A Continuous-Time Binary Consensus Protocol with Hysteretic Units Applied to the Density Classification Problem

In this paper, we present a continuous-time binary consensus protocol whereby entities connected via a directed ring topology solve the one-dimensional density classification problem. In our model, the participating entities behave as non-ideal relays, that is, they have memory of the trajectory of an internal state variable, which gives them hysteretic properties. We show that this feature is necessary for the system to reach consensus on the state shared by the initial majority. The connections between this protocol and collective decisionmaking mechanisms in swarm intelligence systems are also discussed. Introduction The density classification problem (also known as the majority problem), consists in classifying finite linear binary strings according to whether they have a majority of 0’s or 1’s. This problem has been the subject of numerous studies in the cellular automata literature (see, e.g., Mitchell et al. (1994); Land and Belew (1995); Fukś (1997, 2002); AlonsoSanz and Bull (2009); Fates (2013)) because it has a simple formulation and illustrates very well the idea of “emergent computation” since the cells interact only locally and do not have access to the global structure they are trying to classify. In a cellular automata setting, the density classification problem translates into finding evolution rules for cells that make the automata be all equal to 0 if initially there were more than half of the cells in a 0 state, or 1 if initially there were more than half of the cells in a 1 state. The idea of emergent computation is also present in another computational paradigm called swarm intelligence (Bonabeau et al., 1999; Dorigo and Birattari, 2007). In this paradigm, relatively simple agents locally interact with one another and with their environment to produce self-organized spatio-temporal patterns that represent solutions to problems that no individual agent could solve on its own (e.g., finding shortest paths (Goss et al., 1989), sorting (Deneubourg et al., 1990), or constructing nests (Grasse, 1959)). As we explain in the next section, the density classification problem is relevant in swarm intelligence because some of the methods that solve it may be used as collective decision-making mechanisms for swarms. In this paper, we further explore the connection between cellular automata and swarm intelligence by presenting and analyzing a consensus protocol1 on networks of agents derived from previous work on collective decision-making in swarms (Montes de Oca et al., 2012). Our consensus protocol may be seen as encoding evolution rules for continuous-time cellular automata with memory. It may also be thought of as a model of social influence in a group whose members observe the actions performed by other individuals, increasing, as a result, their tendency to perform the observed actions. We are interested in this type of mechanisms because we want to eventually endow swarms of robots or agents with collective-decision mechanisms that are robust, flexible and effective in real environments. Our approach is backed by recent studies on social learning (Rendell et al., 2010), that support the idea that learning from the observation of others’ actions is a mechanism whereby individuals indirectly probe the environment. In a swarm, which is typically composed of many individuals, using the behavior of others as a guide provides each individual with potentially many indirect channels for obtaining information about their environment, and thereby increasing the amount of information they use to make decisions. The key finding presented in this paper is that if agents influence each other as if arranged in a unidirectional ring, and each agent integrates over time information coming to it from its neighbor using a mechanism akin to exponential smoothing (Gardner Jr., 2006), then symmetric blocks of 0’s or 1’s propagate through the network indefinitely. Moreover, we provide evidence that when the symmetry of these blocks is broken (that is, there is a majority of 0’s or 1’s), then the information wave propagates for a finite amount of time and eventually dies out, which translates into the population of agents reaching a consensus. Finally, the state on which the We use the term protocol to comply with literature tradition in communication networks and where interaction rules are called protocols. See, for example, (Mesbahi and Egerstedt, 2010). ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems population reaches a consensus corresponds to that of the initial majority. In other words, we provide evidence that the proposed continuous-time consensus protocol with hysteretic units solves the density classification problem. Ongoing work is aimed at analytically determining how much time is needed for the system to converge. Collective Decision-Making in Swarms In (Montes de Oca et al., 2012), we proposed a social influence model whose dynamics can be used as a collective decision-making mechanism for swarms of robots that need to collectively choose the most efficient of two alternative actions (henceforth referred to as Decision-Making Model or DM model). In the DM model, each of a set of n agents can be in one of two states (represented with a binary variable Xi ∈ {0, 1}, with i = 1, 2, . . . , n). In applications of the DM model, an agent’s state can represent, for example, a robot’s preferred action or current belief of the state of an environmental variable. The DM model is a discrete-time model where at each time step t of the system’s evolution, an agent imight be able to observe the state of another random agent j 6= i. When agent i observes the state of another agent j, the observing agent i updates an internal real-valued variable Si, which we call tendency, as follows: S t+1 i = (1− α)S t i + αX t j , (1) where 0 ≤ α ≤ 1 determines the relative weight given to the agent’s latest observation (X j) and the agent’s accumulated experience (S i ). After updating its tendency, an agent updates its state as follows: X i =  1, if S t+1 i ≥ λ 0, if S t+1 i ≤ μ X i , if μ < S t+1 i < λ , (2) where μ+ λ = 1 (the reason for this constraint will become apparent later). Eq. 2 implements a sort of dynamic memory that allows the agent to integrate its observations over time. By properly choosing values for the parameters α, μ, λ, and the initial conditions X i and S 0 i , one can control the imitation behavior of agent i. While in principle, each agent may have different values for its parameters, in the DM model, α, λ and μ, are constant and common to all agents. An example of the behavior of an individual agent in the DM model is shown in Figure 1. In the DM model the population is reshuffled randomly so that each individual observes a different agent at each time step. Additionally, one single agent may potentially influence more than one other agent in the group. The DM model’s collective dynamics make the population reach a consensus on the state that at time step 0 is shared by most (that is, the majority) of the population. In (Montes de Oca 0 50 100 150 200 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Observation Number T e n d e n c y /S ta te Figure 1: Single agent behavior in the DM model. Starting with an initialization of S = 0.5 and X = 0, an agent observes a stream of state values plotted as dots with values 0 or 1. The black line shows the evolution of the agent’s tendency and the blue line shows the evolution of the agent’s state. In this simulation, α = 0.2, λ = 0.75 and μ = 0.25 (shown as dotted lines). et al., 2012), and (Montes de Oca et al., 2011), we show how this behavior can be used for optimal collective decisionmaking in robot swarms. The DM model may be seen as a method to solve the density classification problem if its definition is relaxed. In particular, if the agents (cells) are allowed to be reshuffled, then the DM may solve it. In this paper, we explore the question of whether it is possible to solve the original density classification problem with a variation of the DM model that does not require reshuffling. In the following sections, we present such a variation as well as theoretical and experimental results that make us believe that the question can be answered positively. Continuous-Time Consensus Protocol with Hysteretic Units The protocol that we propose in this paper, henceforth referred to as Consensus with Hysteresis or CH, is in its basic form the continuous-time equivalent of Eq. 1. However, in the CH protocol, the communication topology of the population does not change over time and each agent influences exactly one other agent. In the remainder of this paper, we assume that individuals are arranged in a directed ring topology with an agent i influenced by agent j, where j = i+1 or j = i− 1 (agents “look” to their right or left, respectively). ALIFE 14: Proceedings of the Fourteenth International Conference on the Synthesis and Simulation of Living Systems