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For non-uniform cellular automata (NUCA) with finite memory over an arbitrary universe with multiple local transition rules, we show that pointwise nilpotency, pointwise periodicity, and pointwise eventual periodicity properties are…
We present a method to eliminate redundancy in the transition tables of Boolean automata: schema redescription with two symbols. One symbol is used to capture redundancy of individual input variables, and another to capture permutability in…
In this work we introduce a deterministic scheme of synchronization of linear and nonlinear cellular automata (CA) with complex behavior, connected through a master-slave coupling. By using a definition of Boolean derivative, we use the…
We show that a behaviour analogous to degenerate hyperbolicity can occur in nearest-neighbour cellular automata (CA) with three states. We construct a 3-state rule by "lifting" elementary CA rule 140. Such "lifted" rule is equivalent to…
Various non-classical approaches of distributed information processing, such as neural networks, computation with Ising models, reservoir computing, vector symbolic architectures, and others, employ the principle of collective-state…
We give a one-dimensional quantum cellular automaton (QCA) capable of simulating all others. By this we mean that the initial configuration and the local transition rule of any one-dimensional QCA can be encoded within the initial…
A new class of deterministic dynamical systems, termed semipredictable dynamical systems, is presented. The spatiotemporal evolution of these systems have both predictable and unpredictable traits, as found in natural complex systems. We…
Cellular automata (CA) are dynamical systems defined by a finite local rule but they are studied for their global dynamics. They can exhibit a wide range of complex behaviours and a celebrated result is the existence of (intrinsically)…
Efficient representations of data are essential for processing, exploration, and human understanding, and Principal Component Analysis (PCA) is one of the most common dimensionality reduction techniques used for the analysis of large,…
Abelian cellular automata (CA) are CA which are group endomorphisms of the full group shift when endowing the alphabet with an abelian group structure. A CA randomizes an initial probability measure if its iterated images weak *-converge…
We rigorously prove a form of disorder-resistance for a class of one-dimensional cellular automaton rules, including some that arise as boundary dynamics of two-dimensional solidification rules. Specifically, when started from a random…
We study self-similarity in one-dimensional probabilistic cellular automata (PCA) using the renormalization technique. We introduce a general framework for algebraic construction of renormalization groups (RG) on cellular automata and apply…
We present a new classification of elementary cellular automata. It is based on the structure of the network of states, connected with the transitions between them; the latter are determined by the automaton rule. Recently an algorithm has…
Cellular automata on the triangular grid are here referred to as Triangular Automata (TA). This paper focuses on the simplest class of TA, called Elementary Triangular Automata (ETA). They are argued to be the two-dimensional counterpart of…
Neural Cellular Automata (NCA) are a powerful combination of machine learning and mechanistic modelling. We train NCA to learn complex dynamics from time series of images and PDE trajectories. Our method is designed to identify underlying…
Every transitive cellular automaton (CA) is sensitive to initial conditions. We study this implication in the more general context of non-uniform cellular automata (NUCA) with finitely many different local update rules assigned to cells. We…
Descriptive complexity may be useful to design programs in a natural declarative way. This is important for parallel computation models such as cellular automata, because designing parallel programs is considered difficult. Our paper…
We prove that the group of reversible cellular automata (RCA), on any alphabet $A$, contains a subgroup generated by three involutions which contains an isomorphic copy of every finitely generated group of RCA on any alphabet $B$. This…
We have discovered a novel transition rule for binary cellular automata (CA) that yields self-replicating structures across two spatial and temporal scales from sparsely populated random initial conditions. Lower-level, shapeshifting…
Quantum cellular automata (QCA) evolve qubits in a quantum circuit depending only on the states of their neighborhoods and model how rich physical complexity can emerge from a simple set of underlying dynamical rules. For instance,…