[29]  B A Huberman,et al.  Evolutionary games and computer simulations. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[30]  Priyadarsan Patra,et al.  Efficient building blocks for delay insensitive circuits , 1994, Proceedings of 1994 IEEE Symposium on Advanced Research in Asynchronous Circuits and Systems.

[31]  Rodney A. Brooks,et al.  Asynchrony induces stability in cellular automata based models , 1994 .

[32]  M. Biafore.  Cellular automata for nanometer-scale computation , 1994 .

[33]  Grégoire Nicolis,et al.  Synchronous versus asynchronous dynamics in spatially distributed systems , 1994 .

[34]  Scott Hauck,et al.  Asynchronous design methodologies: an overview , 1995, Proc. IEEE.

[35]  Chaté,et al.  Universal Critical Behavior in Two-Dimensional Coupled Map Lattices. , 1996, Physical review letters.

[36]  P. D. Tougaw,et al.  A device architecture for computing with quantum dots , 1997, Proc. IEEE.

[37]  Paul Manneville,et al.  Universality in Ising-like phase transitions of lattices of coupled chaotic maps , 1997 .

[38]  Priyadarsan Patra,et al.  Delay insensitive logic for RSFQ superconductor technology , 1997, Proceedings Third International Symposium on Advanced Research in Asynchronous Circuits and Systems.

[39]  Pekka Orponen,et al.  Computing with Truly Asynchronous Threshold Logic Networks , 1997, Theor. Comput. Sci..

[40]  Steven M. Nowick,et al.  An introduction to asynchronous circuit design , 1998 .

[41]  Tomas Bohr,et al.  DIRECTED PERCOLATION UNIVERSALITY IN ASYNCHRONOUS EVOLUTION OF SPATIOTEMPORAL INTERMITTENCY , 1998 .

[42]  Guillermo Abramson,et al.  Globally coupled maps with asynchronous updating. , 1998 .

[43]  H. Blok,et al.  Synchronous versus asynchronous updating in the ''game of Life'' , 1999 .

[44]  Y. Kameda,et al.  Self-timed parallel adders based on DI RSFQ primitives , 1999, IEEE Transactions on Applied Superconductivity.

[45]  B. Schönfisch,et al.  Synchronous and asynchronous updating in cellular automata. , 1999, Bio Systems.

[46]  Toshio Yanagida,et al.  A single myosin head moves along an actin filament with regular steps of 5.3 nanometres , 1999, Nature.

[47]  Norman H. Margolus,et al.  Crystalline computation , 1998, comp-gas/9811002.

[48]  R. Cowburn,et al.  Room temperature magnetic quantum cellular automata , 2000, Science.

[49]  P. Reimann.  Brownian motors: noisy transport far from equilibrium , 2000, cond-mat/0010237.

[50]  J. Gimzewski,et al.  Electronics using hybrid-molecular and mono-molecular devices , 2000, Nature.

[51]  E. D. Paolo,et al.  Rhythmic and non-rhythmic attractors in asynchronous random Boolean networks , 2001 .

[52]  Péter Gács,et al.  Deterministic computations whose history is independent of the order of asynchronous updating , 2001, ArXiv.

[53]  Kiyoshi Oguri,et al.  Asynchronous Circuit Design , 2001 .

[54]  Effect of asynchronicity on the universal behaviour of coupled map lattices , 2002, nlin/0205020.

[55]  D. Eigler,et al.  Molecule Cascades , 2002, Science.

[56]  Nobuyuki Matsui,et al.  Self-Timed Cellular Automata and their computational ability , 2002, Future Gener. Comput. Syst..

[57]  Chrystopher L. Nehaniv.  Self-reproduction in asynchronous cellular automata , 2002, Proceedings 2002 NASA/DoD Conference on Evolvable Hardware.

[58]  David G. Green,et al.  Do artificial ants march in step? ordered asynchronous processes and modularity in biological systems , 2002 .

[59]  Ferdinand Peper,et al.  Embedding Universal Delay-Insensitive Circuits in Asynchronous Cellular Spaces , 2003, Fundam. Informaticae.

[60]  F. Peper,et al.  Laying out circuits on asynchronous cellular arrays: a step towards feasible nanocomputers? , 2003 .

[61]  F. Peper,et al.  Asynchronous game of life , 2004 .

[62]  Ferdinand Peper,et al.  Universal delay-insensitive circuits with bidirectional and buffering lines , 2004, IEEE Transactions on Computers.

[63]  Ferdinand Peper,et al.  On Signals in Asynchronous Cellular Spaces , 2004, IEICE Trans. Inf. Syst..

[64]  S. Goss,et al.  Autocatalysis as a source of synchronised rhythmical activity in social insects , 1988, Insectes Sociaux.