Aracena, J., Demongeot, J., Fanchon, E., & Montalva, M. (2013). On the number of different dynamics in Boolean networks with deterministic update schedules. Math. Biosci., 242(2), 188–194.
Abstract: Deterministic Boolean networks are a type of discrete dynamical systems widely used in the modeling of genetic networks. The dynamics of such systems is characterized by the local activation functions and the update schedule, i.e., the order in which the nodes are updated. In this paper, we address the problem of knowing the different dynamics of a Boolean network when the update schedule is changed. We begin by proving that the problem of the existence of a pair of update schedules with different dynamics is NPcomplete. However, we show that certain structural properties of the interaction digraph are sufficient for guaranteeing distinct dynamics of a network. In [1] the authors define equivalence classes which have the property that all the update schedules of a given class yield the same dynamics. In order to determine the dynamics associated to a network, we develop an algorithm to efficiently enumerate the above equivalence classes by selecting a representative update schedule for each class with a minimum number of blocks. Finally, we run this algorithm on the well known Arabidopsis thaliana network to determine the full spectrum of its different dynamics. (C) 2013 Elsevier Inc. All rights reserved.

Aracena, J., Demongeot, J., Fanchon, E., & Montalva, M. (2013). On the number of update digraphs and its relation with the feedback arc sets and tournaments. Discret Appl. Math., 161(1011), 1345–1355.
Abstract: An update digraph corresponds to a labeled digraph that indicates a relative order of its nodes introduced to define equivalence classes of deterministic update schedules yielding the same dynamical behavior of a Boolean network. In Aracena et al. [1], the authors exhibited relationships between update digraphs and the feedback arc sets of a given digraph G. In this paper, we delve into the study of these relations. Specifically, we show differences and similarities between both sets through increasing and decreasing monotony properties in terms of their structural characteristics. Besides, we prove that these sets are equivalent if and only if all the digraph circuits are cycles. On the other hand, we characterize the minimal feedback arc sets of a given digraph in terms of their associated update digraphs. In particular, for complete digraphs, this characterization shows a close relation with acyclic tournaments. For the latter, we show that the size of the associated equivalence classes is a power of two. Finally, we determine exactly the number of update digraphs associated to digraphs containing a tournament. (C) 2013 Elsevier B.V. All rights reserved.

Perrot, K., MontalvaMedel, M., de Oliveira, P. P. B., & Ruivo, E. L. P. (2020). Maximum sensitivity to update schedules of elementary cellular automata over periodic configurations. Nat. Comput., 19(1), 51–90.
Abstract: This work is a thoughtful extension of the ideas sketched in Montalva et al. (AUTOMATA 2017 exploratory papers proceedings, 2017), aiming at classifying elementary cellular automata (ECA) according to their maximal onestep sensitivity to changes in the schedule of cells update. It provides a complete classification of the ECA rule space for all period sizes n[ 9 and, together with the classification for all period sizes n <= 9 presented in Montalva et al. (Chaos Solitons Fractals 113:209220, 2018), closes this problem and opens further questionings. Most of the 256 ECA rule's sensitivity is proved or disproved to be maximum thanks to an automatic application of basic methods. We formalize meticulous case disjunctions that lead to the results, and patch failing cases for some rules with simple arguments. This gives new insights on the dynamics of ECA rules depending on the proof method employed, as for the last rules 45 and 105 requiring o0011THORN induction patterns.

Ruivo, E. L. P., MontalvaMedel, M., de Oliveira, P. P. B., & Perrot, K. (2018). Characterisation of the elementary cellular automata in terms of their maximum sensitivity to all possible asynchronous updates. Chaos Solitons Fractals, 113, 209–220.
Abstract: Cellular automata are fullydiscrete dynamical systems with global behaviour depending upon their locally specified state transitions. They have been extensively studied as models of complex systems as well as objects of mathematical and computational interest. Classically, the local rule of a cellular automaton is iterated synchronously over the entire configuration. However, the question of how asynchronous updates change the behaviour of a cellular automaton has become a major issue in recent years. Here, we analyse the elementary cellular automata rule space in terms of how many different onestep trajectories a rule would entail when taking into account all possible deterministic ways of updating the rule, for one time step, over all possible initial configurations. More precisely, we provide a characterisation of the elementary cellular automata, by means of their onestep maximum sensitivity to all possible update schedules, that is, the property that any change in the update schedule causes the rule's onestep trajectories also to change after one iteration. Although the onestep maximum sensitivity does not imply that the remainder of the timeevolutions will be distinct, it is a necessary condition for that. (C) 2018 Elsevier Ltd. All rights reserved.
