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This paper depicts and evaluates an evolutionary design process for gen- erating a complex self-organizing multicellular system based on Cellular Automata (CA). We extend the model of CA with a neural network that controls the cell be- havior according to its internal state. The model is used to evolve an Artificial Neu- ral Network controlling the...
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... problem consists typically of a simulation with a generic interface to the control system. The simulation returns a fitness value which is used by the optimization algorithm for evolving the system. We implemented a CA simulator that is controlled by a representation. The fitness function was implemented as a sum of the squared color in- dex differences between the image after a number of iterations and the reference image. In order to approximate human perception of different images (human vision is focused around edges rather than areas with the same or similar color), the summands have been weighted by a function giving higher values for pixels having pixels of different color in their neighborhood. We found flags to be the simplest kind of patterns to emerge for our proposed set-up. Figure 2 depicts the evolution of a Hungarian flag. The reference image consists of a 6x9 image, where the two top rows are red and the top lower rows are green with white in between. Thus the complexity of this reference is equivalent the French flag which was used as reference in several related work articles as described in Section 2. Note that the proceedings in the quality were highly non-linear over the number of generations, because the evolutionary algorithm gets often stuck in local cost minima after about 100 generations. Thus, improvements past these generations happen only very infrequently. The mechanism to recreate an image over several iterations of the CA can be observed by the example of an Austrian flag. The Austrian flag contains only red and white color and was therefore easier to evolve than the three-colored Hungarian one. We achieved a perfect reproduction after running the evolutionary algorithm for 90 generations. Figure 3 shows how the result unfolds over several CA iterations into the intended image. The limits of the approach can be observed when going to more complex images. Figure 4 depicts the results of trying to reproduce a small image of the Mona Lisa painting (left image). The middle image shows the downsized reference image. The best achiev- able result after over 500 generations is depicted in the right image. The overall back- ground color scheme is present, although, unfortunately, Mona is missing. The main rea- son for this result lies in the increased size of the image – while the flags were evolved on a raster of 6x9, the Mona Lisa image is 20x29. Note that initially only cells at the corner and the borders can detect their position in the image - the inner cells must rely on propagating information. For a larger image, the ratio between border and inner cells is more extreme. Another question of interest was how well natural patterns can be evolved. Sev- eral patterns resembling natural ones can be reproduced by CA executing simple state- transition rules of positive and negative feedback [18]. Interestingly, our evolutionary algorithm did not come up with a feasible solution. This is likely because the fitness function was inappropriate for that task, since it com- pared the potential solution pixel by pixel to a reference image. Thus, a similar pattern is not considered as solution, although a human observer might perceive a similar pattern as being closer to the reference than an image that partially reproduces the original layout of objects in the image. Figure 5 shows that although the created image resembles the reference one on a pixel-by-pixel basis, the quality and type of the reference pattern is not matched. We depicted and evaluated a design process that generates a multicellular system out of a genotypical description for a single cell. The mechanisms have been realized via an open source framework for evolutionary design (FREVO). At the beginning of each simulation, all cells had the same state and commenced their operation at the same time this is comparable with a number of people cooperatively drawing an image in the dark. This differentiates our problem from the ones in the literature, where usually a zygote cell is given, from where the other cells grow. Still, the evolutionary process evolved a solution where also some eminent cell (typically a particular corner cell) serves as a zygote. The main contribution of this paper is not presenting an algorithm for "drawing images in the dark" but rather presenting a proof-of-concept on integrating ANNs into a CA in order to initiate a morphogenetic process. Possible applications of this research could be the self-organized pattern formation in swarm robotics. In other words, given a desired pattern, how can robots acquire it? Another application could be smart paint (as indicated in [9]) that would decide on its color based on a morphogenetic process having only a few distinctive sensory inputs, thus not allowing for a zygote approach. The best results have been achieved when evolving simple structures with large areas of a single color as they are present for example in flags. For more complex images, the current setup causes the evolutionary algorithm to get stuck at a suboptimal stage. There is, however, a large space of possibilities for variations of the model which gives rise to future work. E. g., findings on well-suited or less well-suited model configurations could give insight to the understanding of such phenomena as morphogenesis and camouflage mechanisms in nature. Future experiments are planned to involve modifications in the internal ANN structure (e.g., investigate on the optimal number of hidden nodes for different reference images) as well as increasing or decreasing the ability of ANNs to communicate with their neighbors. For evolving structures rather than replications of images, we are planning to design the fitness function in a way to have the fitness based on the type of the emerging structure instead of a pixel-by-pixel comparison. This work was supported by the European Regional Development Fund and the Carinthian Economic Promotion Fund (contract KWF 20214|18128|26673) within the Lakeside Labs project DEMESOS and the follow-up project MESON. We would like to thank Marcin Pilat, Miguel Gonzalez, and Rajesh Krishnan for their input on earlier versions of the paper. Furthermore, we would like to thank the anonymous reviewers for their constructive ...
Citations
... While others have simulated evolutionary growth of neural network-controlled cellular automata with hardwired mechanistic rules, this is the first exploration of the emergent consequences of evolutionary scaling based on dynamic interactions of cells with information-processing homeostatic capacity. The combination of neural networks and cellular automata which grow under evolutionary control is described in the work of Elmenreich & Mordvintsev [47,48]. However, their neural cellular automata do not include any cellular signalling or any use of stress capable of linking the different homeostatic levels for the scaling of goals (metabolic in individual cells and morphogenetic for the tissue). ...
Complex living agents consist of cells, which are themselves competent sub-agents navigating physiological and metabolic spaces. Behaviour science, evolutionary developmental biology and the field of machine intelligence all seek to understand the scaling of biological cognition: what enables individual cells to integrate their activities to result in the emergence of a novel, higher-level intelligence with large-scale goals and competencies that belong to it and not to its parts? Here, we report the results of simulations based on the TAME framework, which proposes that evolution pivoted the collective intelligence of cells during morphogenesis of the body into traditional behavioural intelligence by scaling up homeostatic competencies of cells in metabolic space. In this article, we created a minimal in silico system (two-dimensional neural cellular automata) and tested the hypothesis that evolutionary dynamics are sufficient for low-level setpoints of metabolic homeostasis in individual cells to scale up to tissue-level emergent behaviour. Our system showed the evolution of the much more complex setpoints of cell collectives (tissues) that solve a problem in morphospace: the organization of a body-wide positional information axis (the classic French flag problem in developmental biology). We found that these emergent morphogenetic agents exhibit a number of predicted features, including the use of stress propagation dynamics to achieve the target morphology as well as the ability to recover from perturbation (robustness) and long-term stability (even though neither of these was directly selected for). Moreover, we observed an unexpected behaviour of sudden remodelling long after the system stabilizes. We tested this prediction in a biological system—regenerating planaria—and observed a very similar phenomenon. We propose that this system is a first step towards a quantitative understanding of how evolution scales minimal goal-directed behaviour (homeostatic loops) into higher-level problem-solving agents in morphogenetic and other spaces.
... Research on applying neural nets for learning and designing CA rules can be traced back to Wulff & Hertz (1992), with subsequent notable contributions by Elmenreich & Fehérvári (2011), Nichele et al. (2017), and Gilpin (2019). Recently, Neural Cellular Automata (NCAs) have been proposed as CAs with transition rules encoded as-typically light-weight-neural networks. ...
Cellular automata (CAs) are computational models exhibiting rich dynamics emerging from the local interaction of cells arranged in a regular lattice. Graph CAs (GCAs) generalise standard CAs by allowing for arbitrary graphs rather than regular lattices, similar to how Graph Neural Networks (GNNs) generalise Convolutional NNs. Recently, Graph Neural CAs (GNCAs) have been proposed as models built on top of standard GNNs that can be trained to approximate the transition rule of any arbitrary GCA. Existing GNCAs are anisotropic in the sense that their transition rules are not equivariant to translation, rotation, and reflection of the nodes' spatial locations. However, it is desirable for instances related by such transformations to be treated identically by the model. By replacing standard graph convolutions with E(n)-equivariant ones, we avoid anisotropy by design and propose a class of isotropic automata that we call E(n)-GNCAs. These models are lightweight, but can nevertheless handle large graphs, capture complex dynamics and exhibit emergent self-organising behaviours. We showcase the broad and successful applicability of E(n)-GNCAs on three different tasks: (i) pattern formation, (ii) graph auto-encoding, and (iii) simulation of E(n)-equivariant dynamical systems.
... Chavoya and Duthen used a Genetic Algorithm to evolve Cellular Automata that produced different 2D and 3D shapes [13] and evolved an Artificial Regulatory Network (ARN) for cell pattern generation, resolving the French flag problem [14]. The combination of neural networks and cellular automata which grow under evolutionary control is described in the work of Elmenreich and Mordvintsev [23,65]. A key difference in our work is that the artificial neural network (ANN) inside each cell here models the behavior of a gene-regulatory network and can control gap junctions to deliver the morphogen and the ability to reduce stress (homeostasis) to the other cells. ...
All cognitive agents are composite beings. Specifically, complex living agents consist of cells, which are themselves competent sub-agents navigating physiological and metabolic spaces. Behavior science, evolutionary developmental biology, and the field of machine intelligence all seek an answer to the scaling of biological cognition: what evolutionary dynamics enable individual cells to integrate their activities to result in the emergence of a novel, higher-level intelligence that has goals and competencies that belong to it and not to its parts? Here, we report the results of simulations based on the TAME framework, which proposes that evolution pivoted the collective intelligence of cells during morphogenesis of the body into traditional behavioral intelligence by scaling up the goal states at the center of homeostatic processes. We tested the hypothesis that a minimal evolutionary framework is sufficient for small, low-level setpoints of metabolic homeostasis in cells to scale up into collectives (tissues) which solve a problem in morphospace: the organization of a body-wide positional information axis (the classic French Flag problem). We found that these emergent morphogenetic agents exhibit a number of predicted features, including the use of stress propagation dynamics to achieve its target morphology as well as the ability to recover from perturbation (robustness) and long-term stability (even though neither of these was directly selected for). Moreover we observed unexpected behavior of sudden remodeling long after the system stabilizes. We tested this prediction in a biological system - regenerating planaria - and observed a very similar phenomenon. We propose that this system is a first step toward a quantitative understanding of how evolution scales minimal goal-directed behavior (homeostatic loops) into higher-level problem-solving agents in morphogenetic and other spaces.
... A subsequent approach is that of Elmenreich and Fehérvári [9], who designed an evolutionary algorithm to identify a NN transition rule that would generate a desired pattern (a process called morphogenesis). Similarly, Nichele et al. [10] used the NEAT genetic algorithm [11] and compositional pattern producing NNs [12] for morphogenesis and pattern replication in a CA with discrete states. ...
Cellular automata (CA) are a class of computational models that exhibit rich dynamics emerging from the local interaction of cells arranged in a regular lattice. In this work we focus on a generalised version of typical CA, called graph cellular automata (GCA), in which the lattice structure is replaced by an arbitrary graph. In particular, we extend previous work that used convolutional neural networks to learn the transition rule of conventional CA and we use graph neural networks to learn a variety of transition rules for GCA. First, we present a general-purpose architecture for learning GCA, and we show that it can represent any arbitrary GCA with finite and discrete state space. Then, we test our approach on three different tasks: 1) learning the transition rule of a GCA on a Voronoi tessellation; 2) imitating the behaviour of a group of flocking agents; 3) learning a rule that converges to a desired target state.
... that generate target images [9]. Our method uses backpropagation instead of evolution, and focuses on textures rather than pixel-perfect reconstructions. ...
... Our method uses backpropagation instead of evolution, and focuses on textures rather than pixel-perfect reconstructions. Authors of [9] admit inability of their method to synthesize textures in the last paragraph of section five. ...
Neural Cellular Automata (NCA) have shown a remarkable ability to learn the required rules to "grow" images, classify morphologies, segment images, as well as to do general computation such as path-finding. We believe the inductive prior they introduce lends itself to the generation of textures. Textures in the natural world are often generated by variants of locally interacting reaction-diffusion systems. Human-made textures are likewise often generated in a local manner (textile weaving, for instance) or using rules with local dependencies (regular grids or geometric patterns). We demonstrate learning a texture generator from a single template image, with the generation method being embarrassingly parallel, exhibiting quick convergence and high fidelity of output, and requiring only some minimal assumptions around the underlying state manifold. Furthermore, we investigate properties of the learned models that are both useful and interesting, such as non-stationary dynamics and an inherent robustness to damage. Finally, we make qualitative claims that the behaviour exhibited by the NCA model is a learned, distributed, local algorithm to generate a texture, setting our method apart from existing work on texture generation. We discuss the advantages of such a paradigm.
... However, the SIMEON framework does not provide a strategy for distributed simulation across multiple servers. In addition, the previously mentioned tool FREVO supports creating and evaluating swarm behavior by evolution and has been used in several studies as an evolution tool including robotics [12] and pattern generation [35]. ...
Swarms of cyber-physical systems can be used to tackle many challenges that traditional multi-robot systems fail to address. In particular, the self-organizing nature of swarms ensures they are both scalable and adaptable. Such benefits come at the cost of having a highly complex system that is extremely hard to design manually. Therefore, an automated process is required for designing the local interactions between the cyber-physical systems that lead to the desired swarm behavior. In this work, the authors employ evolutionary design methodologies to generate the local controllers of the cyber-physical systems. This requires many simulation runs, which can be parallelized. Two approaches are proposed for distributing simulations among multiple servers. First, an approach where the distributed simulators are controlled centrally and second, a distributed approach where the controllers are exported to the simulators running stand-alone. The authors show that the distributed approach is suited for most scenarios and propose a network-based architecture. To evaluate the performance, the authors provide an implementation that builds upon the eXstensible Messaging and Presence Protocol (XMPP) and supersedes a previous implementation based on the Message Queue Telemetry Transport (MQTT) protocol. Measurements of the total optimization time show that it outperforms the previous implementation in certain cases by a factor greater than three. A scalability analysis shows that it is inversely proportional to the number of simulation servers and scales very well. Finally, a proof of concept demonstrates the ability to deploy the resulting controller onto cyber-physical systems. The results demonstrate the flexibility of the architecture and its performance. Therefore, it is well suited for distributing the simulation workload among multiple servers.
... Other soft-computing techniques have also been investigated in relation with CA. For example, Elmenreich et al. proposed an technique for calculating the transition function of CA using neural networks [15]. The goal was to train the network by means of Evolutionary Programming [16] in order to develop self-organising structures. ...
A comparative study is presented regarding the evolutionary design of complex multi-state cellular automata.
In particular, two-dimensional cellular automata will be considered in combination with pattern development problem as a~case study.
Two techniques for the representation of transition functions for the cellular automata are proposed: a conventional table-based method and an advanced concept utilising conditionally matching rules.
It will be shown that using a proper settings of Evolution Strategy, various working solutions can be obtained using both representations.
Some observations from an analysis of resulting cellular automata will be presented which indicate that the behavior of the automata is totally different and depends on the representation applied.
Specifically, the table representation exhibit a chaotic development during which a target pattern emerges at a moment.
On the other hand, the conditional rules are able to achieve behavior that progressively constructs the target pattern which, in addition, represents a stable final state.
Moreover, the latter method also exhibits significantly higher success rate which represents one of its advantages and proves an importance of systematic research in this area.
... The main purpose of FREVO is to support the optimization process as it guides through the individual steps of the evolutionary design, whereas it requires work by the software developer to implement the modeled problem details. Besides evolving controllers for CPSs, such as in cooperative robotics or wireless sensor networks, FREVO can as well be used for other problems, such as pattern generation [3] or economical simulations [6]. Figure 1.5 gives an overview of the FREVO architecture. FREVO is implemented in the Java programming language and makes strong use of the object-oriented programming paradigm. ...
Cyber physical systems (CPSs) find their application in different domains, including smart cities, Internet of Things (IoT), and Industry 4.0. The increasing degree of interaction among CPSs leads to unpredictable and partially unexpected behavior. The major steps to manage emerging behavior in CPSs are taken in the design process. Although a high number of methods and tools already exist from related disciplines (including complex system research, embedded system design, and self-organization), there is no comprehensive toolset available to address the extensive CPS design process. This chapter presents a proposal for a common CPS design toolset. It combines existing and emerging tools to design, simulate, evaluate, and deploy solutions for complex, real-world problems using evolutionary algorithms on the example of swarms of unmanned aerial vehicles (UAVs).
... We also show how Morphozoic may be used to reverse engineer a sequence of state changes of a system and derive an approximation to the rules governing that system. Morphozoic may therefore be used for reverse engineering (Gordon & Melvin, 2003; Deutsch, Maini & Dormann, 2007; Elmenreich & Fehervari, 2011; Lobo & Levin, 2015). Because of its local/global construction, Morphozoic may be a step towards meeting the challenge posed by Russ Abbott: " ...when a glider appears in the Game of Life, it has no effect on the how the system behaves. ...
A cellular automaton model, Morphozoic, is presented. Morphozoic may be used to investigate the computational power of morphogenetic fields to foster the development of structures and cell differentiation. The term morphogenetic field is used here to describe a generalized abstraction: a cell signals information about its state to its environment and is able to sense and act on signals from nested neighborhood of cells that can represent local to global morphogenetic effects. Neighborhood signals are compacted into aggregated quantities, capping the amount of information exchanged: signals from smaller, more local neighborhoods are thus more finely discriminated, while those from larger, more global neighborhoods are less so. An assembly of cells can thus cooperate to generate spatial and temporal structure. Morphozoic was found to be robust and noise tolerant. Applications of Morphozoic presented here include: (1) Conway's Game of Life, (2) cell regeneration, (3) evolution of a gastrulation-like sequence, (4) neuron pathfinding, and (5) Turing's reaction-diffusion morphogenesis.
... The 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 10 transition rule updates synchronously the states of each cell at each time step. A large number of quantitative mathematical techniques can be used such in the field of Machine Learning: as Artificial Neural Networks (Almeida et al., 2008), genetic algorithms (Ak et al., 2013), self-organizing systems (Elmenreich and Fehérvári, 2011), Markov chain (Balzter et al., 1998), Monte Carlo simulations (Zio et al., 2006), fuzzy logic (Wu, 1998) to implement this transition function. ...
Forecasting atmospheric dispersion in complex configurations is a current challenge influid dynamics in
terms of calculation time and accuracy. CFD models provide good accuracy but require a great compu-
tation time. Simplified or empirical models are designed to quickly evaluate the dispersion but are not
adapted to complex geometry. Cellular Automata coupled with an Artificial Neural Network (CA-ANN)
are developed here to calculate the atmospheric dispersion of methane (CH4) in 2D. Efforts are made in
reducing computation time while keeping an acceptable accuracy. A CFD simulations database is created
and the Advection-Diffusion Equation is discretized to provide variables for the ANN. Neural network
design is made thanks to best sampling selection, architecture selection and optimized initialization. The
coefficient of determination is over 0.7 for most cases of the test set despite small errors accumulated
through time steps. CA-ANN is faster than CFD models by a factor from 1.5 to 120.
