An Implementation of von Neumann's Self-Reproducing Machine

1995 ◽  
Vol 2 (4) ◽  
pp. 337-354 ◽  
Author(s):  
Umberto Pesavento
Keyword(s):  
Cellular Automaton ◽  
State Transition ◽  
The State ◽  
Transition Rule ◽  
John Von Neumann ◽  
Von Neumann ◽  
Special Case

This article describes in detail an implementation of John von Neumann's self-reproducing machine. Self-reproduction is achieved as a special case of construction by a universal constructor. The theoretical proof of the existence of such machines was given by John von Neumann in the early 1950s [6], but was first implemented in 1994, by the author in collaboration with R. Nobili. Our implementation relies on an extension of the state-transition rule of von Neumann's original cellular automaton. This extension was introduced to simplify the design of the constructor. The main operations in our constructor can be mapped into operations of von Neumann's machine.

scholarly journals Characterization of the Evolution of Nonlinear Uniform Cellular Automata in the Light of Deviant States

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Pabitra Pal Choudhury ◽  
Sudhakar Sahoo ◽  
Mithun Chakraborty
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Boolean Functions ◽  
State Transition ◽  
The State ◽  
Transition Diagram ◽  
One Dimensional ◽  
State Transition Diagram ◽  
Linear Rule

Dynamics of a nonlinear cellular automaton (CA) is, in general asymmetric, irregular, and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable, primarily due to the presence of a matrix handle. In this paper, we present a novel technique of studying the properties of the State Transition Diagram of a nonlinear uniform one-dimensional cellular automaton in terms of its deviation from a suggested linear model. We have considered mainly elementary cellular automata with neighborhood of size three, and, in order to facilitate our analysis, we have classified the Boolean functions of three variables on the basis of number and position(s) of bit mismatch with linear rules. The concept of deviant and nondeviant states is introduced, and hence an algorithm is proposed for deducing the State Transition Diagram of a nonlinear CA rule from that of its nearest linear rule. A parameter called the proportion of deviant states is introduced, and its dependence on the length of the CA is studied for a particular class of nonlinear rules.

scholarly journals EVOLUTION OF TWO-DIMENSIONAL CELLULAR AUTOMATA. NEW FORMS OF PRESENTATION

2021 ◽  
Vol 3 (1) ◽  
pp. 85-90
Author(s):  
S. M. Bilan ◽  
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Color Image ◽  
The State ◽  
Dynamic Processes ◽  
State Change ◽  
Two Dimensional ◽  
Time Step ◽  
Von Neumann ◽  
Elementary Cellular Automata

The paper considers cellular automata and forms of reflection of their evolution. Forms of evolution of elementary cellular automata are known and widely used, which allowed specialists to model different dynamic processes and behavior of systems in different directions. In the context of the easy construction of the form of evolution of elementary cellular automata, difficulties arise in representing the form of evolution of two-dimensional cellular automata, both synchronous and asynchronous. The evolution of two-dimensional cellular automata is represented by a set of states of two-dimensional forms of cellular automata, which complicates the perception and determination of the dynamics of state change. The aim of this work is to solve the problem of a fixed mapping of the evolution of a two-dimensional cellular automaton in the form of a three-dimensional representation, which is displayed in different colors on a two-dimensional image The paper proposes the evolution of two-dimensional cellular automata in the form of arrays of binary codes for each cell of the field. Each time step of the state change is determined by the state of the logical "1" or "0". Moreover, each subsequent state is determined by increasing the binary digit by one. The resulting binary code identifies the color code that is assigned to the corresponding cell at each step of the evolution iteration. As a result of such coding, a two-dimensional color matrix (color image) is formed, which in its color structure indicates the evolution of a two-dimensional cellular automaton. To represent evolution, Wolfram coding was used, which increases the number of rules for a two-dimensional cellular automaton. The rules were used for the von Neumann neighborhood without taking into account the own state of the analyzed cell. In accordance with the obtained two-dimensional array of codes, a discrete color image is formed. The color of each pixel of such an image is encoded by the obtained evolution code of the corresponding cell of the two-dimensional cellular automaton with the same coordinates. The bitness of the code depends on the number of time steps of evolution. The proposed approach allows us to trace the behavior of the cellular automaton in time depending on its initial states. Experimental analysis of various rules for the von Neumann neighborhood made it possible to determine various rules that allow the shift of an image in different directions, as well as various affine transformations over images. Using this approach, it is possible to describe various dynamic processes and natural phenomena.

Bittorio revisited: structural coupling in the Game of Life

2019 ◽  
Vol 28 (4) ◽  
pp. 197-212 ◽  
Author(s):  
Randall D Beer
Keyword(s):  
Cellular Automaton ◽  
State Transition ◽  
The State ◽  
Theoretical Frameworks ◽  
Transition Graph ◽  
Structural Coupling ◽  
Game Of Life ◽  
Interaction Graphs ◽  
State Transition Graph ◽  
Autopoietic System

The notion of structural coupling plays a central role in Maturana and Varela’s biology of cognition framework and strongly influenced Varela’s subsequent enactive elaboration of this framework. Building upon previous work using a glider in the Game of Life (GoL) cellular automaton as a toy model of a minimal autopoietic system with which to concretely explore these theoretical frameworks, this article presents an analysis of structural coupling between a glider and its environment. Specifically, for sufficiently small GoL universes, we completely characterize the nonautonomous dynamics of both a glider and its environment in terms of interaction graphs, derive the set of possible glider lives determined by the mutual constraints between these interaction graphs, and show how such lives are embedded in the state transition graph of the entire GoL universe.

Pelé Meets John Von Neumann in the Penalty Area

2014 ◽  
Author(s):  
Ignacio Palacios-Huerta
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
General Theory ◽  
Minimax Theorem ◽  
Common Interest ◽  
John Von Neumann ◽  
Von Neumann ◽  
Complete Test ◽  
Zero Sum ◽  
Special Case

The movie A Beautiful Mind (2001) portrays the life and work of John F. Nash Jr., who received the Nobel Prize in Economics in 1994. A class of his theories deals with how people should behave in strategic situations that involve what are known as “mixed strategies,” that is, choosing among various possible strategies when no single one is always the best when you face a rational opponent. This chapter uses data from a specific play in soccer (a penalty kick) with professional players to provide the first complete test of a fundamental theorem in game theory: the minimax theorem. The minimax theorem can be regarded as a special case of the more general theory of Nash. It applies only to two-person, zero-sum or constant-sum games, whereas the Nash equilibrium concept can be used with any number of players and any mixture of conflict and common interest in the game.

John von Neumann

2004 ◽  
Vol 174 (12) ◽  
pp. 1371 ◽  
Author(s):  
Mikhail I. Monastyrskii
Keyword(s):  
John Von Neumann ◽  
Von Neumann

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

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