Let AZAZ be the Cantor space of bi-infinite sequences in a finite alphabet AA, and let σσ be the ... more Let AZAZ be the Cantor space of bi-infinite sequences in a finite alphabet AA, and let σσ be the shift map on AZAZ. A cellular automaton is a continuous, σσ-commuting self-map ΦΦ of AZAZ, and a ΦΦ-invariant subshift is a closed, (Φ,σ)(Φ,σ)-invariant subset S⊂AZ. Suppose a∈AZ is S-admissible everywhere except for some small region we call a defect. It has been empirically observed that such defects persist under iteration of ΦΦ, and often propagate like ‘particles’. We characterize the motion of these particles, and show that it falls into several regimes, ranging from simple deterministic motion, to generalized random walks, to complex motion emulating Turing machines or pushdown automata. One consequence is that some questions about defect behaviour are formally undecidable.
If X is a discrete abelian group and B a finite set, then a cellular automaton (CA) is a continuo... more If X is a discrete abelian group and B a finite set, then a cellular automaton (CA) is a continuous map F:B^X-->B^X that commutes with all X-shifts. If g is a real-valued function on B, then, for any b in B^X, we define G(b) to be the sum over all x in X of g(b_x) (if finite). We say g is `conserved' by F if G is constant under the action of F. We characterize such `conservation laws' in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.
Let AZAZ be the Cantor space of bi-infinite sequences in a finite alphabet AA, and let σσ be the ... more Let AZAZ be the Cantor space of bi-infinite sequences in a finite alphabet AA, and let σσ be the shift map on AZAZ. A cellular automaton is a continuous, σσ-commuting self-map ΦΦ of AZAZ, and a ΦΦ-invariant subshift is a closed, (Φ,σ)(Φ,σ)-invariant subset S⊂AZ. Suppose a∈AZ is S-admissible everywhere except for some small region we call a defect. It has been empirically observed that such defects persist under iteration of ΦΦ, and often propagate like ‘particles’. We characterize the motion of these particles, and show that it falls into several regimes, ranging from simple deterministic motion, to generalized random walks, to complex motion emulating Turing machines or pushdown automata. One consequence is that some questions about defect behaviour are formally undecidable.
If X is a discrete abelian group and B a finite set, then a cellular automaton (CA) is a continuo... more If X is a discrete abelian group and B a finite set, then a cellular automaton (CA) is a continuous map F:B^X-->B^X that commutes with all X-shifts. If g is a real-valued function on B, then, for any b in B^X, we define G(b) to be the sum over all x in X of g(b_x) (if finite). We say g is `conserved' by F if G is constant under the action of F. We characterize such `conservation laws' in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.
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