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Internal symmetries of cellular automata
1Instituto de Investigacion en Communicacion Optica, Universidad Autonoma de San Luis Potosi 78000, San Luis Potosi, SLP, Mexico.
Chaos (Woodbury, N.Y.)
|June 5, 2003
Summary
Cellular automaton symmetries can be encoded as group homomorphisms. This research presents algorithms to compute these symmetries and classify automata, revealing dynamical implications.
Area of Science:
- Theoretical Computer Science
- Dynamical Systems Theory
- Algebraic Automata Theory
Background:
- Cellular automata (CA) are discrete dynamical systems with applications in modeling complex phenomena.
- Internal symmetries of CA transformations are crucial for understanding their behavior and structure.
- Existing methods for analyzing CA symmetries are limited.
Purpose of the Study:
- To define and characterize internal symmetries of cellular automata.
- To develop a method for encoding the full group of internal symmetries as a group homomorphism.
- To present algorithms for computing and classifying automata based on their symmetries.
Main Methods:
- Defining internal transformations as bi-infinite sequences of permutations.
- Establishing the condition for a pair of transformations to be an internal symmetry (f=theta(-1)fgamma).
- Representing the symmetry group as a homomorphism (F) where theta=F(gamma).
Main Results:
- The full group of internal symmetries of a cellular automaton can be encoded as a group homomorphism.
- The homomorphism F, which maps gamma to theta, is presented by a local automaton-like rule.
- The domain and image of F generally have infinite order.
Conclusions:
- Internal symmetries of cellular automata can be systematically analyzed and computed using group homomorphisms.
- Algorithms are provided for computing the symmetry homomorphism and classifying automata.
- The study discusses the dynamical implications of these internal symmetries, offering insights into automaton behavior.
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