Firing Squad Synchronization Problem in Cellular Automata

Umeo, Hiroshi

doi:10.1007/978-1-4939-8700-9_211

Cited by 12 publications

(2 citation statements)

References 72 publications

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“…Cellular Automata-CA uses a different mechanism [23][24][25][26][27][28][29][30][31][32][33][34][35] to represent a given function. In a one-dimensional form of CA, a N -length binary sequence is…”

Section: Cellular Automata Representationsmentioning

confidence: 99%

Hierarchical Organization of Variant Logic

Zheng

2018

Variant Construction From Theoretical Foundation to Applications

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In modern logic, various systems have been proposed extending classical Boolean logic & switching theory. Such logic frameworks include multiple-valued logic, probability logic, fuzzy logic, module logic, quantum logic and various other frameworks. Although these extensions have been applied to many applications in mathematics, in science and in engineering, all extensions to Boolean logic invalidates at least one of the six fundamental rules of Boolean logic shown in L1 to L6. We propose a new framework of logic, variant logic, extending Boolean logic whilst satisfying the six fundamental rules (L1-L6). By defining the Variant-Invariant behaviour of logical operations, this framework can be constructed using four types of general operators. Main results of the chapter are summarized in Theorems 8-10, respectively. To show significant differences between classical logic and new variant logic, invariant properties of this hierarchical organization are discussed. Simplest cases of one-variable conditions are illustrated. Variant logic can provide the necessary framework to support analysis and description of Cellular Automata, Fractal Theory, Chaos Theory and other systems dealing with complexity. Such applications of this framework will be explored in future papers. Keywords Switching theory • Boolean/multiple valued/probability/fuzzy logic Variant/invariant property • Hierarchical organization • Variant logic

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Section: Cellular Automata Representationsmentioning

confidence: 99%

Hierarchical Organization of Variant Logic

Zheng

2018

Variant Construction From Theoretical Foundation to Applications

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“…Following the introduction of Conway's Game of Life [10], Stephan Wolfram from the 1980s [11,12] started to apply Boolean algebra to describe the behaviour of Cellular Automata. His approach used a binary counting sequence to naming different rules of behaviour based upon the functions generating the next iteration in the game.…”

Section: Introductionmentioning

confidence: 99%

Variant Logic Construction Under Permutation and Complementary Operations on Binary Logic

Zheng

2018

Variant Construction From Theoretical Foundation to Applications

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This chapter presents a binary logic framework whose function elements are invariant under permutation and complementary operations. The entire framework is described using 4 levels of hierarchy: n variables, 2 n states, 2 2 n functions, and 2 n !2 2 n logic functionals. Under the proposed framework, it is possible to determine higher level function complexity by analysing lower levels of organisation characteristics. These characteristics can be determined quite accurately because the symmetry conditions of variable and state organisations have invariant logic functions and a corresponding logic functional organisation. More symmetrical arrangement at state level creates more symmetrical permutations within the function space. Lower level properties are highly influential on the higher level properties of function components within a logic functional space. The proposed framework provides a logic foundation to describe complex binary systems using lower level properties, making analysis of systems more efficient and less calculation intensive. Different global coding schemes are discussed and typical two-variable cases of logic functionals are illustrated.

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