Maximum sensitivity to update schedules of elementary cellular automata over periodic configurations

Abstract

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...

#P-completeness of counting update digraphs, cacti, and a series-parallel decomposition method

“…They also consider a surprisingly complex question: given an AN f , knowing whether there exist two block-sequential update schedules B, B such that f (B) = f (B ) , is NP-complete. The value of #UD(G) is known to be 3 n − 2 n+1 + 2 for bidirected cycles on n vertices [31], and to equal n! if and only if the digraph is a tournament on n vertices [3].…”

“…It is well known that the schedule of components update may have a great impact on the dynamics [6,10,17,19,29,31]. A block-sequential update schedule B = (B 1 , .…”

Lac Operon Boolean Models: Dynamical Robustness and Alternative Improvements

“…We will analyze its dynamical behavior through a detailed study of all its different dynamics by using the algorithms developed in [22] that, roughly speaking, list all the sets of schemes that generate exactly the same dynamics, significantly reducing the number of dynamics to analyze. These algorithms have been shown to be effective in obtaining new and relevant information for the study of concrete biological networks [34,35] as well as in the field of the cellular automata theory [36][37][38][39][40][41].…”