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- Jarkko Kari, Siamak Taati
- 2013

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)

A 1D Reversible Cellular Automata (RCA) with forward and backward radius-1 2 neighborhoods is called Rectangular. It was previously conjectured that the conservation laws in 1D Rectangular RCA can be described as linear combinations of independent constant-speed flows to the right or to the left. This is indeed the case; so is a similar statement about a… (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)

- Tucs Dissertations, Siamak Taati, Supervisor, Jarkko Kari, Cristopher Moore, Bruno Durand +2 others
- 2009

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)

A conservation law in a cellular automaton is the statement of the invari-ance of a local and additive energy-like quantity. This chapter reviews the basic theory of conservation laws in cellular automata. A general mathematical framework for formulating conservation laws in cellular automata is presented and several characterizations of them are… (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)