Characterizing Autopoiesis in the Game of Life

Characterizing Autopoiesis in the Randall D. Beer* Game of Life Indiana University

Keywords Autopoiesis, organizational closure, artificial chemistry, cellular automata Abstract Maturana and Varelaʼs concept of autopoiesis defines the essential organization of living systems and serves as a foundation for their biology of cognition and the enactive approach to cognitive science. As an initial step toward a more formal analysis of autopoiesis, Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 this article investigates its application to the compact, recurrent spatiotemporal patterns that arise in Conwayʼs Game-of-Life cellular automaton. In particular, we demonstrate how such entities can be formulated as self-constructing networks of interdependent processes that maintain their own boundaries. We then characterize the specific organizations of several such entities, suggest a way to simplify the descriptions of these organizations, and briefly consider the transformation of such organizations over time.

1 Introduction

Everyone agrees that a biological cell is alive, but why? Upon what criteria do we base this assertion? How we choose to answer this fundamental question determines not only what the subject matter of biology actually is, but also how we approach many practical biological endeavors such as investigating possible pathways for the origin of life [36, 41], testing for extraterrestrial life [11, 29], or creating synthetic life [10, 22]. On the rare occasions when this question is even raised in the biological literature, proposed answers typically take the form of either a list of components (e.g., carbon chemistry, RNA/DNA, phospholipid bilayer) or a list of properties (e.g., growth, reproduction, evolution). Neither of these answers is particularly satisfying theoretically. The first list seems overly tied to the specific material con- stitution of life as we have so far encountered it, and exceptions to the second list are easily found. Unsatisfiedwithsuchlists,theChileanbiologists Humberto Maturana and Francisco Varela proposed that what distinguishes a living system from a nonliving one is not its material structure, but the particular organization of processes supported by that structure [39, 40, 51]. On their view, the observed properties of living systems are consequences or further elaborations of this basic organization, which they termed autopoietic (self-producing) [39, p. 78]: An autopoietic machine is a machine organized (defined as a unity) as a network of processes of production (transformation and destruction) of components that produces the components which: (i) through their interactions and transformations continuously regenerate and realize the network of processes (relations) that produced them; and

* Cognitive Science Program, School of Informatics and Computing, 840 Eigenmann, 119 East 10th Street, Indiana University, Bloomington, IN 47406. E-mail: [email protected]

(ii) constitute it (the machine) as a concrete unity in the space in which they (the components) exist by specifying the topological domain of its realization as such a network.

To put it mildly, this is not an easy definition to understand on first reading. It is quite abstract, makes use of somewhat idiosyncratic terminology, depends upon a larger background of unfamiliar concepts, and exhibits an essential circularity that can be difficult to assimilate. There are also key differences in how subsequent authors have interpreted various aspects of this definition, and even Maturana and Varela later diverged somewhat in their opinion of its scope and consequences [25, 44, 55]. Nevertheless, autopoiesis stands as an important example of how to formulate a definition of living systems that goes beyond lists of components or properties. It is also representative of a number of other such attempts [34], for example in chemistry [27], theoretical biology [24, 32, 45], and philosophy [31]. Furthermore, the concept of autopoiesis serves as the foundation for the growing enactive approach to cognition [47, 48, 53]. Thus, despite its difficulties, it is worth trying Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 to understand and explore this notion in depth. Consider the paradigmatic example of an autopoietic system: the living cell. A typical cell consists of a large number of protein and water molecules bounded by a semipermeable membrane formed from a bilayer of phospholipid molecules [4]. The material structure of such a cell is thus given by the physical details of these molecular components: their type, position, orientation, bonds, modes of vi- bration, and so on. However, the cell itself cannot be equated with this totality of material properties, because these properties are constantly changing. Over time, individual molecules move, rotate, and enter and leave the cell, while chemical bonds are continually being broken and re-formed through reactions. And yet, despite this ongoing material change, the cell itself, as a delimited spatiotemporal organization of processes, persists. This persistence of identity in the face of material change is exactly what the notion of autopoiesis attempts to capture. More specifically, Maturana and Varelaʼs definition states two conditions that a network of processes must satisfy in order to be considered autopoietic. The closure condition demands that the network of processes must produce the components whose interactions generate and maintain that very same network. The boundary condition demands that the spatial boundary that distinguishes an autopoietic system from its background must itself be generated and maintained by the network of processes and in turn must play a central role in enabling those same processes. This basic circularity of autopoiesis is illustrated in Figure 1. In order to go beyond this rough characterization of autopoiesis, we need to examine in con- siderably more depth an application of this concept to a simple concrete system. Since even the simplest biological system is still quite complex, we turn to models for this purpose. There is a long history of computational modeling of autopoiesis [7, 13, 21, 42, 52, 60]. However, these models are

Figure 1. An illustration of the central circularity of autopoiesis. A bounded system generates a network of processes that produces the components that determine the very same bounded system. Although organizationally closed, such a system is both materially and interactionally open to its environment. Adapted from [35].

2 Artificial Life Volume 21, Number 1 R. D. Beer Characterizing Autopoiesis in the Game of Life typically designed a priori to satisfy the definition of autopoiesis. In contrast, the goal of this article is to assess the extent to which the definition can be applied to a system that was not designed specifically to exhibit autopoiesis. In particular, building on a previous proposal [8], I investigate the conditions under which spatiotemporal patterns that arise in the well-known Game of Life (GoL) cellular automaton can be considered to be autopoietic. Note that the intent here is not to argue that such patterns are “really” alive or that they capture all relevant characteristics of living systems. Rather, I wish merely to utilize GoL as a simple model universe in which we can explore in detail the many issues raised when attempting to apply the definition of autopoiesis to a concrete system. This article is organized as follows. Section 2 introduces the game of Life and suggests a way to describe the patterns that arise within this cellular automaton as collections of interacting processes. Sections 3 and 4 then show how to formulate the closure and boundary conditions, respectively, of autopoiesis in such a way that they can be applied to the patterns that arise in GoL. Section 5 then examines simplified descriptions of GoL organizations that are more analogous to the formalism of Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 theoretical chemistry, and Section 6 briefly explores the transformation of organizations over time. Finally, Section 7 considers the broader implications of our analysis for the concept of autopoiesis and its various applications and extensions.

2 The Game of Life

Cellular automata are a well-known class of simple models of spatiotemporal processes in which state, time, and space are all discrete. Conwayʼs game of Life is a binary outer totalistic cellular automaton defined on a rectangular lattice L, whose exploration has been widely publicized [2, 12, 28, 43]. Descriptions of GoL are typically couched in biological terms (e.g., individual elements of the lattice are “cells” that can be “born” due to “reproduction” and can “die” due to “overcrowding” or “loneliness”). However, this biological vocabulary is strictly metaphorical. For our purposes here, it is better to think of GoL as a simple discretized physics [50] whose single dynamical law can be tþ1 P t P t L ¼ y ; þ L y ; L most compactly written as x; y 3 x; y x; y 2 x; y ,where x;y is the state of the lattice cell (x, y) ∑ ʼ at time t, x, y is the number of 1-cells in that cell s Moore neighborhood (the eight lattice cells sur- y = rounding it), and i, j is the Kronecker delta function (which takes on the value 1 when i j and 0 otherwise). On this view, any potential “biological” entities arise as self-perpetuating networks of processes grounded in the underlying GoL physics. A typical GoL time evolution is shown in Figure 2A for a 100 × 100 lattice with periodic boundary conditions. On the left is a random initial configuration with a 1-density U0 of 0.5. On the right is the same lattice after 1000 applications of the GoL physics; here the 1-density has decayed to U = 1000 0.037. Statistics from a large number of repetitions of this experiment demonstrate that U this pattern is typical (Figure 2B). For a wide range of 0-values, the initial density decays exponen- tially to an asymptotic mean value of U∞ ≈ 0.029 [46]. Interestingly, if the initial density is too small or too large, even this tiny asymptotic 1-density cannot be sustained and U∞ quickly collapses to 0 [6]. Thus, GoL physics exhibits a very strong decay to quiescence. As we will see, persistent GoL patterns that resist this decay do so only by virtue of their self-sustaining (autopoietic) organization. And persistent patterns certainly abound in GoL, as is visually obvious to any observer after even a quick glance at the right side of Figure 2A. Consider the three circled patterns. From left to right, they are known as a block,ablinker,andaglider. We will focus our analysis on these three patterns because (1) they represent three distinct classes within GoL (called still lifes, oscillators,andspaceships, respectively); (2) they are each the simplest members of their respective classes; (3) they are each the most common members of their respective classes. Because such compact and spatiotemporally coherent patterns are resistant to the general decay to quiescence, we tentatively interpret them as the GoL analogues of biological organisms. In order to characterize the organizations of the networks of processes that underlie the persistence of such entities, it is necessary to think about the GoL physics at a somewhat higher level of description more akin to an artificial “chemistry” [20]. When a lattice cell in a given state is surrounded by

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Figure 2. The game of Life. (a) After 1000 time steps, an initially random 100 × 100 toroidal lattice with a 1-density of 0.5 (left) evolves into a highly structured lattice with a 1-density of 0.037 (right), including, from left to right, a block, a blinker, and a glider (circled). (b) The mean variation of 1-density at time t with initial 1-density U0 t = 1, 10, 100, 1000, 10000, on a 100 × 100 toroidal lattice. For each initial 1-density, the values plotted represent the mean of 104 random initial conditions drawn from a uniform distribution. Convergence was tested by running selected initial density values out to t = 100,000 and verifying that the resulting mean densities were the same as for t = 10,000 to two significant digits. other lattice cells in particular states, the GoL physics dictates the subsequent “reaction” that takes place between them. Since a given cell can be in one of two states and is surrounded by a Moore neighborhood of eight cells, we have 2 × 28 = 512 different such local reactions. We can divide these 512 processes into four classes: production, destruction, maintenance (which can itself be subdivided into 0-maintenance and 1-maintenance), and null (Figure 3). Although the null process is a kind of 0-maintenance process, we distinguish it here because it forms the quiescent background against which all other GoL processes take place. Enumerating the full set of possibilities, we find that there are 56 production processes, 84 1-maintenance processes, 172 destruction processes, and 199 0-maintenance processes, as well as 1 null process. Examples of each, along with the graphical notation for processes that we will use throughout this article, are given in Figure 3. Of course, symmetry considerations allow us to further divide these process classes. We will examine the advantages and disadvantages of such a subdivision in Section 5. Note that processes resulting in 0-components outnumber those resulting in 1-components by nearly three to one, at least partly explaining the strong decay to quiescence in GoL. In order to emphasize the chemistry analogy, we will often refer to lattice cells in the 1 state as 1-components and lattice cells in the 0 state as 0-components. It may seem odd to consider 0-cells as components, but we will see in Section 4 that this is necessary.

4 Artificial Life Volume 21, Number 1 R. D. Beer Characterizing Autopoiesis in the Game of Life

A GoL model universe is obviously highly simplified compared to real physics, chemistry, and biology. Real cells are composed of diverse molecular components and processes occurring over many spatial and temporal scales. Real chemistry is stochastic, with multiple reactants and products whose reactions, which occur at multiple rates and involve the making and breaking of different types of chemical bounds, are determined by thermodynamic constraints. Real physics involves multiple particles and forces whose interactions are subject to conservation laws and quantum mechanical constraints. Thus, there are many complexities of the physical instantiation of autopoiesis that we cannot hope to address with this simple model. However, it is not the purpose of this article to provide a physical account of living systems. Rather, my goal is far more modest. Autopoiesis targets the relational organization of living systems, not their physical structure. Accordingly, what is important about the GoL model for our purposes is that it shares with the physical world the following characteristics: (1) it consists of a spatial medium whose time evolution is governed by a universal law that is local in both space Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 and time; (2) there is a general tendency for organized structures to dissipate; (3) it supports a variety of distinct local processes of production/maintenance/destruction with selective preconditions; (4) self-sustaining, self-delimited networks of such processes can form and disintegrate. By analyzing in depth this simple model, we gain insight into the logical structure of the concept of autopoiesis

Figure 3. The different classes of local processes that can occur in the game of Life. The left column gives the name of each class. The right column shows an example process of each class acting on a particular cell (outlined in bold). By convention, components are outlined in brown and processes are outlined in orange. A processʼs class is given by the color of its central cell, while the colors of the cells in its Moore neighborhood indicate the requirements for that process to occur (gray cells must be 1 and white cells must be 0). Note that, although the null process is technically a 0-maintenance process, we distinguish it because it forms the background against which all other processes occur. (Color in electronic version.)

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Figure 4. The closed organization of a block. (a) Processes can be interpreted as mapping components to components. (b) Components can be interpreted as linking processes to processes whenever the conditions for one process depend upon the products of others. (c) The process dependency network for a block makes explicit which preconditions depend upon which products. Because the dependency network involves only processes that mutually enable one another, we say that this organization is closed. Note that, for a block, all processes are 1-maintenance processes. (d) The skeletal process graph for a block is a simplified representation of the process dependency network in which the details of which product plays which role in each process are suppressed. Because every interaction is bidirectional in this graph, arrows are not shown. (Color in electronic version.) in a way that can hopefully inform future studies of the additional complications imposed by physical instantiation.

3 The Closure Condition

In order to demonstrate that a GoL entity satisfies the closure condition of autopioesis, we must do three things. First, we must identify the set of production, destruction, and maintenance processes that underlie it. Second, we must characterize the interdependencies between these processes. Third, we must show that these dependencies form a closed network. Let us begin with a block, the simplest of the three GoL entities that we will consider. Traditionally, a block is defined as a static 2 × 2 square of 1-components that persists over time (the role of 0-components will be considered in the next section). As Figure 4A shows, we can think of the 1-maintenance processes involved in a block as mapping 1-components to 1-components. Shifting

6 Artificial Life Volume 21, Number 1 R. D. Beer Characterizing Autopoiesis in the Game of Life our frame of reference from component-centric to process-centric, we can also say that the components link processes to processes, since subsequent 1-maintenance processes depend upon the products of earlier ones (Figure 4B). Thus, we can remove the components completely and just draw a dependency link directly from the product cell of each process to the Moore neighborhood cells of the other processes that depend on that product, making explicit how the enabling conditions of a later process are satisfied by the products of earlier ones. The totality of such dependencies for a set of processes form a process dependency network, which we will define to be the organization of that set of processes [57]. There is something rather special about a blockʼs process dependency network. In general, we would expect the processes enabled by the spatial arrangement of a set of components to be different from the processes that originally produced those components. However, in Figure 4B we see that the “after” processes on the right are exactly the same as the “before” processes on the left. In other words, the set of processes underlying a block constantly reenable themselves.Tomakethis explicit, we can redraw Figure 4B with corresponding “before” and “after” processes equated, pro- Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 ducing a closed network of dependencies (Figure 4C). This is the simplest example of organizational/ operational1 closure in GoL. Thus, underlying the persistence of a block we see a particular set of four 1-maintenance processes that produces a spatial arrangement of 1-components that reenable the original set of 1-maintence processes that produced them in a self-perpetuating cycle. This clearly satisfies the closure condition of autopoiesis. An important feature of this representation is that it makes explicit the essential dependency relations defining a blockʼs organization while abstracting over the inessential structural details of any particular instantiation, such as the specific lattice cells that it occupies at any given point in time. In other words, process dependency networks capture only what is common to all structural instantiations of a block, regardless of their location, orientation, or chirality in the lattice.2 Interest- ingly, although they abstract over absolute spatial characteristics, process dependency networks do fully preserve the relative spatial relationship between processes, since these latter relationships play a central role in defining the dependencies. Indeed, one can easily reconstruct an instantiation of a block at any desired lattice location from the information contained in Figure 4C. As process dependency networks become more complicated in later examples, it will be useful to simplify them for purposes of visualization. In a skeletal process graph, each process node in the dependency network becomes a simple disk colored by the type of the process that it represents (blue = production, red = destruction, black = 1-maintenance, and white = 0-maintenance). Like- wise, each dependency link between a product cell and a Moore neighborhood cell in the process dependency network becomes a simple link between the corresponding process nodes. The skeletal process graph for a block is shown in Figure 4D. Thus, skeletal process graphs abstract over the fine details of how individual lattice cells of particular processes interrelate in order to highlight the overall topology of dependency relations between types of processes. However, it is important to emphasize that it is only the process dependency network that fully captures the organization of a GoL entity. Skeletal process graphs do not replace process dependency networks; they are merely simplified representations of them. Although the block example is extremely simple, it is not quite as trivial as it might seem. Of the 15 possible nonempty configurations of a 2 × 2 square of lattice cells, this is the only one that persists. The 10 arrangements that involve only one or two 1-components all decay immediately to quiescence. The remaining 4 arrangements, which consist of a block with a missing corner, are more interesting in that they turn out to be block precursors; they are patterns that evolve into a block after one time step. This occurs because there are production processes that have exactly the same triggering conditions as the 1-maintenance processes that normally maintain a blockʼscorners(bothcanbeseen in the examples in Figure 3). Thus, if any corner of a block is destroyed, it is immediately regenerated.

1 Varela used these two terms interchangeably. Following Thompson [48, p. 45], we will use the term “organizational closure” to refer to the topology of the network of dependency relations, and the term “operational closure” to refer to the dynamics of such a network. 2 Of course, due to its high degree of symmetry, there are no interesting variations in orientation or chirality for a block. However, more complicated entities such as gliders can appear with different orientations and chiralities.

Artificial Life Volume 21, Number 1 7 R. D. Beer Characterizing Autopoiesis in the Game of Life

In GoL, even seemingly static entities are in fact actively maintained as attractors of the dynamics. Contrast this with what we might call RockWorld, a binary cellular automaton whose law of physics Ltþ1 ¼ Lt x;y x;y exhibits no decay to quiescence and allows all patterns to trivially persist.

4 The Boundary Condition

In order to demonstrate that a GoL entity satisfies the boundary condition of autopioesis, we must show that it generates and maintains its own spatial boundary, which both serves to distinguish it from the rest of the GoL lattice and plays a necessary role in the persistence of the whole. Our analysis so far has focused only on processes involving the maintenance, production, or destruction of 1-components. This accords well with our normal intuition about GoL: 1-components are

pattern, 0-components are background. However, this section will argue that some 0-components, Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 and the processes that involve them, must be considered on an equal footing with the 1-components, and that a subset of those 0-components serves as a spatial boundary [8]. Consider once again the simple block. Surrounding the four 1-components that we have so far examined are twelve 0-components (left side of Figure 5A). There are three reasons for including these 0-components as a part of a block. First, since they fall into the Moore neighborhoods of the 1-components, there are non-null processes associated with these 0-components (right side of Figure 5A). Second, if these were 1-components, an observer would fail to identify this entity as a block. A block just is a 2 × 2 square of 1-components surrounded by a ring of 0-components; the 0-components are part of its identity. Third, if these 0-components were 1-components, the entity would fail to persist as a block, because its processes of self-maintenance would be disrupted. That is, the triggering conditions for the 1-maintenance processes of a block involve not only the other 1-components sustained by the 1-maintenance processes, but also the 0-components maintained by the surrounding 0-maintenance processes. Thus, these 0-maintenance processes fall within the operational limits of a block considered as a closed network of processes [57]. For all of these reasons, we are forced to define the full organization of a block to be the process dependency network containing 84 dependency links between these 16 1-maintenance and 0-maintenance processes, whose skeletal process graph is shown in Figure 5B. Note that this extended process graph is still organizationally closed. There is one complication with this definition of a blockʼs boundary that we must deal with. Con- sider the 0-maintenance process occurring at the upper right-hand corner of a block organization (right side of Figure 5A). The block organization requires two things of this process. First, it must of course be a 0-maintenance process. Second, its dependencies on other processes within the block must be as shown (i.e., the gray cell in its Moore neighborhood must be on, and the two white cells must be off ). However, the organization itself does not impose any further conditions on the lattice cells of this

Figure 5. The full closed organization of a block. (a) A schematic representation of the full set of process dependencies involved in a block (including the 12 0-maintenance processes that constitute its boundary). Yellow cells indicate donʼt- care conditions on processes. (b) The corresponding skeletal process graph. Because every interaction is bidirectional in this graph, arrows are not shown. (Color in electronic version.)

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Figure 6. The closed organization of a blinker. (a) A schematic representation of the full set of process dependencies involved in a blinker. (b) The corresponding skeletal process graph. Because every interaction is bidirectional in this graph, arrows are not shown. (Color in electronic version.) process that are highlighted in yellow. Indeed, the states of these yellow cells form the immediate environment in which the block exists and are therefore driven in part by processes external to the block [9]. Thus, any 0-maintainence process consistent with the remainder of the block organization is acceptable. For example, the upper right corner cell in its Moore neighborhood could be either on or off, and this process would still play its proper role in the block organization. But if exactly two of the yellow cells were on, then it would become a production process and the block organization would be destroyed. From this point forward we will highlight such partial donʼt-care conditions of boundary processes in yellow. The same strategy that we have employed to characterize the closure and boundary of a block can also be applied to blinkers and gliders. By the same reasoning as above, we must include not only the 1-components of blinkers and gliders, but also the 0-components in the Moore neighborhoods of those 1-components and the processes involving them. Interestingly, unlike those of a block, the resulting spatial boundaries of both a blinker and a glider vary in time. A blinker is a configuration of three 1-components that alternates between a vertical and a horizontal arrangement (Figure 6). Blinkers are interesting because, unlike blocks, they are dynamic entities.

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Figure 7. The closed organization of a glider. (a) A schematic representation of the full set of process dependencies involved in a glider. (b) The corresponding skeletal process graph. Note that, although the dependencies are directional, arrows are not shown, because they would obscure the links. (Color in electronic version.)

As shown schematically in Figure 6A, there are two production processes, two destruction processes, one 1-maintenance process, and ten 0-maintenance processes involved in each transition between the two blinker states,3 for a total of 30 processes and 162 dependency links. Note that four of a blinkerʼs boundary processes are not 0-maintenance processes, but production processes. These form the “buds” from which sprout the “caps” at the next iteration. The skeletal process graph representation of a blinkerʼs organization is shown in Figure 6B. Note that every dependency link is bidirectional, because the blinker alternates between these two sets of processes. Finally, let us turn to the organization of a glider, the most interesting of our three GoL entities (Figure 7). A glider is a repeating sequence of four configurations that is not only dynamic, but also moves diagonally at a rate of one lattice cell every four iterations. As shown schematically in Figure 7A, each of the four transitions of a glider involve two production processes, two destruction processes, three 1-maintenance processes, and fifteen 0-maintenance processes (one of which is internal and thus does not form a part of the boundary). Note that, although the glider has shifted position after four time steps, the arrangement of its components still enables the same set of processes as before.

3 Note that the total number of production and destruction processes must always balance for entities that maintain their size on average.

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Thus, although a glider is never made of the same components twice, its organization persists. These 88 processes form a rather complicated dependency network with 566 dependency links, whose skeletal process graph is shown in Figure 7B. Here the dependency links are directed, because dependency paths only close after four steps. These results demonstrate that GoL entities such as blocks, blinkers, and gliders satisfy the boundary condition of autopoiesis. A careful consideration of the closure of these entities uniquely determines a spatial extent for each of them that includes both 1-components and 0-components. The boundary components participate in the other processes of maintenance of the entity, and the nonboundary components actively participate in the maintenance of the boundary. The perimeters of these regions constitute boundaries that delineate each entity from its background in a systematic way, providing a kind of “insulation” that buffers, but by no means prevents, influences from the surrounding environment from affecting the core components. Indeed, in a nonempty environment, these boundaries are the interface through which external perturbations act on the entity [9]. Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021

5 Equivalence Classes of Processes

As is perhaps most obvious in Figure 7B, even the skeletal process graph representation of process dependency networks can quickly become quite complicated. The reason for this complexity is that our current representation preserves the spatiotemporal identity of each process. That is, the same process occurring at two different relative spatial locations or at two different relative times within the network is represented twice. The advantage of this seeming redundancy is that the organization captures not only the dependency relations between processes, but also the spatial and temporal relations. In contrast, chemical organizations are typically represented more abstractly in theoretical chemistry [19, 24]. There, each node is a particular molecular species, and links between nodes represent reactions that transform molecules of one species to another. In this description, each molecular species and each reaction is represented only once, abstracting over the particular location and time at which any given instance of a reaction takes place. This is closely related to assuming a mass-action formulation under well-stirred conditions in which all reactions that can happen are assumed to do so at their equilibrium rates. Although it leads to simpler descriptions, it has the significant disadvantage that it fails to capture both the spatiotemporal proximity of molecules required for a reaction between them to occur [5] and the compartmentalized nature of cellular biochemistry [4]. If we wish to simplify our process graphs in a similar way, we must also abstract our processes over space and time. We can accomplish this by merging all instances of the same process into a single node whose dependencies are the union of the dependencies of all of the original nodes. Consider, for example, any of the corner 0-maintenance processes in a blinker (Figure 6A). Each of these corner processes appears twice: once in the horizontal-to-vertical transition of the blinker and once in the vertical-to-horizontal transition. This means that they each also appear twice in the corresponding skeletal process graph (Figure 6B). Specifically, each equivalent pair of processes forms one of the truncated corners of the process graph. Thus, we could collapse each of these pairs, reducing the graph by four nodes and eight edges (since each edge is bidirectional in this graph, a reduction by four edges corresponds to the removal of eight dependencies). When collapsing multiple processes into one in our process graph representation, we must be precise about the conditions under which we wish to consider two processes to be the “same.” Equivalence classes can be defined by abstracting over symmetries in an object of interest. For example, our collapsing pairs of corner processes as described above is grounded in the translational symmetry of GoL in space and time (a GoL process has the same effect no matter where in the lattice it occurs and at what time). GoL also exhibits two other spatial symmetries: rotation and reflection. What role might these symmetries play? Consider our characterization of the closed glider organization (Figure 7). This glider moves diagonally toward the lower right. If we rotate Figure7Aclockwisebyninetydegrees,weob- tain a glider that moves toward the lower left. This second glider has a process graph that is

Artificial Life Volume 21, Number 1 11 R. D. Beer Characterizing Autopoiesis in the Game of Life isomorphic to the first. However, the set of processes in these two networks are different because they have been rotated. A similar conclusion can be drawn if we flip Figure 7A left to right. Thus, if we wish to consider gliders moving in all four possible directions as instances of the same organization, we must stipulate that all translations, rotations, and reflections of any given process fall into the same equivalence class. Under this equivalence relation, the 512 possible GoL processes reduce to only 102 distinct classes, with a null process, 10 distinct production processes, 16 distinct 1-maintenance processes, 35 distinct destruction processes, and 40 distinct 0-maintenance processes. To see how these equivalence classes allow us to simplify skeletal process graphs, let us return to the full blinker organization (Figure 6). Removing duplicated processes from this graph leaves us with 26 distinct processes. If we group these processes by equivalence under rotation and reflection, we find that they fall into one of only six classes (shown as distinct rows in Figure 8): a single production class, a single destruction class, a single 1-maintenance class, and three 0-maintenance classes. Collapsing the full blinker process graph (Figure 6) by these equivalences, we obtain the Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 reduced process graph shown in Figure 9B. By a similar procedure, reduced process graphs can also be obtained for the full block organization (Figure 9A) and the full glider organization (Figure 9C). Note that this reduction only makes sense for skeletal process graphs; since process dependency networks make explicit the relationship between individual product and neighborhood cells (Figure 4C), they cannot be reduced in this way.

6 The Dynamics of Organizations

To this point, we have only examined isolated autopoietic organizations in GoL. However, such organizations do not in general exist in a vacuum. Rather, they arise from ongoing activity in a GoL lattice, persist for some period of time amidst this activity, and then are ultimately destroyed by it. Consider, for example, the scenario shown at the top of Figure 10. Here, an initial lattice configuration

Figure 8. The 30 processes in the full blinker organization (Figure 6) can be grouped into six equivalence classes based upon rotational and reflection symmetry. Each row corresponds to a distinct equivalence class. Note that duplicates have been removed, leaving 26 distinct processes. (Color in electronic version.)

12 Artificial Life Volume 21, Number 1 R. D. Beer Characterizing Autopoiesis in the Game of Life Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021

Figure 9. Simplified skeletal process graphs for (a) a block, (b) a blinker, and (c) a glider. Each class of processes that occurs in each entity is represented only once, abstracting over all translational, rotational, and reflectional symmetries of processes. Only unidirectional dependencies are shown with arrows; non-arrowed links correspond to bidirectional dependencies. (Color in electronic version.) containing no autopoietic organizations gives rise to a block that persists for several time steps and then is destroyed by a perturbation from the ongoing activity that surrounds it. How can we apply the idea of a closed network of process dependencies to such transient situations? The skeletal process graph for this scenario is shown at the bottom of Figure 10. Here, each non-null process involved in transforming the lattice between one configuration and the next is shown as a node, with dependencies between process edges. This graph contains 164 processes and 803 dependency links. Note that different numbers and types of processes are active in the lattice at different times, giving rise to changing dependencies. Due to these ongoing changes, which continue after the rightmost configuration shown, there is no closure of the entire lattice. How- ever, as highlighted in red, we do find the closed block organization embedded in this larger process graph for a period of time. That is, the red subgraph of Figure 10 is isomorphic to the closed process graph shown in Figure 5B (the same processes stand in the same dependency relationships in both graphs). Interestingly, although the block is observed to exist structurally over three iterations, organizationally it can only be said to exist over one iteration. This is because we must examine the dependencies between two adjacent sets of processes (i.e., two transformations between three subsequent structural configurations) in order to identify the block organization. Once we consider autopoietic organizations embedded within nonempty lattices, several new issues arise. First, such entities can undergo external perturbations, which can be either destructive or nondestructive [9]. Second, hierarchical organizations can occur. As a simple example, consider an array of blocks with overlapping boundaries. Not only do the individual blocks satisfy the definition of autopoiesis, but so does the entire array. Third, entities that require a continuous stream of organized components from their environment to sustain themselves become conceivable. While blocks, blinkers, and gliders can all persist indefinitely in an otherwise empty lattice, we could imagine a nonequilibrium entity that required, say, a continuous stream of incoming 1-components in order to persist. Such an entity

Artificial Life Volume 21, Number 1 13 R. D. Beer Characterizing Autopoiesis in the Game of Life would depend upon the source of these 1-components, but that source would not in turn depend on the entity. This could be interpreted as the GoL analogue of the flow of matter and energy required by a living cell. Many examples of patterns that can consume other patterns are known to exist; they are called “eaters” in GoL parlance. Likewise, GoL patterns known as “puffers” leave behind debris as they move. Such debris could be interpreted as the analogue of the waste products produced by living cells. Finally, Figure 10 suggests a complementary way of viewing GoL dynamics. GoL is traditionally conceived as a dynamics over structural configurations of the lattice. However, since each structural configuration corresponds to a set of dependencies between subsequent sets of processes, we can equally well consider GoL as a dynamics over process dependency networks (i.e., as a dynamics over the space of spatially embedded organizations). On this view, autopoietic systems within GoL are invariants with respect to this organizational dynamics; they are spatially localized, metastable attractors in the space of organizations. They are localized because their present operation is independent of any other organizations simultaneously occurring within the same lattice but with which they Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 currently have no causal contact. They are only metastable because over time they can come into contact with other organizations and suffer perturbations that destroy them. They are attractors to the extent that some transient organizations transform into them over time, constituting a basin of attraction. Although it is beyond the scope of this article, it would be very interesting to try to formally characterize the structure of the space of possible organizations in GoL, the dynamics induced on that space by the GoL physics, and the nature of its attractor landscape.

Figure 10. Transient skeletal process graph for a lattice in which a block comes into existence, persists for a short time, and then disintegrates as a result of an external perturbation. All non-null processes in the lattice are shown, with the subgraph instantiating the closed block organization highlighted in red. (Color in electronic version.)

14 Artificial Life Volume 21, Number 1 R. D. Beer Characterizing Autopoiesis in the Game of Life

7 Discussion

The concept of autopoiesis is necessarily quite abstract; it aims to capture only the essential organizational characteristics of living systems while generalizing over the many structural details of any particular instantiation. Of course, the claim is not that these details are uninteresting or unimportant, but simply that they may obscure an underlying commonality: Of all the spatiotemporal networks of chemical dynamics that can physically occur, living systems are precisely those that, through their closed and bounded organization of molecular transformation, are self- creating. Unfortunately, this abstractness can make it easy to misinterpret or trivialize what is being said (e.g., Dennettʼs [16] dismissal of autopoiesis as just the cell theory). In order to counter such misunderstandings, it is helpful to have as simple and concrete an illustration of the central ideas of autopoiesis as possible. But the utility of such a formulation goes significantly beyond mere pedagogy. The autopoiesis literature is full of verbal descriptions and computational models, but Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 quite short on formal theory. Careful analysis of a minimal concrete model can provide a more rigorous foundation for the concept of autopoiesis and highlight its ambiguities and difficulties. As is well known [30, 43, 59], cellular automata (CA) provide a powerful framework for studying complex spatiotemporal dynamics whose analysis is nevertheless made tractable by its discreteness in time, space and state. Of course, as in the application of any mathematical formalism, how we interpret the CA lattice states is a free choice of the modeler. Given our interests here in the organization of self-creating systems, we must reject the most common biological interpretation of GoL states as being simply “dead” or “alive,” since that would trivialize the entire enterprise. Instead, we interpret Conwayʼs update rule as a kind of physical law that applies uniformly throughout space and time, somewhat analogous to the quantum field theories that underlie the Standard Model in physics. Localized components and processes in GoL then arise through the local action of the Conway physics in a way loosely analogous to how particles and their interactions (and ultimately, all the phe- nomena of chemistry) arise through the interaction dynamics of local excitations of the corresponding quantum fields. Finally, the GoL analogues of biological cells arise as closed, bounded networks of interdependencies between local processes, just as autopoiesis claims living cells do. This article has argued that such GoL entities as blocks, blinkers, and gliders are autopoietic with respect to the GoL physics because they satisfy both the closure and boundary conditions of Maturana and Varelaʼs definition. The closure condition is satisfied by these entities because each constitutes a mutually enabling network of component-producing processes within the GoL lattice that collectively ensure the persistence of the whole. The boundary condition is satisfied because, in each case, this network of processes determines a spatial boundary that both delineates the entity from its environment and plays a necessary enabling role in its continuation. Beyond simply demon- strating that these GoL entities are autopoeitic, we have also characterized the specific closed organizations and boundaries that define each of them, as well as proposed a strategy for simplifying the description of these closed organizations through equivalence classes. Finally, we have suggested the possible utility of reconceiving GoL as a dynamics in the space of organizations. A significant advantage of the kind of cellular automaton model of autopoiesis that we have examined here is that it allows us to capture the precarious nature of biological entities. This has become a central concern of enactive approaches [7, 17]. Di Paolo [18, p. 16] defines precarious circumstances as “those in which isolated constituent processes will tend to run down or extinguish in the absence of the organization of the system in an otherwise equivalent physical situation.” In other words, the key idea is that precarious systems are not pre-given, but rather emergent dynamic networks that can fall apart. For example, separated from the organization as a whole, the individual constituent processes of a glider (Figure 7A) will in general either decay to quiescence or generate transient organizations unrelated to a glider. This can only occur when organizations emerge from local interactions rather than being hardcoded a priori into the model. Because such ephemeral networks of interdependent processes in GoL can come into being, persist for a while, and then disintegrate, characterizing the organization of these entities becomes a nontrivial and insightful exercise of direct relevance to theoretical biology and enactive cognitive science.

Artificial Life Volume 21, Number 1 15 R. D. Beer Characterizing Autopoiesis in the Game of Life

Examining the application of autopoiesis to a model that was not designed a priori to exhibit it forces us to be explicit about a host of important issues. By what criteria do we as observers distinguish the simple unities that we wish to study? Here, our mark of distinction has been the persistence of bounded, recurrent spatiotemporal patterns. How do we recursively distinguish within these unities components whose interactions generate them? Here we have enumerated a set of local processes of production, maintenance, and destruction that are grounded in the Conway physics of GoL. How do we describe the relations between processes defining the invariant organization that gives such a composite unity its identity, while at the same time abstracting over all inessential details of its particular instantiation? Here we have formalized an organization as the network of dependencies between the products and enabling conditions of the constituent processes. When is such an organization closed? Here organizational closure arises when the same set of processes and dependencies recur in a causal chain, allowing us to collapse that chain into a closed network. How do we determine the spatial boundary of such a closed network? Here we have identified this Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 boundary based not on the type of individual components, but rather on the role that they play in the processes that maintain the whole. Our analysis of autopoiesis in GoL has repeatedly forced us to confront the issue of choosing appropriate levels of description. Maturana and Varela are very careful to distinguish between structure and organization in their work. In GoL, the lowest possible level of structural description is obviously given by the states of all the individual lattice cells and the Conway physics that determines their dynamics.4 But at what level should organization be described? For example, we can treat every GoL process as distinct, group them into classes by effect (production, destruction, etc.), group them into classes by various combinations of symmetries (translational in space and time, rotational, and reflectional), and abstract over absolute location and time only, or relative spatiotemporal relationships as well. Formally, these decisions come down to what equivalence relations we choose to impose on processes and dependency networks. Particular choices have been made in this article, but other choices are possible, and whatever choices are made have consequences. To take just one example, consider the decision made here to associate the identity of a GoL entity with the invariance of its specific closed organization. An unfortunate consequence of this decision is that an entity whose closed organization is disrupted, however briefly, must always be considered to have disintegrated. For example, on this definition, a block that loses a corner 1-component is no longer a block, even though another block will appear in the same location one time step later. In contrast, if we associate a GoL entityʼs identity with the basin of attraction of its specific closed organization, then we can consider the loss of a corner 1-component to be a temporary perturbation to a block, which is subsequently “repaired.” In this case, the block would retain its identity across the perturbation. Here, the invariant would be the potential to return to a specific closed organization rather than that specific closed organization itself. A problem with this definition is that sequences of perturbations could take the system arbitrarily far away from the closed organization associated with a block, postponing the return indefinitely. This would make it difficult to justify calling the entity a block in the first place. A third possibility would be to associate identity with the persistence of any organizational closure, not just a specific one. For example, under this definition, a blinker that was transformed into a glider by some environmental perturbation would be considered to be the same entity, even though the two organizational closures are completely different. The point is not that the definition adopted here is the best one, but rather that analysis of the GoL model allows these different notions to be concretely explored and evaluated. The approach taken here to the analysis of GoL entities is a two-step process [8]. First, as described in this article, we characterize its autopoietic organization in an otherwise empty lattice. Sec- ond, we place it in a nonempty environment and study its interactions [9]. Although this approach works well for the specific GoL entities considered here, it is clearly unsatisfactory in general. Ideally,

4 Of course, the Conway physics of GoL is itself implemented as a program that executes on a microprocessor whose operation is ultimately grounded in “real” physics. However, these even lower-level details are irrelevant by construction in a properly implemented model.

16 Artificial Life Volume 21, Number 1 R. D. Beer Characterizing Autopoiesis in the Game of Life we would like to be able to directly identify and characterize autopoietic organizations embedded in and moving through arbitrary environments, even if those organizations depend upon some aspects of those environments for their persistence. This seems like a very difficult problem in general, but a few comments about how one might extend the work here in this direction are in order. Assume that we are given a transient process graph possibly containing an autopoietic entity (e.g., Figure 10) and that any entity in the graph persists long enough for its closure to be manifest. Then finding this entity amounts to finding either a process-equivalent subgraph of the process graph that is isomorphic to a known autopoietic organization or a causally connected pair of process-equivalent subgraphs that are isomorphic to one another. In either case, aside from the computational cost of identifying subgraphs (subgraph isomorphism is an NP-complete problem [15]), the central open issue is how best to define the appropriate equivalence classes over processes. Although we have argued that GoL entities such as gliders are autopoietic with respect to the Conway physics of GoL, we do not claim that they are alive. Autopoiesis is a formal property Downloaded from http://direct.mit.edu/artl/article-pdf/21/1/1/1665049/artl_a_00143.pdf by guest on 23 September 2021 that can be satisfied by any kind of components as long as they participate in the necessary relations [52]. However, Maturana and Varela are quite clear that only autopoiesis in the physical space has the necessary and sufficient conditions for life [39, p. 82; 51, p. 41]. Thus, this article has analyzed a model of—not an instance of—life. In order to explore some of the challenges of achieving autopoiesis in physical systems, the model can be extended in a variety of ways. Perhaps the simplest limitation to overcome is that of determinism. Physical systems are stochastic, and it is straightforward to extend GoL to a temperature-dependent probabilistic transition function whose zero-temperature limit is the deter- ministic Conway rule [30, 46]. One can then study such issues as the “melting points” at which various GoL organizations such as blocks, blinkers, and gliders disintegrate [1]. A second direction for extension would be to incorporate more realistic energetic requirements. Although cellular automata are a popular model within statistical physics [33, 58] and we demonstrated that GoL exhibits an almost universal decay to a quiescent equilibrium state, GoL does not possess a realistic thermodynamics or anything like conservation of energy or mass. However, other CA rules can certainly produce more physically realistic behavior, and CAs in general have been suggested as a fruitful modeling methodology for physics [14, 38, 49, 54]. Finally, the number of possible states and the spatial and temporal resolution could be increased, bringing CA models of autopoiesis closer to partial-differential-equation approaches such as reaction-diffusion models [26, 56]. Evans [23] and Pivato [37] have taken important steps in this direction for GoL. Indeed, numerically solving a set of PDEs on a digital computer essentially involves implementing a CA. Motivated in part by a desire to extend the analysis in this article to more realistic models of autopoiesis, we have recently developed such a model of metabolism-boundary co-construction [3]. Despite its many limitations, the attempt to characterize autopoiesis in cellular automata such as the game of Life can serve as a powerful vehicle for clarifying and formalizing the notion of autopoiesis. In addition, it can ground the analysis of other concepts from Maturana and Varelaʼs biology of cognition [39, 51], which are of direct relevance to the enactive approach to cognition [47, 48, 53]. For example, by studying the effects of all possible local perturbations to a GoL entity, we can determine the so-called cognitive domain of nondestructive interactions in which an entity can participate and analyze the structural coupling between it and its environment that results. These and other related topics are explored at length for a glider in a companion article [9].

Acknowledgments I would like to thank Eran Agmon, Ezequiel Di Paolo, Matthew Egbert, Tom Froese, Alex Gates, Eduardo Izquierdo, and Paul Williams for discussion and feedback. This work was supported in part by NSF grant IIS-0916409.

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