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Boolean automata networks (aka Boolean networks) are space-time discrete dynamical systems, studied as a model of computation and as a representative model of natural phenomena. A collection of simple entities (the automata) update their 0-1 states according to local rules. The dynamics of the network is highly sensitive to update modes, i.e., to the schedule according to which the automata apply their local rule. A new family of update modes appeared recently, called block-parallel, which is dual to the well studied block-sequential. Although basic, it embeds the rich feature of update repetitions among a temporal updating period, allowing for atypical asymptotic behaviors. In this paper, we prove that it is able to breed complex computations, squashing almost all decision problems on the dynamics to the traditionally highest (for reachability questions) class PSPACE. Despite obtaining these complexity bounds for a broad set of local and global properties, we also highlight a surprising gap: bijectivity is still coNP.

Among the fundamental questions in computer science is that of the impact of synchronism/asynchronism on computations, which has been addressed in various fields of the discipline: in programming, in networking, in concurrence theory, in artificial learning, etc. In this paper, we tackle this question from a standpoint which mixes discrete dynamical system theory and computational complexity, by highlighting that the chosen way of making local computations can have a drastic influence on the performed global computation itself. To do so, we study how distinct update schedules may fundamentally change the asymptotic behaviors of finite dynamical systems, by analyzing in particular their limit cycle maximal period. For the message itself to be general and impacting enough, we choose to focus on a ``simple'' computational model which prevents underlying systems from having too many intrinsic degrees of freedom, namely elementary cellular automata. More precisely, for elementary cellular automata rules which are neither too simple nor too complex (the problem should be meaningless for both), we show that update schedule changes can lead to significant computational complexity jumps (from constant to superpolynomial ones) in terms of their temporal asymptotes.

In this paper, we explore the dynamic behavior of threshold networks on undirected signed graphs. Much attention has been dedicated to understand the convergence and long-term behavior of this model. Yet, an open question persists: How does the underlying graph structure impact network dynamics? Similar studies have been carried out for threshold networks and other types of networks, but these primarily focus on unsigned networks. Here, we address the question on signed threshold networks. We introduce the stability index of a graph, related to the concepts of frustration and balance in signed graphs, to establish a connection between the structure and the dynamics of such networks. We show that graphs which present a negative stability index exhibit stable dynamics, i.e., the dynamics converges to fixed points regardless of its threshold parameters. Conversely, if at least one subgraph has a positive stability index, oscillations in long term behavior may appear. Furthermore, we generalize the analysis to network dynamics under periodic update modes and explore the case of the existence of some subgraph with a positive stability index, for which we find that attractors of super-polynomial period in the size of the network may appear.

We state an algorithm that, given an automata network and a block-sequential update schedule, produces an automata network of the same size or smaller with the same limit dynamics under the parallel update schedule. Then, we focus on the family of automata cycles which share a unique path of automata, called tangential cycles, and show that a restriction of our algorithm allows to reduce any instance of these networks under a block-sequential update schedule into a smaller parallel network of the family and to characterize the number of reductions operated while conserving their limit dynamics. We also show that any tangential cycles reduced by our main algorithm are transformed into a network whose size is that of the largest cycle of the initial network. We end by showing that the restricted algorithm allows the direct characterization of block-sequential double cycles as parallel ones., Comment: Accepted at CIE 2023

When we focus on finite dynamical systems from both the computability/complexity and the modelling standpoints, automata networks seem to be a particularly appropriate mathematical model on which theory shall be developed. In this paper, automata networks are finite collections of entities (the automata), each automaton having its own set of possible states, which interact with each other over discrete time, interactions being defined as local functions allowing the automata to change their state according to the states of their neighbourhoods. The studies on this model of computation have underlined the very importance of the way (i.e. the schedule) according to which the automata update their states, namely the update modes which can be deterministic, periodic, fair, or not. Indeed, a given network may admit numerous underlying dynamics, these latter depending highly on the update modes under which we let the former evolve. In this paper, we pay attention to a new kind of deterministic, periodic and fair update mode family introduced recently in a modelling framework, called the block-parallel update modes by duality with the well-known and studied block-sequential update modes. More precisely, in the general context of automata networks, this work aims at presenting what distinguish block-parallel update modes from block-sequential ones, and at counting and enumerating them: in absolute terms, by keeping only representatives leading to distinct dynamics, and by keeping only representatives giving rise to distinct isomorphic limit dynamics. Put together, this paper constitutes a first theoretical analysis of these update modes and their impact on automata networks dynamics.

In automata networks, it is well known that the way entities update their states over time has a major impact on their dynamics. In particular, depending on the chosen update schedule, the underlying dynamical systems may exhibit more or less asymptotic dynamical behaviours such as fixed points or limit cycles. Since such mathematical models have been used in the framework of biological networks modelling, the question of choosing appropriate update schedules has arised soon. In this note, focusing on Boolean networks, our aim is to emphasise that the adequate way of thinking regulations and genetic expression over time is certainly not to consider a wall segregating synchronicity from asynchronicity because they actually complement rather well. In particular, we highlight that specific update schedules, namely block-parallel update schedules, whose intrinsic features are still not known from a theoretical point of view, admit realistic and pertinent properties in the context of biological modelling and deserve certainly more attention from the community.

In this paper we study the dynamical behavior of Boolean networks with firing memory, namely Boolean networks whose vertices are updated synchronously depending on their proper Boolean local transition functions so that each vertex remains at its firing state a finite number of steps. We prove in particular that these networks have the same computational power than the classical ones, ie any Boolean network with firing memory composed of $m$ vertices can be simulated by a Boolean network by adding vertices. We also prove general results on specific classes of networks. For instance, we show that the existence of at least one delay greater than 1 in disjunctive networks makes such networks have only fixed points as attractors. Moreover, for arbitrary networks composed of two vertices, we characterize the delay phase space, \ie the delay values such that networks admits limit cycles or fixed points. Finally, we analyze two classical biological models by introducing delays: the model of the immune control of the $\lambda$-phage and that of the genetic control of the floral morphogenesis of the plant \emph{Arabidopsis thaliana}.

Our daily social and political life is more and more impacted by social networks. The functioning of our living bodies is deeply dependent on biological regulation networks such as neural, genetic, and protein networks. And the physical world in which we evolve, is also structured by systems of interacting particles. Interaction networks can be seen in all spheres of existence that concern us, and yet, our understanding of interaction networks remains severely limited by our present lack of both theoretical and applied insight into their clockworks. In the past, efforts at understanding interaction networks have mostly been directed towards applications. This has happened at the expense of developing understanding of the generic and fundamental aspects of interaction networks. Intrinsic properties of interaction networks (eg the ways in which they transmit information along entities, their ability to produce this or that kind of global dynamical behaviour depending on local interactions) are thus still not well understood. Lack of fundamental knowledge tends to limit the innovating power of applications. Without more theoretical fundamental knowledge, applications cannot evolve deeply and become more impacting. Hence, it is necessary to better apprehend and comprehend the intrinsic properties of interaction networks, notably the relations between their architecture and their dynamics and how they are affected by and set in time. In this chapter, we use the elementary mathematical model of Boolean automata networks as a formal archetype of interaction networks. We survey results concerning the role of feedback cycles and the role of intersections between feedback cycles, in shaping the asymptotic dynamical behaviours of interaction networks.

Boolean networks are discrete dynamical systems where each automaton has its own Boolean function for computing its state according to the configuration of the network. The updating mode then determines how the configuration of the network evolves over time. Many of updating modes from the literature, including synchronous and asynchronous modes, can be defined as the composition of elementary deterministic configuration updates, i.e., by functions mapping configurations of the network. Nevertheless, alternative dynamics have been introduced using ad-hoc auxiliary objects, such as that resulting from binary projections of Memory Boolean networks, or that resulting from additional pseudo-states for Most Permissive Boolean networks. One may wonder whether these latter dynamics can still be classified as updating modes of finite Boolean networks, or belong to a different class of dynamical systems. In this paper, we study the extension of updating modes to the composition of non-deterministic updates, i.e., mapping sets of finite configurations. We show that the above dynamics can be expressed in this framework, enabling a better understanding of them as updating modes of Boolean networks. More generally, we argue that non-deterministic updates pave the way to a unifying framework for expressing complex updating modes, some of them enabling transitions that cannot be computed with elementary and non-elementary deterministic updates., Comment: Accepted at AUTOMATA 2021

This paper details a method for optimising the size of Boolean automata networks in order to compute their attractors under the parallel update schedule. This method relies on the formalism of modules introduced recently that allows for (de)composing such networks. We discuss the practicality of this method by exploring examples. We also propose results that nail the complexity of most parts of the process, while the complexity of one part of the problem is left open., Comment: 14 pages

Automata networks are a very general model of interacting entities, with applications to biological phenomena such as gene regulation. In many contexts, the order in which entities update their state is unknown, and the dynamics may be very sensitive to changes in this schedule of updates. Since the works of Aracena et. al, it is known that update digraphs are pertinent objects to study non-equivalent block-sequential update schedules. We prove that counting the number of equivalence classes, that is a tight upper bound on the synchronism sensitivity of a given network, is #P-complete. The problem is nevertheless computable in quasi-quadratic time for oriented cacti, and for oriented series-parallel graphs thanks to a decomposition method.

Boolean networks are a general model of interacting entities, with applications to biological phenomena such as gene regulation. Attractors play a central role, and the schedule of entities update is a priori unknown. This article presents results on the computational complexity of problems related to the existence of update schedules such that some limit-cycle lengths are possible or not. We first prove that given a Boolean network updated in parallel, knowing whether it has at least one limit-cycle of length $k$ is $\text{NP}$-complete. Adding an existential quantification on the block-sequential update schedule does not change the complexity class of the problem, but the following alternation brings us one level above in the polynomial hierarchy: given a Boolean network, knowing whether there exists a block-sequential update schedule such that it has no limit-cycle of length $k$ is $\Sigma_2^\text{P}$-complete.

Modules were introduced as an extension of Boolean automata networks. They have inputs which are used in the computation said modules perform, and can be used to wire modules with each other. In the present paper we extend this new formalism and study the specific case of acyclic modules. These modules prove to be well described in their limit behavior by functions called output functions. We provide other results that offer an upper bound on the number of attractors in an acyclic module when wired recursively into an automata network, alongside a diversity of complexity results around the difficulty of deciding the existence of cycles depending on the number of inputs and the size of said cycle., Comment: 21 pages

Boolean automata networks (BANs) are a generalisation of Boolean cellular automata. In such, any theorem describing the way BANs compute information is a strong tool that can be applied to a wide range of models of computation. In this paper we explore a way of working with BANs which involves adding external inputs to the base model (via modules), and more importantly, a way to link networks together using the above mentioned inputs (via wirings). Our aim is to develop a powerful formalism for BAN (de)composition. We formulate two results: the first one shows that our modules/wirings definition is complete, the second one uses modules/wirings to prove simulation results amongst BANs.

In this article we study the minimum number $\kappa$ of additional automata that a Boolean automata network (BAN) associated with a given block-sequential update schedule needs in order to simulate a given BAN with a parallel update schedule. We introduce a graph that we call $\mathsf{NECC}$ graph built from the BAN and the update schedule. We show the relation between $\kappa$ and the chromatic number of the $\mathsf{NECC}$ graph. Thanks to this $\mathsf{NECC}$ graph, we bound $\kappa$ in the worst case between $n/2$ and $2n/3+2$ ($n$ being the size of the BAN simulated) and we conjecture that this number equals $n/2$. We support this conjecture with two results: the clique number of a $\mathsf{NECC}$ graph is always less than or equal to $n/2$ and, for the subclass of bijective BANs, $\kappa$ is always less than or equal to $n/2+1$.

We prove that the fully asynchronous dynamics of a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ without negative loop can be simulated, in a very specific way, by a monotone Boolean network with $2n$ components. We then use this result to prove that, for every even $n$, there exists a monotone Boolean network $f:\{0,1\}^n\to\{0,1\}^n$, an initial configuration $x$ and a fixed point $y$ of $f$ such that: (i) $y$ can be reached from $x$ with a fully asynchronous updating strategy, and (ii) all such strategies contains at least $2^{\frac{n}{2}}$ updates. This contrasts with the following known property: if $f:\{0,1\}^n\to\{0,1\}^n$ is monotone, then, for every initial configuration $x$, there exists a fixed point $y$ such that $y$ can be reached from $x$ with a fully asynchronous strategy that contains at most $n$ updates., Comment: To appear in the proceedings of ACA 2016 (fourth International Workshop on Asynchronous Cellular Automata and Asynchronous Discrete Models)

Boolean automata networks (BANs) are a well established model for biological regulation systems such as neural networks or genetic networks. Studies on the dynamics of BANs, whether it is synchronous or asynchronous, have mainly focused on monotonic networks, where fundamental questions on the links relating their static and dynamical properties have been raised and addressed. This paper explores analogous questions on asynchronous non-monotonic networks, xor-BANs, that are BANs where all the local transition functions are xor-functions. Using algorithmic tools, we give a general characterisation of the asynchronous transition graphs for most of the cactus xor-BANs and strongly connected xor-BANs. As an illustration of the results, we provide a complete description of the asynchronous dynamics of two particular classes of xor-BAN, namely xor-Flowers and xor-Cycle Chains. This work also leads to new bisimulation equivalences specific to xor-BANs.

The understanding of Boolean automata networks dynamics takes an important place in various domains of computer science such as computability, complexity and discrete dynamical systems. In this paper, we make a step further in this understanding by focusing on their cycles, whose necessity in networks is known as the brick of their complexity. We present new results that provide a characterisation of the transient and asymptotic dynamics, i.e. of the computational abilities, of asynchronous Boolean automata networks composed of two cycles that intersect at one automaton, the so-called double-cycles. To do so, we introduce an efficient formalism inspired by algorithms to define long sequences of updates, that allows a better description of their dynamics than previous works in this area.

In this paper are presented first results of a theoretical study on the role of non-monotone interactions in Boolean automata networks. We propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours according to two axes. The first one consists in supporting the idea that non-monotony has a peculiar influence on the sensitivity to synchronism of such networks. It leads us to the second axis that presents preliminary results and builds an understanding of the dynamical behaviours, in particular concerning convergence times, of specific non-monotone Boolean automata networks called XOR circulant networks., Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.2498

This paper aims at setting the keystone of a prospective theoretical study on the role of non-monotone interactions in biological regulation networks. Focusing on discrete models of these networks, namely, Boolean automata networks, we propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours. More precisely, in this paper, we start by detailing some motivations, both mathematical and biological, for our interest in non-monotony, and we discuss how it may account for phenomena that cannot be produced by monotony only. Then, to build some understanding in this direction, we propose some preliminary results on the dynamical behaviour of some specific non-monotone Boolean automata networks called XOR circulant networks.

This paper investigates questions related to the modularity in discrete models of biological interaction networks. We develop a theoretical framework based on the analysis of their asymptotic dynamics. More precisely, we exhibit formal conditions under which agents of interaction networks can be grouped into modules. As a main result, we show that the usual decomposition in strongly connected components fulfils the conditions of being a modular organisation. Furthermore, we point out that our framework enables a finer analysis providing a decomposition in elementary modules.

Although models are built on the basis of some observations of reality, the concepts that derive theoretically from their definitions as well as from their characteristics and properties are not necessarily direct consequences of these initial observations. Indeed, many of them rather follow from chains of theoretical inferences that are only based on the precise model definitions and rely strongly, in addition, on some consequential working hypotheses. Thus, it is important to address the question of which features of a model effectively carry some modelling meaning and which only result from the task of formalising observations of reality into a mathematical language. In this article, we address this question with a theoretical point view that sets our discussion strictly between the two stages of the modelling process that require knowledge of real systems, that is, between the initial stage that chooses a global theoretical framework to build the model and the final stage that exploits its formal predictions by comparing them to the reality that the model was designed to simulate. Taking Boolean automata networks as instances of models of systems observed in reality, we analyse in this setting the remaining stages of the modelling process and we show how the meaning of theoretical concepts can subtly rely on formal choices such as definitions and hypotheses.

In this report, we present a formal approach that addresses the problem of emergence of phase transitions in stochastic and attractive nonlinear threshold Boolean automata networks. Nonlinear networks considered are informally defined on the basis of classical stochastic threshold Boolean automata networks in which specific interaction potentials of neighbourhood coalition are taken into account. More precisely, specific nonlinear terms compose local transition functions that define locally the dynamics of such networks. Basing our study on nonlinear networks, we exhibit new results, from which we derive conditions of phase transitions.

In this paper, we propose and evaluate a distributed protocol to manage trust diffusion in ad hoc networks. In this protocol, each node i maintains a \trust value" about an other node j which is computed both as a result of the exchanges with node j itself and as a function of the opinion that other nodes have about j. These two aspects are respectively weighted by a trust index that measures the trust quality the node has in its own experiences and by a trust index representing the trust the node has in the opinions of the other nodes. Simulations have been realized to validate the robustness of this protocol against three kinds of attacks: simple coalitions, Trojan attacks and detonator attacks.