Jarkko Kari

Theory of cellular automata: a survey

This article surveys some theoretical aspects of cellular automata CA research. In particular, we discuss classical and new results on reversibility, conservation laws, limit sets, decidability questions, universality and topological dynamics of CA. The selection of topics is by no means comprehensive and reflects the research interests of the author. The main goal is to provide a tutorial of CA theory to researchers in other branches of natural computing, to give a compact collection of known results with references to their proofs, and to suggest some open problems.

Reversibility and surjectivity problems of cellular automata

The problem of deciding if a given cellular automaton (CA) is reversible (or, equivalently, if its global transition function is injective) is called the reversibility problem of CA. In this article we show that the reversibility problem is undecidable in case of two-dimensional CA. We also prove that the corresponding surjectivity problem-the problem of deciding if the global function is surjective-is undecidable for two-dimensional CA. Both problems are known to be decidable in case of one-dimensional CA. The proofs of the theorems are based on reductions from the well-known tiling problem of the plane, known also as the domino problem.

Image compression using weighted finite automata

Abstract We introduce Weighted Finite Automata (WFA) as a tool to define real functions, in particular, greyness functions of grey-tone images. Mathematical properties and the definition power of WFA have been studied by Culik and Karhumaki. Their generative power is incomparable with Barnsleys Iterative Function Systems. Here, we given an automatic encoding algorithm that converts an arbitrary grey-tone-image (a digitized photograph) into a WFA that can regenerate it (with or without information loss). The WFA seems to be the first image definition tool with such a relatively simple encoding algorithm.

The nilpotency problem of one-dimensional cellular automata

The limit set of a celullar automaton consists of all the configurations of the automaton that can appear after arbitrarily long computations. It is known that the limit set is never empty—it contains at least one homogeneous configuration. A CA is called nilpotent if its limit set contains just one configuration. The present work proves that it is algorithmically undecidable whether a given one-dimensional cellular automaton is nilpotent. The proof is based on a generalization of the well-known result about the undecidability of the tiling problem of the plane. The generalization states that the tiling problem remains undecidable even if one considers only so-called NW-deterministic tile sets, that is, tile sets in which the left and upper neighbors of each tile determine the tile uniquely. The nilpotency problem is known to be undecidable for d-dimensional CA for

Synchronizing Finite Automata on Eulerian Digraphs

d \geq 2

A small aperiodic set of Wang tiles

. The result is the basis of the proof of Rice’s theorem for CA limit sets, which states that every nontrivial property of limit set...

Rice's theorem for the limit sets of cellular automata

?ernýs conjecture and the road coloring problem are two open problems concerning synchronization of finite automata. We prove these conjectures in the special case that the vertices have uniform in- and outdegrees.

Representation of reversible cellular automata with block permutations

Abstract A new aperiodic tile set containing only 14 Wang tiles is presented. The construction is based on Mealy machines that multiply Beatty sequences of real numbers by rational constants.

Reversible cellular automata

Abstract Rices theorem is a well-known result in the theory of recursive functions. A corresponding theorem for cellular automata limit sets is proved: All nontrivial properties of limit sets of cellular automata (CAs) are shown undecidable. The theorem remains valid even if only one-dimensional CAs are considered.

The topological entropy of cellular automata is uncomputable

We demonstrate the structural invertibility of all reversible one- and two-dimensional cellular automata. More precisely, we prove that every reversible two-dimensional cellular automaton can be expressed as a combination of four block permutations, and some shift-like mappings. Block permutations are very simple functions that uniformly divide configurations into rectangular regions of equal size and apply a fixed permutation on all regions.