Stationary Distribution and Eigenvalues for a de Bruijn Process

We define a de Bruijn process with parameters n and L as a certain continuous-time Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as vertices. We determine explicitly its steady state distribution and its characteristic polynomial, which turns out to decompose into linear factors. In addition, we examine the stationary state of two specializations in detail. In the first one, the de Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de Bruin process, the distribution has constant density but nontrivial correlation functions. The two point correlation function is determined using generating function techniques.Comment: Dedicated to Herb Wilf on the occasion of his 80th birthda

Complex dynamics emerging in Rule 30 with majority memory

In cellular automata with memory, the unchanged maps of the conventional cellular automata are applied to cells endowed with memory of their past states in some specified interval. We implement Rule 30 automata with a majority memory and show that using the memory function we can transform quasi-chaotic dynamics of classical Rule 30 into domains of travelling structures with predictable behaviour. We analyse morphological complexity of the automata and classify dynamics of gliders (particles, self-localizations) in memory-enriched Rule 30. We provide formal ways of encoding and classifying glider dynamics using de Bruijn diagrams, soliton reactions and quasi-chemical representations

Probabilistic cellular automata with conserved quantities

We demonstrate that the concept of a conservation law can be naturally extended from deterministic to probabilistic cellular automata (PCA) rules. The local function for conservative PCA must satisfy conditions analogous to conservation conditions for deterministic cellular automata. Conservation condition for PCA can also be written in the form of a current conservation law. For deterministic nearest-neighbour CA the current can be computed exactly. Local structure approximation can partially predict the equilibrium current for non-deterministic cases. For linear segments of the fundamental diagram it actually produces exact results.Comment: 17 pages, 2 figure

Random walks on semaphore codes and delay de Bruijn semigroups

We develop a new approach to random walks on de Bruijn graphs over the alphabet $A$ through right congruences on $A^k$, defined using the natural right action of $A^+$. A major role is played by special right congruences, which correspond to semaphore codes and allow an easier computation of the hitting time. We show how right congruences can be approximated by special right congruences.Comment: 34 pages; 10 figures; as requested by the journal, the previous version of this paper was divided into two; this version contains Sections 1-8 of version 1; Sections 9-12 will appear as a separate paper with extra material adde

Response Curves and Preimage Sequences of Two-Dimensional Cellular Automata

We consider the problem of finding response curves for a class of binary two-dimensional cellular automata with $L$-shaped neighbourhood. We show that the dependence of the density of ones after an arbitrary number of iterations, on the initial density of ones, can be calculated for a fairly large number of rules by considering preimage sets. We provide several examples and a summary of all known results. We consider a special case of initial density equal to 0.5 for other rules and compute explicitly the density of ones after $n$ iterations of the rule. This analysis includes surjective rules, which in the case of $L$-shaped neighbourhood are all found to be permutive. We conclude with the observation that all rules for which preimage curves can be computed explicitly are either finite or asymptotic emulators of identity or shift.Comment: 7 pages, 3 figure