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Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. Characterizing all the invariant measures of a cellular automaton could be challenging. Nevertheless, we establish a connection between the invariance of Gibbs measures (used in statistical(More)
We study the group-valued and semigroup-valued conservation laws in cellular automata (CA). We provide examples to distinguish between semigroup-valued, group-valued and real-valued conservation laws. We prove that, even in one-dimensional case, it is undecidable if a CA has any non-trivial conservation law of each type. For a fixed range, each CA has a(More)
The problem of describing the dynamics of a conserved energy in a cellular automaton in terms of local movements of " particles " (quanta of that energy) has attracted some people's attention. The one-dimensional case was already solved by Fuk´s (2000) and Pivato (2002). For the two-dimensional cellular automata, we show that every (context-free)(More)
We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional(More)
Conservation laws in physics are numerical invariants of the dynamics of a system. In cellular automata (CA), a similar concept has already been defined and studied. To each local pattern of cell states a real value is associated , interpreted as the " energy " (or " mass " , or. . .) of that pattern. The overall " energy " of a configuration is simply the(More)
Conservation laws in cellular automata (CA) are studied as an abstraction of the conservation laws observed in nature. In addition to the usual real-valued conservation laws we also consider more general group-valued and semigroup-valued conservation laws. The (algebraic) conservation laws in a CA form a hierarchy, based on the range of the interactions(More)
We analyze a basic building block of gene regulatory networks using a stochastic/geometric model in search of a mathematical backing for the discrete modeling frameworks. We consider a network consisting only of two interacting genes: a source gene and a target gene. The target gene is activated by the proteins encoded by the source gene. The interaction is(More)