Modelling cellular automata with partial differential equations

Omohundro, Stephen M.

doi:10.1016/0167-2789(84)90255-0

Cited by 77 publications

(27 citation statements)

References 3 publications

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“…Computations may be supported by many different systems (1,2), including physical systems like the digital computer, Fredkin logic gates (3), billiard-ball collisions (4), enzymes operating on a polymer chain (1,5), and more abstract systems like cellular automata (6)(7)(8), partial differential equations that simulate cellular automata (9), generalized shifts (4), and neural networks (10)(11)(12)(13). Some of these systems can be computationally universal and thus are formally equivalent with a universal Turing machine (10,14).…”

Section: Introductionmentioning

confidence: 99%

Chemical implementation of neural networks and Turing machines.

Hjelmfelt

Weinberger

Ross

1991

Proc. Natl. Acad. Sci. U.S.A.

245

175

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We propose a reversible reaction mechanism with a single stationary state in which certain concentrations assume either high or low values dependent on the concentration of a catalyst. The properties of this mechanism are those of a McCulloch-Pitts neuron. We suggest a mechanism of interneuronal connections in which the stationary state of a chemical neuron is determined by the state of other neurons in a homogeneous chemical system and is thus a "hardware" chemical implementation of neural networks. Specific connections are determined for the construction of logic gates: AND, NOR, etc. Neural networks may be constructed in which the flow oftime is continuous and computations are achieved by the attainment of a stationary state of the entire chemical reaction system, or in which the flow of time is discretized by an oscillatory reaction. In another article, we will give a chemical implementation of finite state machines and stack memories, with which in principle the construction of a universal Turing machine is possible.Computations may be supported by many different systems (1, 2), including physical systems like the digital computer, Fredkin logic gates (3), billiard-ball collisions (4), enzymes operating on a polymer chain (1, 5), and more abstract systems like cellular automata (6-8), partial differential equations that simulate cellular automata (9), generalized shifts (4), and neural networks (10-13). Some of these systems can be computationally universal and thus are formally equivalent with a universal Turing machine (10,14). We may inquire about whether computationally universal devices may be constructed solely from chemical reaction mechanisms in a homogeneous medium. All living entities process information to varying degrees, and this can occur only by chemical means. It is for this reason alone that the subject is ofinterest. In this article, we discuss the construction of chemical networks where coupled reaction mechanisms implement "programmed" computations as the concentrations evolve in time. It has already been noted that bistable chemical systems are in many ways analogous to a flip-flop circuit, by coupling bistable reactions it is possible to build universal automata (15,16), and that various chemical mechanisms share a formal relationship with electronic devices (17, 18). We address the construction of computational devices from the viewpoint of neural networks. We propose a chemical reaction network, which is a "hardware" implementation of a neural network, and hence the network can in principle be as powerful as a universal Turing machine (10).Neural networks are a versatile basis for computation (19). Any finite state machine, and hence the finite state part of a universal Turing machine, can be simulated by a neural network (10,20). Neural networks also form the basis of many collective computational systems such as feedforward networks or Hopfield's network (11-13). A chemical neural network may serve as the "hardware" for any of the approaches to computation. We present hardw...

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Section: Introductionmentioning

confidence: 99%

Chemical implementation of neural networks and Turing machines.

Hjelmfelt

Weinberger

Ross

1991

Proc. Natl. Acad. Sci. U.S.A.

245

175

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“…A system of nine first order quasi-linear partial differential equations has been offered as a computationally-universal system [32]. If a deterministic dynamical system has a generating partition [21] then the symbolic dynamics can in principle be solved and the future behavior can be 2 2…”

Section: 21mentioning

confidence: 99%

Nonintegrability, chaos, and complexity

McCauley

1997

Physica A: Statistical Mechanics and its Applications

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Two dimensional driven-dissipative flows are generally integrable via a conservation law that is singular at equilibria. Nonintegrable dynamical systems are confined to n≥3 dimensions. Even driven-dissipative deterministic dynamical systems that are critical, chaotic or complex have n-1 local time-independent conservation laws that can be used to simplify the geometric picture of the flow over as many consecutive time intervals as one likes. Those conservation laws generally have either branch cuts, phase singularities, or both. The consequence of the existence of singular conservation laws for experimental data analysis, and also for the search for scale-invariant critical states via uncontrolled approximations in deterministic dynamical systems, is discussed. Finally, the expectation of ubiquity of scaling laws and universality classes in dynamics is contrasted with the possibility that the most interesting dynamics in nature may be nonscaling, nonuniversal, and to some degree computationally complex.

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“…CAs and coupled map lattices are used in modeling a broad class of physical phenomena [33]. Coupled map lattices can be considered time and space discretizations of PDEs, which are thus computationally universal as well [34].…”

Section: Discrete Time Modelsmentioning

confidence: 99%

Analog computation with dynamical systems

Siegelmann

Fishman

1998

Physica D: Nonlinear Phenomena

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A b s t r a c tPhysical systems exhibit various levels of complexity: their long term dynamics may converge to fixed points or exhibit complex chaotic behavior. This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes Pd, Co-RPd, NPd and EXP,t, corresponding to their standard counterparts, according to the complexity of their long term behavior. The complexity of chaotic attractors relative to regular ones leads to the conjecture Pa :fi NPj. Continuous time flows have been proven useful in solving various practical problems. Our theory provides the tools for an algorithmic analysis of such flows. As an example we analyze the continuous Hopfield network. © 1998 Elsevier Science B.V. All rights reserved. 05.45; 89.80 Kevwords: Theory of analog computation; Dynamical systems PACS: I n t r o d u c t i o nDigital computers can be viewed as dynamical systems. A dynamical system is defined by a set of equations that describe the evolution of a vector of variables called the state of the system [1][2][3][4][5][6]. It is a map of a phase space onto itself. Considering the configuration of a Turing machine (the mathematical abstraction of a digital computer) [7] as a state in some space, the update rules describing the behavior of the machine constitute a dynamical system. The input for the machine is the initial condition for the dynamical system. The series of consecutive configurations during the compu-* Corresponding author. E-mail: iehava@ie.technion.ac.il. 1 E-mail: fishman@physics.technion.ac.il. tation process define a trajectory in the space of Turing machine configurations. This trajectory may converge to a halting configuration from which the output can be read. The behavior of physical systems can also be modeled by dynamical systems. A physical system is prepared in some initial state from which it evolves in its phase space according to some equations of motion. Many physical systems are dissipative and hence converge to attractors; the attractor may be a simple fixed point, a limit cycle or it may be chaotic with a fractal dimension. This correspondence between computers and physical systems, through the common description by dynamical systems prompts us to view the evolution of physical systems as a process of computation. Analyzing this process and its complexity can contribute to the understanding of computation in nature.0167-2789/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII: S0167-2789(98)00057-8 H. T. Siegelmann, S. Fishman/Physica D 120 (1998) In the theory of computation problems are clas...

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Modelling cellular automata with partial differential equations

Cited by 77 publications

References 3 publications

Chemical implementation of neural networks and Turing machines.

Chemical implementation of neural networks and Turing machines.

Nonintegrability, chaos, and complexity

Analog computation with dynamical systems

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