### A cellular automaton model of brain tumor treatment and resistance.

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The problem of synchronizing a network of identical processors that work synchronously at discrete steps is studied. Processors are arranged as an array of m rows and n columns and can exchange each other only one bit of information. We give algorithms which synchronize square arrays of (n × n) processors and give some general constructions to synchronize arrays of (m × n) processors. Algorithms are given to synchronize in time n2, $n\lceil logn\rceil $, $n\lceil \sqrt{n}\rceil $ and 2n a square array of (n × n) processors. Our approach...

This paper is a contribution to the general tiling problem for the hyperbolic plane. It is an intermediary result between the result obtained by R. Robinson [Invent. Math.44 (1978) 259–264] and the conjecture that the problem is undecidable.

We show that the theorems of Moore and Myhill hold for cellular automata whose universes are Cayley graphs of amenable finitely generated groups. This extends the analogous result of A. Machi and F. Mignosi “Garden of Eden configurations for cellular automata on Cayley graphs of groups” for groups of sub-exponential growth.

Here is presented a 6-states non minimal-time solution which is intrinsically Minsky-like and solves the three following problems: unrestricted version on a line, with one initiator at each end of a line and the problem on a ring. We also give a complete proof of correctness of our solution, which was never done in a publication for Minsky's solutions.

We present a unified and systematic approach to basic principles of Arbology, a new algorithmic discipline focusing on algorithms on trees. Stringology, a highly developed algorithmic discipline in the area of string processing, can use finite automata as its basic model of computation. For various kinds of linear notations of ranked and unranked ordered trees it holds that subtrees of a tree in a linear notation are substrings of the tree in the linear notation. Arbology uses pushdown automata...

The aim of this paper is to evaluate the growth order of the complexity function (in rectangles) for two-dimensional sequences generated by a linear cellular automaton with coefficients in $\mathbb{Z}/l\mathbb{Z}$, and polynomial initial condition. We prove that the complexity function is quadratic when l is a prime and that it increases with respect to the number of distinct prime factors of l.

We present several solutions to the Firing Squad Synchronization Problem on grid networks of different shapes. The nodes are finite state processors that work in unison with other processors and in synchronized discrete steps. The networks we deal with are: the line, the ring and the square. For all of these models we consider one- and two-way communication modes and we also constrain the quantity of information that adjacent processors can exchange at each step. We first present synchronization...