Open-Endedness in Genelife

We explore open-ended properties of Genelife (McCaskill & Packard, 2019), an evolutionary extension of the famous 2D cellular automaton Conway’s Game of Life (GoL; Gardner, 1970). The state space of the GoL is a 2D square lattice with a binary variable at each cell, with 1 representing a live cell and 0 representing a dead (or empty) cell. A configuration of 1s and 0s on the lattice evolves in time through the action of a local rule applied to all sites synchronously: If a cell is dead, it will become live if it has three live neighbors (a birth), and if a cell is alive, it will stay alive if it has two or three live neighbors; otherwise, it will die. The state space of Genelife associates a finite-length binary sequence with each live cell, which is copied from one of the mandatory live cells in the neighborhood during birth processes and hence represents a genome for the live cell. Reproduction is thus asexual, with the possibility of mutation but without recombination, in the current work. Although nontrivial, even adhering homogeneously to the GoL rule, the genome will typically encode the local rule used at its lattice site; thus Genelife becomes a spatially inhomogeneous cellular automaton. A spatial configuration may thus be regarded as a population of agents, with an agent in each live cell. When the local dynamics cause a cell to transition from 1 (live) to 0 (dead), the agent dies, and its genetic information is removed. As with the GoL, our interest is in the propagation and survival of local spatial patterns involving these cell states (e.g., gliders in the GoL), separated from other patterns by empty (lifeless) cells. In Genelife, one can think of these patterns as spatially structured colonies or multicellular organisms or ecologies of live cells, the genomes of which may be identical (e.g., all copied from a single live cell without mutation) or different (through mutation, differentiation, or ecological interactions of different genomes). Fundamentally, as in biology, the information in live cells comes from other live cells in the vicinity.

Conway’s GoL has long been a paradigmatic model for complex systems, as an example of complex dynamics, including universal computation, coming from a simple local deterministic rule. The GoL is subcritical in the sense that random patterns eventually converge to simple dynamics, fixed patterns with simple local oscillations. The subcriticality was extremely useful in GoL engineering of complex dynamical structures that eventually produced universal computation and construction. But as a model for biological complexity and open-endedness, both the subcriticality of the GoL and the lack of robustness toward small perturbations are problematic. We have created Genelife (McCaskill & Packard, 2019) as an evolutionary extension of the GoL, with the goal of providing a simple model for evolution with potential open-endedness, just as the GoL was a simple model for complex dynamics.

Cellular automata have been the focus of many studies using the tools of statistical physics (Bak et al., 1989; Packard & Wolfram, 1985; Schulman & Seiden, 1978; Wolfram, 1983), but none of these studies has pushed in the direction of understanding the potential of these systems to elucidate models for living systems and evolution. The basic idea of Genelife is to modify Conway’s GoL by endowing each live cell with a genome that can optionally modify the local dynamical rule according to the values (sequences) of the genes. Even without modification of the local dynamical rule, the genetic extension demonstrates interesting inheritance dynamics (McCaskill & Packard, 2019). With modification, the cellular automaton local rule governing a cell’s life or death state becomes spatially inhomogeneous, with different local rules determined by different genomes living at different spatial locations. Our approach in this article is to consider evolution in a range of (phenotypic) rule spaces, which may be organized in terms of the complexity of a local rule’s dependence on neighboring cell states. Live cells become agents in Genelife, and each agent observes its neighbors to determine its future state. Conferring live cells with agency, with local dynamics controlled by genomes, is the essential feature of Genelife that turns Conway’s GoL into a platform for studying evolution.

Agents perceive their neighbors; observation of neighbor cells’ states can be very coarse (e.g., an agent’s local rule might depend only on the sum of neighbor states) or fine-grained (e.g., dependence on more complex details of neighbor state configurations). Genelife provides structure to the rule spaces, which may in turn be interpreted as classes of agents with different sensory complexities.

A rich variety of evolutionary dynamics may be observed with details depending on the precise specifications of the local rules governing the cell state and the genome state. Here we explore a restricted subset of these to evaluate the open-endedness properties of Genelife. The regulation of birth and survival (and their absence, which is death), as a function of local density of live states near to the threshold for proliferation, can be done more finely using the additional symmetries introduced in McCaskill and Packard (2019). In addition to exploring the implications of these for open-ended evolution in this article, we consider two enhancements in this regulation, allowing (a) the genomes to specify the symmetry employed and (b) explicit overarching local density regulation. Both of these extensions are found to have a positive influence on the open-ended evolution.

There have been other extensions of Conway’s GoL to make it evolutionary: SproutLife (Shapiro, 2016) replaces seed configurations (e.g., a 2 × 2 set of live cells) with a sprout (e.g., a pentamino) and grows organisms from the sprout using GoL dynamics; HetCA (Medernach et al., 2013; Ryan et al., 2016) uses genomes for live cells to encode transition rules based on linear genetic programming, in contrast to Genelife’s more direct and symmetry-classified look-up table genetic specification of transition rules; Evolife (Zamaraev, 2019) also includes a genetic encoding of transition rules, with the addition of an energy variable for each living cell that controls its survival. Turney (2019, 2020, 2021) has explored evolutionary exploration of initial seed configurations in the context of a two-player version of the GoL called the Immigration Game (Woods, 1971). All of these GoL extensions have shown indications of open-endedness. What sets Genelife apart is (a) the dissection of a general family of rule extensions by successive symmetry breaking and allowance masks for rule extensions depending on numbers of local live cells; (b) the existence of a fully deterministic dynamics for many such rule extensions (apart from gene mutation, which could also be realized deterministically, if desired, using simple linear feedback shift registers, sufficiently uncorrelated with other dynamics; McCaskill & Packard, 2019); and (c) powerful analytical measuring and graphical tools that keep dynamical track of connected spatial patterns and their movements (e.g., gliders), genealogies, activities, and novelty. This makes Genelife a suitable simulation tool for studying the basis of open-ended evolution.

Genelife is available at a public repository (Packard & McCaskill, 2018) as a set of Jupyter notebooks using a Python interface with C code for simulation speed. The reader is encouraged to explore examples presented here with the simulation. Many of the dynamical phenomena are poorly captured by static images.

The state space of Conway’s GoL is a set of possible configurations of 0s and 1s on a lattice, that is, {0,1}^ℤ2. For every point in the lattice x ∈ℤ², a configuration’s local value s_x is either 0 (for a dead cell) or 1 (for a live cell). Genelife’s configurations add a genome with a positive integer value for every live cell, so the state space is $0,ϕ,1,NZ2$ with states $sx,gx$⁠. It is convenient also to consider projections of these states into the usual binary lattice configurations, with s_x = 0 if the local state is 0, that is, for $0,ϕ$⁠, and s_x = 1 if the local value is 1, in which case, there is an additional local variable comprising heritable information in the form of a single genetic sequence or genome represented by a nonnegative integer g_x ∈ℕ. For the simulations presented here, we shall restrict possible genomes to be binary numbers with ≤ 64 digits, and the lattice ℤ² will be approximated with a finite subset, usually 256 × 256 sites, bounded on each side (unless otherwise indicated) with periodic (toroidal) boundary conditions.

In general, taking this genetic information into account hugely expands the local state space at each cell, but we retain the dynamical simplicity of the GoL by retaining only the three elementary biological and GoL processes of survival, death, and birth—with genetic information unchanged in survival, annihilated in death, and inherited as a copy of one of the live neighbors in birth. As in the GoL, each of these processes can indeed be made deterministic as described in McCaskill and Packard (2019), and indeed, Genelife, with appropriate genomes specifying only the GoL rule, can proceed with identical state dynamics to the GoL. Note that the live cells may now be identified by their genetic information and hence can also be interpreted as agents in the simulation. These agents sense their local neighborhoods insofar as their survival or proliferation follows rules that depend on the local neighborhood.

On a finite square lattice, we resolve these neighbors at the borders of the array using periodic boundary conditions. The GoL rule has the values R_GoL(1,2) = R_GoL(1,3) = R_GoL(0,3) = 1, and 0 for all other arguments.

To capture the simple essence of genetics modulating the cellular automaton rules, the specification of the dynamics can be separated into three steps:

1. Examine the current live states s_xδ^t in the local neighborhood to determine the current local dynamical configuration, which is the arrangement of neighboring live states, distinguished in the chosen symmetry. This step depends only on live states, not on their genomes. It may be quantified in terms of integer variables characterizing the state configuration. For less than semi-totalistic symmetry, this involves, in addition to their sum s, either $se=sesxδ$⁠, the number of edge-centered live states in the Moore neighborhood, or $crsxδ$⁠, the rotation needed to map the configuration into its canonical form (see later), or both.

2. Determine whether birth or survival occurs. Examine the current live state $sxt$ to determine whether a birth (if $sxt=0$⁠) or survival rule (if $sxt=1$⁠) is in contention. Then, for survival to occur, the genome at the central site x must contain a gene g specifying survival for the current local live state configuration found in Step 1. For birth to occur, examine each of the live genomes in the neighborhood for the presence of a gene specifying birth for this current local live state configuration. If a majority (for the majority variant of Genelife explored in this article) of the live genomes contain this gene, then birth occurs. For all the symmetry cases studied here, this can be formalized for survival and birth rules as (the set of cases with next state $sxt+1=1$⁠)

   $RsS=RsSsxt,sxδt,gxt=sxt=1andgS,s,se,cr∈gxtRsB=RsBsxt,sxδt,gxδt=sxt=0andgB,s,se,cr∈gxδt≥s2$

   where $$ represents the number of elements in the set.

3. If survival occurs, the state and gene survive unchanged at the next time step. If birth occurs, choose one of the live neighbor genomes to copy (possibly with mutation) as the genome for the live central state at the next time step. If neither birth nor survival occurs, the central cell is “dead,” that is, $0,ϕ$ at the next time step.

The choice of live neighbor gene studied in this article is either neutral (independent of neighbor genome values) or based on the integer value of the neighbor genomes. Specifically, we use the term neutral to refer to an ancestor choice algorithm that chooses an ancestor without reference to the value of the genomes. One possibility for a neutral algorithm would be to choose an ancestor at random from the live neighbors; we prefer, however, a deterministic algorithm that enables a position-based unique ancestor choice to preserve determinism in Genelife’s dynamical rules (i.e., for mutation to be the only nondeterministic element of the evolutionary dynamics). The algorithm used for neutral ancestor choice is based on the value of s_xΣ; for s_xΣ = 1, 3, 5, 7, a position-based unique ancestor can be identified as the ancestor in the “most different” position, based on symmetry of the ancestor positions. The idea is illustrated in Figure 1 for a few ancestor configurations, for the case s_xΣ = 3.

The basic idea illustrated in Figure 1 is straightforwardly generalized for all ancestor configurations having s_xΣ = 1, 3, 5, 7 (McCaskill & Packard, 2019). For s_xΣ taking even values, there is not a unique most different ancestor position; in the present work, this ambiguity is resolved by prohibiting birth in these cases (the Genelife simulation has various other neutral algorithm choices). This neutral ancestor choice algorithm makes Genelife deterministic and symmetric like the GoL, in the absence of mutation, when birth is restricted to these cases, even for the lower-symmetry cases introduced later.

Each genome is interpreted as encoding two sets of nonzero next binary state (i.e., live) rules, one for current central state 0 (birth rules) and one for central state 1 (survival rules), with the set covering the possible distinguished live neighbor state configurations in the chosen symmetry. For the semi-totalistic case, live neighbor configurations are distinguished only by their sum s_xΣ (with values 0–8), and we choose to disallow live rules with s_xΣ = 0, leaving eight distinguished live rules possible separately for survival and birth. However, in the neighborhood of a particular site, there are potentially multiple live states and hence multiple genomes, and so we must also specify which information is used to specify the dynamical next state rule. For birth rules, that is, for local configurations without a central genome, we retain totalistic symmetry for the genetic encoding by requiring a majority ≥1/2 of the genomes at neighboring live cells to encode a live rule for the current neighbor live state configuration. Symmetric alternatives not pursued in the current article include requiring just one (OR) or all (AND) the neighboring live cells to specify the live rule. A nontotalistic alternative is to designate a particular live state as directing the rule (e.g., the one at the most different position among the eight nearest neighbors, as discussed in McCaskill & Packard, 2019).

The GoL for three live neighbors has the same outcome whether or not the central cell is alive: Both survival and birth lead to a central live state (1). Nothing changes there if one regards survival with two live neighbors rather as a birth overwriting the central live state. Thus there are now two possible interpretations of cases in which a live cell persists from one time step to the next: It survived or was repopulated. We consider birth rules for repopulation as an optional additional birth process in our model that can change the live cell’s genome. This option turns out to yield interesting evolutionary dynamics and has been used throughout the present work. The Genelife simulation enables conditioning of overwrite according to the value of s_xΣ; we consider here only the unconditional case (overwrite is allowed for all values of s_xΣ). Overwrite uses the same algorithm as birth to choose $gxt+1$⁠. In this article, the selection/inheritance algorithm is either neutral (all live genome neighbors “equally fit,” i.e., are equally eligible for selection if appropriately positioned) or biased by the numerical value of the genome. The birth process then chooses that genome from among the “equally fit” subset (see later) of the live neighbors { g_{x +δ}^t} that is at the most different position (the Genelife simulation provides other algorithms as well, determined by bits 0–3 of an option mask called repscheme), resolving infrequent remaining ambiguities in this deterministic choice (potentially occurring for symmetric even live neighbor configurations) by aborting the birth process. Interestingly, complex evolutionary dynamics may be obtained with purely deterministic genetic dynamics, that is, with a nonstochastic version of R_gB.

The genome inheritance function R_gB thus involves a two-stage process (genetic and positional) for selecting a genome for birth among the live neighbors, which is executed only if the current configuration of live states is allowed. Previous work (McCaskill & Packard, 2019) has explored a variety of R_gB functions; here we restrict our attention to two choices for the “equally fit” subset used to define the genetic first stage of selection in R_gB: (a) R_gN, a neutral choice of all live neighbors as candidates, as described in the discussion of Figure 1 (see also McCaskill & Packard, 2019, and the Genelife simulation tool, Packard & McCaskill, 2018, for details), and (b) R_g↑, a purposely biased choice of the “equally fit” subset of g_{x +δ}^t, based on the numerical value of the integer, that is, those with the equal highest numerical value of g_{x +δ}^t. We have specified some constraints on the local rules R_s and R_g that we will enforce for the work presented here, but most of these constraints may be relaxed in the Genelife simulation tool.

In addition to the deterministic genetic dynamics just described, we typically add mutations to the genome, random changes with probability μ, which will be held at 2⁻⁸ = 1/256 unless otherwise noted. Multiple mutations are allowed, as a realized mutation is followed by a further mutation attempt with the same probability. The point in the genome sequence for mutation is chosen at random from the available positions. Note that if, on average, a fraction 2⁻⁸ of sites are mutated, this is one mutation for every 2⁸ live sites, and the minimum size of a square patch with that many live sites is 2⁴ × 2⁴ = 16 × 16, but more typically at lower live cell densities, 32 × 32. This means the spatial density of mutations is, on average, such that structures up to that size might form before being perturbed by a mutation. Conversely, persistent random structure on length scales less than that size must be due to randomness (or deterministic chaos) of the evolutionary dynamics rather than randomness due to mutations, even though short-lived structure may be caused by mutations.

The semi-totalistic rules of the GoL neglect all pattern features in the arrangement of live neighbors other than their sum (or, equivalently, their density): The sum of neighbors may take on nine possible values {0, …,8}, and the survival and birth functions R_sS and R_sB may be considered as look-up tables (LUTs) with these nine entries. Enforcing the constraint R_sS(0) = R_sB(0) = 0 reduces the size of the LUTs to eight entries. The GoL corresponds to a particular pair of LUTs, as shown in Table 1.

In the semi-totalistic case, Genelife’s genomes interact with dynamics by determining directly the LUT entries with binary genes, eight genes for R_sS and eight genes for R_sB. Thus there are 2¹⁶ = 65,536 different genome phenotypes (rules), encoded by a larger number of possible genomes. If we represent a semi-totalistic genome with four hex digits (two lower digits for R_sS and two upper digits for R_sB), the GoL corresponds to the semi-totalistic genome 0×0406.

Because the number of different rules in the semi-totalistic case is so restricted, the problem of reaching a balance between proliferation and extinction has little scope for a solution. Indeed, as mentioned already, the GoL is subcritical, settling down to rather uninteresting behavior with residual isolated periodic structures after a transient. However, simply ignoring all symmetries introduces both unphysical asymmetries and too many individual rules for compact genetic encoding and evolutionary exploration. An established simple first way to break the semi-totalistic symmetry is by separating the effect of the closest (edge-centered) neighbors and the farther (diagonal) neighbors (McCaskill & Packard, 2019). This may be neatly accomplished by expanding the semi-totalistic LUTs to include entries for all possible values of the sum of neighbors weighted by distance, with the distance of the edge-centered neighbors being 1 and the distance of the diagonal neighbors being $2$⁠. Excluding neighborhood configurations of all 0s or all 1s, there are 23 independent weighted configurations; that is, the weighted sum takes on 23 different values. Thus 46 bits are needed to specify both R_sS and R_sB (for further discussion, see McCaskill & Packard, 2019). We refer to this space of rules as quarter-totalistic rules and the corresponding 46-bit genomes as quarter-totalistic genomes, a space of 2⁴⁶ ∼7$1013$ possibilities. Quarter-totalistic rules may also be characterized as depending on an additional argument, the number of edge-centered live nearest neighbors s_xe, in addition to the central state s_x and total number of live neighbors s_xΣ that are the arguments of semi-totalistic rules. Possible values of s_xe range from 0 to $minsxΣ,4$⁠.

Another way to break semi-totalistic symmetry introduced in McCaskill and Packard (2019) is to consider the sequence of eight neighbor states around a cell as a pattern that can be rotated cyclically without changing the rule outcome: We label this 1/8 or canon or rot symmetry. Not all 8-bit occupation patterns are distinct. Clearly if the summed occupation s is different, then the states are distinct. But there may be different patterns with the same value of s. For s = 1, all single occupation patterns are related by rotation and hence equivalent (not distinct): We do not distinguish corner and edge-centered neighbors in this symmetry. However, for s = 2, the two live neighbors may be separated by 0, 1, 2, or 3 unoccupied cells, and these configurations cannot be mapped into one another by rotation. As shown in McCaskill and Packard (2019), we can define a canonical pattern for each distinct set of neighbor patterns, of which there are {1, 1, 4, 7, 10, 7, 4, 1, 1} different ones for s in {0, 1, 2, … , 8}.

Distinguishing the distance between cells (as in the quarter-totalistic case) as well as differences in patterns under rotation and reflection reduces the symmetry further to the well-known case of physical spatial symmetry in 2D. In this case (as analyzed in McCaskill & Packard, 2019), there are 1, 2, 6, 10, 13, 10, 6, 2, 1 distinguished patterns for s in {0, 1, 2, … , 8}. Each of these distinct patterns can have a separate survival and/or birth rule.

In this article, in addition to examining the evolution of Genelife with a particular one of these symmetries, we also consider a genetic encoding of rules in which the gene itself encodes the symmetry with which the cellular automaton rule will be decoded. A minimum of two bits in the genome are required to specify one of the four symmetries introduced earlier, to which we refer as sum (for sum encoding of neighbors), dist (for distance encoding of neighbors), rot (for encoding with canonical rotation), and 2D (for full 2D symmetry encoding), or equivalently as 1/2, 1/4, 1/8, and 1/16 symmetries, respectively. This is explored in the section 6, Genetic Control of Symmetry Breaking.

Successive breaking of symmetry may be identified with an increase in sensory resolution, if we imagine live cells containing agents that are sensing data in their neighborhoods.

Figure 2 illustrates how quarter-totalistic agents have greater resolution of their neighborhood data than semi-totalistic agents. Likewise, breaking totalistic (sum) symmetry but preserving cyclic rotational symmetry (1/8 symmetry), as described earlier, results in increased sensory resolution (Figure 2(c)). Then, patterns of 1s in the gray area of Figure 2(a) that can be rotated cyclically to match one another are subject to the same automata rule and have just one entry each for birth and survival in the LUT. Finally, in 1/16 symmetry, we have distinct entries in the genetically encoded LUT for each pattern of 1s in Figure 2(d) (distinguishing distances, i.e., gray and dark gray cells) that cannot be mapped onto one another by rotation or reflection.

We will find it useful to control the effects of genetic variation (mutation) by constraining genome locations where mutation can take place. We specify the constraints as a mask, with 0s in genomic locations where mutation cannot take place and 1s in the locations where mutations can take place. The mask for $(sxt,sxt+1)=(0,1)$ is called the birth mask, M_b, and the mask for $(sxt,sxt+1)=(1,1)$ is called the survival mask, M_s. Evolution with no constraints is obtained with masks having 1s in all available locations; for the semi-totalistic genomes, the masks would have values M_s = M_b = 0xff. Constrained evolution occurs when some of the bits in M_s and M_b are set to zero.

Note that for different rule spaces (semi-totalistic, quarter-totalistic, symmetry coded), the genome for each rule space is a different size, and so the masks M_s and M_b will have different numbers of bits. Masks M_s and M_b cannot be compared from one rule space to another.

In the special case where symmetry breaking is genetically encoded (considered later), the upper bits of M_s and M_b are used to control access to the symmetry-breaking genes rather than the LUT, as described subsequently.

For each rule space, we focus on a standard set of meta-parameter variations, with a simple template of results repeated for a run using each meta-parameter set. The standard template of results will enable comparison between rule spaces and between meta-parameter variations.

The set of variations we consider is summarized in Figure 3. For each rule space, semi-totalistic and quarter-totalistic, we explore four meta-parameter settings. For the symmetry-coded rule space, we explore only the neutral case, unconstrained and constrained. For each of these 10 cases, we display a template of results: two spatial configurations at t = 100 and t = 1,000, with two activity statistics described in the following section.

After displaying the results for these 10 cases, we select three exemplars, the neutral unbiased cases for each rule space, to study the effects of density regulation.

The sensory resolution of agents whose local rule lies in the different symmetry classes ranges from low resolution (sum) to high resolution (2D). Correspondingly, the number of genome bits needed to encode low resolution is lower than the number needed for high resolution. The present work explores sum and dist rule spaces independently, then all four symmetry classes together in the form of mixed populations.

Evolutionary activity is a simple measure of how much innovation is being generated in an evolving population. Evolutionary activity was introduced as a statistical measure to quantify the degree to which a population evolves. The measure was originally used in a study of sensory-motor agents (Bedau & Packard, 1992) and has since been used to quantify evolution in other models for evolution, for example, Geb (Channon, 2006) and StringMol (Droop & Hickinbotham, 2012). It has also been proposed as a litmus test for lifelike properties of Artificial Life models (Bedau et al., 2000; Bedau et al., 1998).

The activities $ãit$ as a function of time t are sometimes referred to as activity waves; an example of 1,000 waves taken from a representative simulation is shown in Figure 4. Note that the measurement of activity is nonperturbative in the sense that the evolving population is unaffected by the measurement.

Evolutionary activity can also be defined for any quantitative functionality or phenotype of genomes, for example, for the coding of a particular birth rule or combination of birth rules in the genome or for the genome specifying a superset of the GoL rules. Then its use as a measure of innovation acquires functional significance. In this work, we restrict attention to the simplest genotypic evolutionary activity.

We may think of measurements of $Ãt$ as an empirical sample of the activity distribution. We represent the time evolution of the activity distribution $Ãt$ by taking deciles of the distribution at each time step and graphing each decile as a time series, as seen in Figure 5.

Different types of evolutionary activity are obtained from the evolving distribution $Ãt$⁠. If the population dies, $Ãt$ collapses to a delta function at zero. If some genomes survive forever, $Ãt$ will have a top quantile that grows indefinitely. If all new genomes die out very quickly, $Ãt$ will collapse to all quantiles having small values.

A_new^t is shown for this example in Figure 7, using $amin$ and $amax$ calculated as described from centiles $c40t$ and $c80t$⁠.

The key criteria for open-endedness are that the activity profile be sustained over time and that new activity remain positive. Ideally, but not necessarily, it continues to increase. The new activity going to zero in Figure 7 is a signal that this example is not open-ended. This is best regarded as a necessary criterion, but not necessarily sufficient, as discussed in the Conclusions. Note that according to this criterion, neutral evolution in a population will sustain genome changes indefinitely and asymptote to a nonzero value of new activity. In the simple, purely genomic context, this does constitute a simple form of open-ended evolution, even though no developments in functional complexity are occurring. For Genelife, a population cannot sustain genome changes indefinitely because the space of genomes is finite, though large (e.g., 2⁴⁶ quarter-totalistic rules). With the mutation rate used here of 2⁻⁸, and 2¹⁶ lattice sites, a simulation time of ∼2³⁸ would be needed to exhaust innovative genetic possibilities, so for simulation times used here (10³–10⁴), access to genomic innovation may be considered effectively open-ended. If experiments were to approach the bound created by the finiteness of the space of genomes, Genelife could be generalized to include an infinite space of genomes, for example, if local neighborhoods were allowed to expand in size.

Measurement of evolutionary activity usually requires reference to a neutral model, in order to measure activity with respect to that produced by a version of the model where no selection is taking place, providing a baseline of activity production (Bedau et al., 1997). The neutral model consists of a population of genomes maintained in parallel with the evolving genomes, uncoupled with the evolving population but constrained to match birth, death, and mutation processes, without reference to a genome’s survivability in the evolving population. Activity statistics are computed for the parallel neutral model just as for the evolving population. The neutral model should be used to set the lower bound, currently set to be the 40th centile $c40t$⁠, to properly avoid counting new activity that could be attributable to neutral drift. However, in the present work, scanning many rule spaces, we have chosen the simpler use of $c40t$ to avoid the complication of separately computed neutral models for each case. A spot check for the quarter-totalistic biased constrained evolution, used as the example in this section, indicates that $c40t$ may actually be lower than the base level set by a neutral model, so reported measurements of A_new^t will be systematically overestimated, containing some activity attributable to neutral evolution. Future work on particular rules should include computation of a neutral model; algorithms for doing so are included in the code release (Packard & McCaskill, 2018).

We now present results of several simulation runs, comparing results and open-endedness properties as reflected in their evolutionary activity. We begin with the simplest case of semi-totalistic rules; the Genelife variants explored in this section are listed in Table 2.

The simulations for the runs illustrated were identical, except for the indicated changed parameters. The lattice was 256 on each side in these and subsequent simulations, unless otherwise specified, and the initial condition was a random population within a centrally placed square of size 100 sites on each side. Different colors represent different genomes in this section.

The figures for these cases (Figures 8–11) will follow identical templates: (a) the population after 100 time steps, (b) the population after 1,000 time steps, (c) the activity production, and (d) the new activity. For (c) and (d), deciles of the respective distributions evolving in time are shown.

Brief comments on the dynamics of each simulation run are contained in each caption; comments comparing the runs and conclusions to be drawn from the comparisons follow the figures.

The simulations in this section, enumerated in Table 3, demonstrate that the more differentiated rules that distinguish orthogonal from diagonal neighbors indeed provide a greater scope for a richer repertoire of evolving patterns than in the semi-totalistic case (Figures 12–15). They also demonstrate that genetically open-ended evolution, as characterized by new activity, is, at least on this timescale, though not a generic, at least a not uncommon feature of the dynamics.

Having established that the symmetry breaking indeed provides scope for a richer evolution of patterns in Genelife, in this section, we examine the consequences of allowing the genes not only to influence the local rules but also to determine which spatial symmetry is employed to do so. All four symmetries that we have implemented are considered: 1/2 (sum, referred to as semi-totalistic), 1/4 (dist, referred to as quarter-totalistic), 1/8 (rot, as described earlier), and 1/16 (2D, for full 2D symmetry, as described previously).

In our foregoing explanation of rules in different spatial symmetry classes, we noted that the symmetry classes (sum, dist, rot, 2D) may be interpreted as providing the Genelife agents implementing local rules in each respective symmetry class with successively higher resolution of the agents’ sensory apparatuses. Genetic control of symmetry breaking enables the preparation of mixtures of symmetries and observation of their evolutionary competition. One might naively expect that agents with higher-resolution sensory apparatuses (agents with 2D-symmetric local rules) should dominate, but we will see that the actual story is more complicated and nuanced.

Recent work by Roli and Kauffman (2022) holds that “the opportunities—and possible harms—that each species presents to the others” (termed affordances by the authors) is characteristic of evolution in nature, but not of evolution in “programs and robots.” Contrary to the authors’ claims, Genelife, with its ability to explore a space of agents with variable sensory resolution, provides an explicit example of affordances in a purely artificial context.

In the current Genelife implementation of genetically controlled symmetry breaking, the 1/2 (sum), 1/4 (dist), 1/8 (rot), and 1/16 (2D) symmetries are allowed for s = 1–4 only. This choice was made so that all distinguished configurations of the full 2D symmetry case can be genetically encoded (31 bits each for birth and death), leaving 2 bits (in the 64-bit genome) to specify the symmetry. Note that the GoL employs no survival or birth rules for s > 3, and the choice to ignore very dense neighborhoods for proliferation or survival corresponds to a biological scenario in which excessive crowding of live genes prevents both birth and survival (compare this with the following section on explicit density regulation). When this variable symmetry mode is chosen, the 8 bits of the survival and birth masks enable the s = 1–4 cases in the lower mask bits and the four symmetries 1/2 (sum), 1/4 (dist), 1/8 (rot), and 1/16 (2D), in that order, in the upper mask bits. In particular, if M_s = 0xff and M_b = 0xff, then all symmetries and values of s between 1 and 4 inclusive are allowed. In the Genelife implementation, the gene coding is positional, with 2 bits specifying which symmetry is to be employed with decoding. When using genetic control of symmetry breaking, genetic change to the upper bits of the LUTs for s > 4 is not available.

We observe that a population with a mixture of symmetries quickly evolves into domains, or patches, which then compete for dominance. An example of this competition is shown in Figure 16. The competition proceeds and can result in a single symmetry class dominating, or else in an ongoing mixture of different classes. This situation is illustrated in Figure 17, where the population of each symmetry class is tracked separately.

The evolution of populations containing a mix of symmetries is shown in Figure 17 for three different evolutionary dynamics. The details of which subpopulation dominates are highly dependent on the initial condition, so the distribution is shown for 10 runs. In Figure 17(a), for unconstrained neutral dynamics, we see that the simplest agents (sum) usually dominate, though often with a mix of the next most complex agents (dist). The more complex agents (rot and 2D) typically die off; they may occur occasionally as a result of mutation, but they typically die off quickly. Figure 17(a) and 17(b) show the same system, constrained in two different ways (using the birth mask M_b. We see that the constraints on evolution affect the outcome substantially; in Figure 17(a), the two most complex subpopulations (rot and 2D) vie for dominance; in Figure 17(b), the distribution of subpopulation configurations is much broader, with the simple agent populations (sum) tending to dominate slightly.

It may be surprising that the more complex agents don’t always dominate. If an agent can perceive more from its local environment, why shouldn’t it be able to find a way to use the information to win in the fight for selection of the fittest? There are two contributing factors. First, there is a cost also in Genelife to increasingly complex sensor distinctions of local neighborhoods. It is that all of these distinctions when encoded in the genome lead to longer stretches of genetic information that must be inherited for function preservation, and as we know from extensive work on population dynamics, for example, with the quasi-species model (Eigen et al., 1989), for a given mutation rate and selective advantage, there is a maximum length of information that can be preserved by evolution. Second, the advantage obtained by an agent being able to sense and respond to more detail in its local neighborhood depends on resolvable local configurations occurring with enough frequency to make a difference. It is often the case that the spatial dynamics relaxes onto a subset of configurations (an attractor) that may exclude many local configurations that may be relevant to providing an advantage for the more complex agents.

In Figures 18 and 19, we give two examples using genetic encoding of symmetry breaking, showing the activity profiles and measurement of new activity.

Limitations on richness of the dynamical structures formed by Genelife evolution often seem to be caused by evolution toward dense populations. For the case of unconstrained evolution, in both the semi-totalistic and quarter-totalistic cases, we see evolution to largely static populations, that is, evolutionary dead ends.

The introduction of density regulation is consistent with Conway’s original rationale for the GoL: Anthropomorphic descriptions of GoL describe live cells with more than three live neighbors “dying from overcrowding.” This is indeed what happens when R_default = R_death. (Note that using R_death for the default rule ensures that whenever a live cell is observed, it has come from $Rssxt,sxδ;gxt,gxδ$ and not from the R_death default.) Because the GoL has subcritical dynamics, this also applies (less abruptly) for the case R_default = R_GoL. The significant difference introduced by the density regulation here is that no specific choice of rules need be made: All possible local rules are allowed by genetics and available for open-ended evolution. As long as the subcritical density condition is met, even high nearest neighbor occupation states may play an occasional role in the dynamics when larger than nearest neighbor neighborhoods are employed in the density regulation. Thus, and although the condition is an override of all genetically determined rules, this allows a wider range of local rules to be employed, and the overcrowding density ρ may effectively be computed more finely using a larger neighborhood.

We now observe the effect of density regulation, using ρ₀ = 0.25for three cases: semi-totalistic neutral unbiased, quarter-totalistic neutral unbiased, and symmetry-encoded neutral unbiased (Figures 20–22).

We have introduced density regulation as a mechanism to open up new possibilities for novel spatial pattern formation. We investigated this for the rule spaces in semi-totalistic, quarter-totalistic, and gene-determined symmetries. We have seen some level of success in this endeavor, illustrated, for example, by heterogeneous spatial patterns seen in density-regulated symmetry-coded neutral unconstrained evolution in Figure 22(b).

We have the several situations in which ongoing production of new activity is observed. However, the spatial dynamics in many of these cases reaches an asymptotic state that seems qualitatively repetitive, not innovative, as might be expected for open-ended evolutionary dynamics. It may be that evolutionary innovation is indeed happening but is simply not visible in our view of the spatial dynamics. An example of structure that might be hidden is evident in some cases of populations with mixed symmetries; we may see complex spatial dynamics of interacting regions containing subpopulations of different symmetry classes but that generate qualitatively similar spatial dynamics (compare Figure 16).

We have noted, in defining evolutionary activity, that there may be different versions of activity, each version stemming from observation of different system attributes to measure the occurrence of innovation. The dependence of evolutionary activity on particular empirical choice may be seen as a shortcoming of the statistical approach, but on the other hand, it offers the flexibility to define multiple versions to observe different forms of innovation. In our earlier definition of new activity, we defined two versions, A_new^t for measurement of innovation in the population of genomes and $Ânewt$ for measurement of innovation in the population of local spatial patterns.

The measurements of new activity in the preceding studies have been measurements of new genetic activity, A_new^t. We now contrast these measurements with measurements of new spatial pattern activity, $Ânewt$⁠, for a few illustrative cases.

The examples shown in Figure 23(a)–23 (c) are representative of most of the cases in which ongoing new genetic activity is observed: New spatial activity diminishes asymptotically. For symmetry-encoded neutral unconstrained dynamics (Figure 23(b)), the new pattern activity may asymptote to a low but nonzero value. The only example that seems to show robust nonzero asymptotic behavior for both new gene activity and new pattern activity is shown in Figure 23(d), symmetry-encoded neutral constrained dynamics with density control. For reference, we include a pair of space-time snapshots for Figure 23(d) in Figure 24.

The prevalence of the low new pattern activity in the presence of high new gene activity begs an explanation. Evidently, evolutionary dynamics is producing an ongoing stream of new genomes, but the new genomes produce no (or little) novel spatial dynamics. The new genomes are neutral with respect to spatial dynamics, and because it is change in spatial dynamics that must cause preferential selection of genomes (fitness), we often appear to have ongoing new gene activity that is exploring a subset of rules (genomes) that have comparable fitness.

It is also worth understanding the aspects of the rule space illustrated in Figure 24 that lead to its double open-endedness. Spatial variety is probably inhibited by overcrowding, so density control is essential to provide necessary voids for spatial pattern innovation. The constraints provided by survival and birth masks also act to reduce density. Though many of the rule spaces explored in the present work appear to lack asymptotic pattern activity, Genelife has ample controls that should enable more robust versions of open-ended pattern activity, with further exploration.

The Genelife approach presents, within the limitations of discrete 2D cellular automata models for studying evolution, a rather natural and simple way to examine the influence of genetic information on nonlinear spatial dynamics. The key idea of associating a genetic sequence, which can modify the local cellular automata rules, with each active/live cell in cellular automata like the GoL opens a large family of models for dynamical evolution, depending on the way in which genetic information affects the rules and is transmitted from local gene states when new live states are formed. The Genelife platform is sufficiently flexible to explore a reasonably complete set of the more natural implementation options for this. It includes an array of tools for recording and identifying both genetic and spatial pattern novelty as well as population diagnostics, genealogical trees, and activity statistics. The natural question that we seek to address in this article is whether this simple genetic control of rules is sufficient to initiate an unlimited evolutionary process with ongoing increases in complexity.

The spatial complexity associated with the GoL is thought to reside in the delicate balance between proliferation and decay of live states caused by the very limited configurations (three live neighbors) allowed for proliferation—that is, conditions that are also not usually propagated to the next time step. Although the GoL rule is subcritical and maybe not optimal for pattern evolution, the number of such delicately balanced rules within semi-totalistic rules, of which with nearest neighbors there are only a quite limited number (512), is so small as to provide very limited scope for evolution. For this reason, and to allow a larger space for evolution, we relaxed the semi-totalistic symmetry of the rules, reducing it in stages en route to the familiar physical 2D symmetry of the local neighborhoods in isotropic space. Although the number of configurations for birth and survival rules is then much larger, one still needs to constrain the world in which evolution occurs so as not to allow facile and ubiquitous proliferation and/or survival for spatial pattern complexity to play a role in evolution. Allowing genes to activate any proliferation and survival rule without constraint will rapidly cause the space to be filled with live states: In some sense, there is no significant problem for evolution to resolve, and hence the outcome of evolution converges rapidly to a certain type of configuration.

Our work revealed one interesting example of evolution in this unconstrained case even for the semi-totalistic rule space. Accessing arbitrary genetically encoded LUTs, especially for survival R_S, leads to the lattice becoming structured with relatively stable domains that won the competition for survival of the fittest during an early evolutionary transient. The reason for this long, persistent structure is connected with an implementation option employed in this example of not allowing occupied cells to be born into (no overwrite). The domain contents may themselves be dynamic, with ongoing evolution producing new activity within the domains (e.g., the semi-totalistic evolution with unconstrained neutral genetic dynamics shown in Figure 8), or they may be static, with a variety of textures that often have interstices that support localized evolutionary dynamics that can produce ongoing new activity notwithstanding the large-scale static structures (e.g., the quarter-totalistic evolution with unconstrained neutral genetic dynamics of Figure 12). Overall, however, these simple versions of Genelife have no way to reward configurations with lower densities, so high-density life takes over. Biological life may have some analogs to this, for example, a lichen filling a stone face and remaining for extremely long periods, but most biological populations are much more dynamic.

We have addressed the basic issue of constraining the possibility of proliferation to depend on specific subset patterns in two different ways, for the context of semi-totalistic and quarter-totalistic rule spaces: (a) imposition of constraints through masks M_s and M_b, limiting the locations in the genome that are accessible for mutational modification, and (b) imposing a density constraint—when local density exceeds a threshold, the genetically determined local rule is overridden by a default, for example, death or the GoL. Both of these mechanisms enable evolutionary dynamics to find a path to ongoing open-endedness, as measured by new activity.

The effect of adding constraints is shown in the comparison of Figures 8 and 9 for the semi-totalistic neutral case and of Figures 12 and 13 for the quarter-totalistic neutral case. Note that the dynamical variety increases with the addition of evolutionary constraints but that the level of new activity actually diminishes, consistent with the overall decrease in population density.

Alternative modifications to Genelife’s evolutionary dynamics could be explored to address the issue of providing a fundamental limit to simple proliferation; one interesting and natural direction would involve a resource species, that is, food for Genelife agents, without which they would die or be unable to reproduce. Such a limitation was provided in the form of an energy dependence in Evolife (McCaskill et al., 2007). For this article, we prefer the comparative simplicity of the control of density, directly, as earlier, or through explicit rule constraints, which already results in rich evolutionary dynamics. When we do study resource species in a forthcoming article, it would be advantageous for the study of open-ended evolution to allow a large variety of resources and, in particular, by-products of birth or survival processes in the form of inanimate (no-birth) species that can potentially be employed as new resources. This clearly goes beyond the scope of the current work.

We have explored an important contrast between two forms of evolutionary dynamics: “neutral,” where decisions for inherited genomes are made as neutrally as possible, and “biased,” where an explicit bias is built into the inheritance process (in the simplest case, a preference for more 1 bits in the genome). Evolutionary bias appears to limit open-endedness; as might be expected, bias pulls evolutionary dynamics toward a “genetic attractor” that typically restricts ongoing evolution in the population following optimization. For more complex rule spaces, that is, for agents with higher sensory resolution, we find that a complete takeover of the population by the genetic attractor may be avoided, but the population still simplifies, typically to domains containing genomes near to the genetic attractor.

Notwithstanding the ultimately negative impact of evolutionary bias on open-endedness, the presence of evolutionary bias generates a strongly structured spatial dynamic during the evolutionary transient toward the genetic attractor. Domains emerge, and instead of being relatively static, with randomly moving boundaries, as is typical in neutral cases, we observe domains with very dynamic boundaries, reminiscent of reaction-diffusion systems. This is of course by no means coincidental, as the complex local dynamics of proliferation and survival, even when fully deterministic, creates a rather isotropic pseudo-random spread of genes, as in a replication-diffusion system (Bauer et al., 1989; McCaskill & Bauer, 1993). Such systems have been studied in a range of modeling platforms, from cellular automata (Boerlijst & Hogeweg, 1991) to systolic bit string processor arrays (McCaskill, 1994, 1997) and probabilistic simplex models (McCaskill et al., 2001). In the competition between genetic species, domains expand rapidly with a moving (locally) circular front to take over other domains. These dynamical features are not so apparent in the static pictures presented here but are very striking in the simulations, available online in Packard and McCaskill (2018).

It is tempting to leverage the dynamical complexity shown by evolutionary bias in architecting nonbiased evolutionary dynamics that could produce the same interesting dynamics. For instance, the genetic attractor could be changed externally, for example, Genelife imposes an exogenous diurnal cycle that changes the genetic attractor from all 1s to all 0s every 500 time steps. This would certainly keep up the interesting spatial dynamics, but not endogenously. Instead of having an exogenous genetic attractor imposed, we would like to see a genetic attractor emerge and even compete with other genetic attractors.

Clearly this is possible in principle with the population-dependent selection already present in Genelife: The takeover of a population by a genetic variant can lead to a changed effective environment and hence a new genetic optimum sequence for invading species. In the standard options of Genelife employed here, a majority in neighboring genes determines the local rules. This means that genes encoding higher-s number rules need to act more cooperatively with neighbors (more of them need to carry the rule) to have these rules implemented than genes for low-s number rules.

We also know that Genelife can support selection that leaves the genomes not subject to any complete ordering (not well ordered) so that no single species has an optimum genome sequence. An example already reported (McCaskill & Packard, 2019) is the famous three-trait game known as scissors-paper-stone, with scissors > paper > stone > scissors, or its extension to the four-trait game that we have explored with Genelife scissors-paper-stone-well in two variants (scissors-stone-well-paper: each winning over the single, or in the second version, respectively, the 1-1-2-2, species cyclically to its left). In this case, in common with predator–prey scenarios, the evolution can maintain all four or subsets of two cooperating species, and it does so in Genelife by generating interesting spatiodynamic patterns in the form of populations of colliding gliders. In general, if the patch size of game-playing subpopulations is large, the settling time for such multispecies attractors becomes larger. These interesting possibilities suggest genetic encoding for evolutionary exploration of them as a path to open-endedness—another topic for future work.

We have explored different levels of sensory complexity through the expansion of the genetic specification of local rules through symmetry breaking. Symmetry breaking inevitably expands the size of the space of possible rules Genelife’s evolution will explore. Generally, we find that more complex sensory mechanisms for agents lead to more complex evolution. Evolutionary activity is usually increased, and the complexity of the dynamics is increased. When domains are formed, they tend to have increased geometrical complexity with different local patterns. There often appears to be ways for static domains to be gradually eaten away by a dynamic interloper. This all results in longer, more complex transients, as well as more complex spatial states during ongoing evolutionary dynamics. Note that there is a cost also in Genelife to increasingly complex sensor distinctions of local neighborhoods. It is that all of these distinctions when encoded in the genome lead to longer stretches of genetic information that must be inherited for function preservation, and as we know from extensive work on population dynamics, for example, with the quasi-species model (Eigen et al., 1989), for a given mutation rate and selective advantage, there is a maximum length of information that can be preserved by evolution. The evolutionary burden of increased sensory distinction is one reason that an evolving population is not always dominated by agents with the most complex sensory apparatuses (compare Figure 17(b) and 17(c)).

Our investigation of density regulation aimed to constrain the system to interesting spatial pattern–generating domains by penalizing through local density feedback the crowded, completely connected configurations that result from unbridled proliferation. We investigated this for the rule spaces in semi-totalistic, quarter-totalistic, and gene-determined symmetries. We found that density regulation indeed succeeded in opening up spatial dynamics for rich local pattern formation and hence additional evolutionary variety. New activity was observed to increase up to an order of magnitude over values observed in non-density-regulated analogs.

We have used evolutionary activity (new activity, in particular) as an indicator of open-endedness. As we have employed them, evolutionary activity measures can be defined for different kinds of information in the system. We have investigated the two most fundamental ones in Genelife, namely, genetic and spatial pattern information. Both of these measures exhibit that Genelife can support ongoing innovative evolution—in genes and patterns. It is, however, difficult to distinguish ongoing variations on a common theme, itself a kind of neutral evolution, from more fundamental and transformative innovations.

Although there are further opportunities to refine these measures, especially for new activity, for example, by using absorbing boundaries in the activity statistics to accurately quantify the flux from new innovations to successful, longer-lived structures in the population, as already discussed, it appears that the major limitation to our use of activities to capture the open-endedness of evolution in Genelife is the restriction to genetic and spatial information, rather than examining the activity of functional information. One way to capture functional information is by examining the utilization of specific rules, or classes of rules, and collecting activity statistics on this usage. Because these local rules are also encoded by the genome in Genelife, this then amounts to examining how evolution explores different parts of the genome, making use of innovative combinations of rules.

Genelife provides a platform with remarkable success in obtaining different versions of open-endedness. We have pursued an increase in sensory resolution, in combination with density regulation, to arrive at a model class that displays robust open-endedness.

Past discussions of open-endedness in Artificial Life models (Packard et al., 2019; Taylor et al., 2016) have highlighted a multiplicity of ways that systems can display open-endedness. The Tokyo categories refer to “ongoing generation of …(1) interesting new kinds of entities and interactions, (2) evolution of evolvability, (3) major transitions, and (4) semantic evolution” (Packard et al., 2019). The present study, focused on measurements of new evolutionary activity as indicators of open-endedness, primarily probes open-endedness in the first Tokyo category. In particular, we use gene activity to measure ongoing generation of interesting new entities and, because of the strong coupling given by the local dynamical rules, new interactions. However, the spatial dynamics of the evolving population of GoL agents often seemed to reach statistical equilibrium, with an observed stasis in qualitative nature of the spatial dynamics; that is, no striking innovation was visible in the dynamics, notwithstanding the ongoing genetic innovation evident in positive gene activity measurements. This observation exposes the weakness of using the word “interesting” in the first Tokyo category of open-endedness and led us to examine pattern activity (activity measuring local pattern innovation) as an alternative. As expected from qualitative observations, the lack of visible innovation in spatial dynamics was reflected in pattern activity going to zero over time (compare Figure 23). Using density regulation, we found one case in which both new gene activity and new pattern activity appeared to have ongoing positive values (compare Figure 24). The Genelife simulation (Packard & McCaskill, 2018) enables visualization of local pattern innovation, as seen in Figures 24(c) and 24(d). Future work on this candidate for open-endedness should press beyond the timescales of the present study (10⁴ time steps, ∼10⁹ reproduction events).

The present work with Genelife touches on Tokyo category 2 open-endedness, the evolution of evolvability. In our study of sensory complexity, we created a hierarchy of agents that have increasing levels of sensory complexity, reflected in an increase in the number of bits in their genome affecting the local dynamics (low sensory complexity requires few bits; higher sensory complexity requires more bits). One might expect that the availability of increasing agent sensory complexity could create an evolutionary path with agents evolving more complex senses having access to even more evolutionary possibilities, thus leading to evolution of evolvability. However, this intuition is not borne out in the observed results, which reveals a central puzzle of the current work, as discussed earlier. Figure 17 shows that evolving populations of agents with heterogeneous mixtures of sensory complexity may lead to dominance of agents with the highest sensory complexity (2D symmetry) but often may lead instead to dominance of agents with low sensory complexity (sum symmetry).

An early attempt to classify open-ended evolutionary dynamics (Bedau et al., 1998) included measurement of diversity and cumulative activity, as well as the new activity that is the focus of this work. In that work, an unbounded increase in diversity was included in the statistical signature for open-endedness. To explore this signature with Genelife, scaling of diversity and activity with lattice size (compare studies of indefinite scalability; Ackley & Small, 2014; Channon, 2019) and measurement over longer timescales should be explored.

In this work, we have restricted our attention to systematically exploring rule spaces with a hierarchy of symmetry breaking; Genelife has access to many other rule spaces that might also be fruitful for exploring open-endedness.

Genelife’s open-endedness has not, however, been shown here to generate new hierarchical structure of the kind well recognized in biology and captured in the Tokyo category 3 open- endedness. It would indeed be remarkable if, in the limited lattice size, extremely limited genome length, most limited neighborhood discrete simulations addressed so far with Genelife, more open-endedness were possible. The limitations of local discreteness can be addressed by expansions in these three parameters, perhaps going as far in this as the model Smooth Life (Rafler, 2011). We suspect, however, that the kinds of nonlocal order and rearrangements in physical systems exhibiting equilibration, as in the containment induced by spherical closure and the morphological transitions in bilayer lipid membrane systems, are important to ensuring an expanding space of options for hierarchical evolution.

The authors thank the reviewers for numerous helpful suggestions.