arXiv subjects

Martin Schaller

Publications and source records attributed to Martin Schaller.

4 recordsLinked to original sources

arXiv2025

Irreducible Rules and Equivalence Classes of One-dimensional Cellular Automata

One-dimensional cellular automata are discrete dynamical systems that operate on an infinite lattice of sites and are characterized by the locality and uniformity of their update rule. Permutations of the state set and isometric transformations of the lattice induce symmetry transformations on the set of local rules and the set of global maps of cellular automata, resulting in a partitioning of the set of cellular automata into equivalence classes. The concept of an irreducible local rule that depends on all its coordinates is used to analyse the equivalence classes and results on the number of equivalence classes of irreducible binary local rules and binary global maps are presented. Finally, another symmetry operator based on the scaling of neighbourhoods is introduced and the change in the number of equivalence classes is analysed.

nlin.CG↗

arXiv2025

Orbits of One-Dimensional Cellular Automata Induced by Symmetry Transformations

Using a group-theoretic approach, a method for determining the equivalence classes (also called orbits) of the set of rules of one-dimensional cellular automata induced by the symmetry operations of reflection and permutation and their product is presented. Orbits are classified by their isomorphism type. Results for the number of orbits and the number of orbits by type for state sets of size two and three are included.

nlin.CG↗

arXiv2009

Zeno Squeezing of Cellular Automata

We have recently introduced the two new computing models of self-similar cellular automata and self-similar Petri nets. Self-similar automata result from a progressive, infinite tessellation of space and time. Self-similar Petri nets consist of a potentially infinite sequence of coupled transitions with ever increasing firing rates. Both models are capable of hypercomputations and can, for instance, ``solve'' the halting problem for Turing machines. We survey the main definitions and propositions and add new results regarding the indeterminism of self-similar cellular automata.

physics.gen-ph↗

arXiv2009

Scale-invariant cellular automata and self-similar Petri nets

Two novel computing models based on an infinite tessellation of space-time are introduced. They consist of recursively coupled primitive building blocks. The first model is a scale-invariant generalization of cellular automata, whereas the second one utilizes self-similar Petri nets. Both models are capable of hypercomputations and can, for instance, "solve" the halting problem for Turing machines. These two models are closely related, as they exhibit a step-by-step equivalence for finite computations. On the other hand, they differ greatly for computations that involve an infinite number of building blocks: the first one shows indeterministic behavior whereas the second one halts. Both models are capable of challenging our understanding of computability, causality, and space-time.

nlin.CG↗