An exact characterization of the different dynamical behavior that exhibit the space phase of a reversible and conservative cellular automaton, the so called Q2R model, is shown in this paper. Q2R is a cellular automaton which is a dynamical variation of the Ising model in statistical physics and whose space of configurations grows exponentially with the system size. As a consequence of the intrinsic reversibility of the model, the phase space is composed only by configurations that belong to a fixed point or a limit cycle. In this work we classify them in four types accordingly to well differentiated topological characteristics. Three of them -which we call of type S-I, S-II and S-III-share a symmetry property, while the fourth, which we call of type AS, does not. Specifically, we prove that any configuration of Q2R belongs to one of the four previous limit cycles. Moreover, at a combinatorial level, we are able to determine the number of limit cycles for some small periods which are almost always present in the Q2R. Finally, we provide a general overview of the resulting decomposition of the arbitrary size Q2R phase space, in addition, we realize an exhaustive study of a small Ising system (4 × 4) which is fully analyzed under this new framework.

In Veliz-Cuba and Stigler 2011, Boolean models were proposed for the lac operon in Escherichia coli capable of reproducing the operon being OFF, ON and bistable for three (low, medium and high) and two (low and high) parameters, representing the concentration ranges of lactose and glucose, respectively. Of these 6 possible combinations of parameters, 5 produce results that match with the biological experiments of Ozbudak et al. 2004. In the remaining one, the models predict the operon being OFF while biological experiments show a bistable behavior. In this paper, we first explore the robustness of two such models in the sense of how much its attractors change against any deterministic update schedule. We prove mathematically that, in cases where there is no bistability, all the dynamics in both models lack limit cycles while, when bistability appears, one model presents 30% of its dynamics with limit cycles while the other only 23%. Secondly, we propose two alternative improvements consisting of biologically supported modifications; one in which both models match with Ozbudak et al. 2004 in all 6 combinations of parameters and, the other one, where we increase the number of parameters to 9, matching in all these cases with the biological experiments of Ozbudak et al. 2004.

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This work is a thoughtful extension of the ideas sketched in [10], aiming at classifying elementary cellular automata (ECA) according to their maximal one-step 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 [11], 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 (0011) * induction patterns. Keywords Synchronism sensitivity • Elementary cellular automata • update digraph 1 Introduction Cellular automata (CAs) are discrete dynamical systems (both in time and space) where each cell of a lattice is given a state from a finite set. States evolve according to local interaction rules among neighbors. The neighborhood and rule are uniform, they can be described as a set of relative positions and a function giving a new state for a cell from states on that set of relative positions. Even with basic local rules, cellular automata may display very complex global behavior: well-known binary cellular automata rules such as the game of life [8] and rule 110 [4] are known to be capable of simulating Turing machines. Classically, CA cells are updated synchronously at every time step: the state of a cell at time t + 1 is a function of its neighbors' at time t [9]. However, the introduction of variations in the schedule (order) according to which cells are updated has been gaining attention in recent years, unveiling richer behaviors. This has been observed both on the dynamics and problem-solving ability, and regarding the simulation of real-world phenomena; see [7] for a comprehensive review. The present work focuses on deterministic evolution, i.e. introduction of variations in a deterministically followed update schedule, and not a stochastic one. In this context a basic question to answer is: given a CA, how much do changes in the schedule of cells update carry changes in its dynamical behavior? We approach this qualitatively by the notion of maximum update schedule sensitivity and the classification that follows. In the more general framework of automata networks, Aracena el al. [2, 3, 1] gave a relevant notion of equivalence classes of update schedule, with a constructive characterization of the equivalence relation. We base our work on an application of this characterization to elementary cellular automata (ECA), and measure the sensitivity to update schedules as the number of different dynamics obtained over the set of update schedule equivalence classes. Given two non-equivalent update schedules ∆ ≡ ∆ , the two obtained dynamics differ if and only if they diffe...

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