A Classification of Weak Asynchronous Models of Distributed Computing

We conduct a systematic study of asynchronous models of distributed computing consisting of identical finite-state devices that cooperate in a network to decide if the network satisfies a given graph-theoretical property. Models discussed in the literature differ in the detection capabilities of the agents residing at the nodes of the network (detecting the set of states of their neighbors, or counting the number of neighbors in each state), the notion of acceptance (acceptance by halting in a particular configuration, or by stable consensus), the notion of step (synchronous move, interleaving, or arbitrary timing), and the fairness assumptions (non-starving, or stochastic-like). We study the expressive power of the combinations of these features, and show that the initially twenty possible combinations fit into seven equivalence classes. The classification is the consequence of several equi-expressivity results with a clear interpretation. In particular, we show that acceptance by halting configuration only has non-trivial expressive power if it is combined with counting, and that synchronous and interleaving models have the same power as those in which an arbitrary set of nodes can move at the same time. We also identify simple graph properties that distinguish the expressive power of the seven classes

A guided tour of asynchronous cellular automata

Research on asynchronous cellular automata has received a great amount of attention these last years and has turned to a thriving field. We survey the recent research that has been carried out on this topic and present a wide state of the art where computing and modelling issues are both represented.Comment: To appear in the Journal of Cellular Automat

Most Permissive Semantics of Boolean Networks

As shown in (http://dx.doi.org/10.1101/2020.03.22.998377), the usual update modes of Boolean networks (BNs), including synchronous and (generalized) asynchronous, fail to capture behaviors introduced by multivalued refinements. Thus, update modes do not allow a correct abstract reasoning on dynamics of biological systems, as they may lead to reject valid BN models.This technical report lists the main definitions and properties of the most permissive semantics of BNs introduced in http://dx.doi.org/10.1101/2020.03.22.998377. This semantics meets with a correct abstraction of any multivalued refinements, with any update mode. It subsumes all the usual updating modes, while enabling new behaviors achievable by more concrete models. Moreover, it appears that classical dynamical analyzes of reachability and attractors have a simpler computational complexity:- reachability can be assessed in a polynomial number of iterations. The computation of iterations is in NP in the very general case, and is linear when local functions are monotonic, or with some usual representations of functions of BNs (binary decision diagrams, Petri nets, automata networks, etc.). Thus, reachability is in P with locally-monotonic BNs, and P$^{\text{NP}}$ otherwise (instead of being PSPACE-complete with update modes);- deciding wherever a configuration belongs to an attractor is in coNP with locally-monotonic BNs, and coNP$^{\text{coNP}}$ otherwise (instead of PSPACE-complete with update modes).Furthermore, we demonstrate that the semantics completely captures any behavior achievable with any multilevel or ODE refinement of the BN; and the semantics is minimal with respect to this model refinement criteria: to any most permissive trajectory, there exists a multilevel refinement of the BN which can reproduce it.In brief, the most permissive semantics of BNs enables a correct abstract reasoning on dynamics of BNs, with a greater tractability than previously introduced update modes

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin

Coalescence in fully asynchronous elementary cellular automata

International audienceCellular automata (CA) are discrete mathematical systems formed by a set of cells arranged in a regular fashion. Each of these cells is in a particular state and evolves according to a local rule depending on the state of the cells in its neighbourhood. In spite of their apparent simplicity, these dynamical systems are able to display a complex emerging behaviour, and the macroscopic structures they produce are not always predictable despite complete local knowledge. While studying the robustness of CA to the introduction of asynchronism in their updating scheme, a phenomenon called coalescence was observed for the first time: for some asynchronous CA, the application of the same local rule on any two di↵erent initial conditions following the same sequence of updates quickly led to the same non-trivial configuration. Afterwards, it was experimentally found that some CA would always coalesce whilst others would never coalesce, and that some of them exhibit a phase transition between a coalescing and non-coalescing behaviour. However, a formal explanation of non-trivial rapid coalescence has yet to be found, and this is the purpose of this project, where we try to characterise and explain this phenomenon both qualitatively and analytically. In particular, we analytically study trivial coalescence, find lower bounds for the coalescence time of ECA 154 and ECA 62, and give some first steps towards finding their upper bounds in order to prove that they have, respectively, quadratic and linear coalescence time