**Abstract** : Cellular automata (CA) are formal models for the simulation and the study of many complex systems encountered in natural or even social sciences. The success of this model is essentially due to its three main properties: locality, synchronicity and uniformity. Over the years it became apparent that many phenomena, such as the chemical reactions occurring in a living cell, can still be termed complex systems although they are neither synchronous nor uniform. These new modeling requirements led to new variants of the classical cellular automata model. Each new variant is obtained by relaxing one or more of the three main properties above.
This special issue is concerned with non-uniform cellular automata (NUCA), i.e., cellular automata in which each single cell is allowed to have its own local updating rule, which may be different from the one of its neighbors. The articles in this issue will show that this change in the formal model deeply impacts all our previous knowledge on the subject and most of the classical results have to be either reproved or disproved. Indeed, the set of objects under investigation has dramatically changed in nature from several points of view. There are countably many CA while the number of NUCA is uncountable. The CA are the shift commuting continuous functions from a symbolic space to itself, while NUCA constitute the whole set of continuous functions over the space. This special issue is at the same time a witness of the growing interest in this matter and an invitation to further develop these studies.
The contributions are the result of a selection process which followed an international call for papers. Fifteen papers were submitted and seven were selected. Each submission received two reviews from internationally renowned scientists in the domain. We believe that the selected papers are a good representative of the current trends in the study of non-uniform cellular automata, understood in a broad sense.
Below we briefly review the content of the papers. Each time we point out the different facets of non-uniformity the paper addresses.
In “ Computational complexity of threshold automata networks under different updating schemes”, by Goles and Montealegre, the authors investigate finite networks of automata in which any single automaton might have a different local rule. Threshold local rules are considered here. In this setting there is also another concept which comes into play, namely, the updating policy, i.e., the temporal sequence of updates among the automata in the network. The authors prove that the complexity of the prediction problem belongs to different complexity classes according to the update policy that is chosen.
Along the same line of thought, “ Attractor stability in nonuniform Boolean networks”, by Kuhlman and Mortveit, considers automata networks with Boolean states. The authors address the structure of the phase space of such systems. In particular, they build a hierarchy of networks where the attractor graph may have an arbitrary number of ergodic sets (connected components of the attractor graph). These results provide a starting point for additional studies into the attractor structure of Boolean networks, their basins of attractions, and conditions for stability under state noise.
In “ Limit cycle structure for dynamic bi-threshold systems”, by Wu, Adiga and Mortveit, the authors investigate the dynamics of automata networks governed by dynamic bi-threshold functions under synchronous and sequential updating policies. Their results generalize the celebrated result by Goles and Olivos on synchronous neural networks (1981), the work by Kuhlman et al. (2012) on static bi-threshold systems, and the work by Chang et al. (2013) on dynamic standard threshold systems.
“ Around probabilistic cellular automata”, by Mairesse and Marcovici, addresses another kind of non-uniformity by considering probabilistic CA. According to this model, at a given time and for a given site, the local rule is applied with some probability, otherwise the identity is applied. The authors survey results coming from several approaches, ranging from combinatorics, statistical physics, to computer science.
In “ Three research directions in non-uniform cellular automata” by Dennunzio, Formenti and Provillard, “standard” NUCA are addressed. After reviewing known results about structural stability, the authors show that sensitivity to initial conditions is not structurally stable. The second part of the paper reports complexity results about the main dynamical properties. The exploration of the fixed points set of non-uniform cellular automata is addressed in the final part of this work.
“ Realization problems for nonuniform cellular automata”, by Salo, proves a realization result: for any subshift of finite type (SFT), a set of local rules is provided such that the sequences in the SFT correspond precisely to the arrangements of local rules that make the NUCA surjective (injective). Surjectivity of subclasses of nonuniform cellular automata, and realizability questions for other properties, in particular number conservation and chain transitivity, are also considered.
“ The firing squad synchronization problem on CA with multiple updating cycles”, by Manzoni and Umeo, introduces another kind of non-uniformity, by considering CA in which each cell may have a different clock. The authors show that also in this particular situation the classical firing squad problem can be solved in linear time (w.r.t. the size of the automaton).
The editors wish to thank all the authors who made this special issue possible. We also warmly thank all the reviewers for their valuable help in selecting the best submissions and in improving the overall paper quality.