1 Introduction

“Since the study of life began, many have asked: is it unique in the universe, or are there other interesting forms of life elsewhere? Before we can answer that question, we should ask others: What makes life special? If we happen across another system with life-like behavior, how would we be able to recognize it?” – David Eppstein [14].

In 1970, the late english mathematician John Horton Conway have introduced the most popular Game of Life (or, we can simply call Life) cellular automata which can able to generate many complex phenomenon and computational universality [3, 6, 12, 17]. Hereafter, the Life research community have continuously asked the fundamental questions – what makes Life too special? In this direction, the early work of Torre and Mártin [13] have displayed the gallery of extended Life games (rules) and their connection with patterns of nature, like ‘labyrinths’, ‘island with borders’, ‘ferromagnetic domains’, ‘solid-liquid mixture’. On the other hand, Life community have identified many Life-like games which shows dynamics most similar to Life [2, 14, 18,19,20]. In terms of density, some Life-like games are associated with low-density (like, Eight Life, High Life, Honey Life, Pedestrian Life etc.) and some games are with high-density (like Dry Life, Drigh Life etc.). In a recent work, Peña and Sayama [26] have explored the quantitative comparison of Life-like games using the notion of information gain complexity or conditional entropy. According to [26], Life shows a consistent amount of complexity throughout its evolution in comparison with low and high density Life-like games. A good survey about Life-like games and their capability of generating gliders, bombers, rakes, small oscillators, still life, spaceships and other patters can found in [14].

Fig. 1.
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Space-time diagrams of Life for changing value of \(\alpha \) after transient time steps.

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On the other hand, the question of randomness (or noise) is fundamental to understand the laws of real life [32]. In this direction, cellular automata research community have also explored the relationship between Life rules and real life (natural systems). Specifically, the fundamental question – ‘how real the Life rules are?’. On other words, the subject of study is to understand the effect of perturbation (or noise) in Life [1, 10, 15, 16, 24, 27, 30]. In the first attempt, Schulman and Seiden [30] have introduced the noise with the notion of stochastic component ‘temperature’ in Life. However, in the construction of [30], the transition probabilities are dependent on the global density, i.e. the rules are not local. In the same direction, Adachi, Peper and Lee [1] have observed a phase transition in Life (from ‘1/f’ phase to ‘Lorentzian spectrum’ phase) for ‘temperature’ based continuous perturbation. Under fully asynchronous updating scheme, Bersini and Detours [7] have also observed the phase transition in Life. Fatès [15, 16] have asked the important question – ‘does Life resist asynchrony?’ many times. In our early work [27], we have observed the continuous and abrupt change in phase during evolution of Life with probabilistic loss of information and delay (during information sharing between two neighbouring cells) perturbation respectively. In this context, following are the other notable contributions – \(\gamma \)-asynchronous Life [15, 16]; Life with short-term memory [5]; quantum Life [8, 25]; probabilistic Life [4]; temporally stochastic Life [22, 29]. Moreover, during the evolution of stochastic Life rules, Monetti and Albano [23, 24] have observed first-order irreversible phase transition from steady state (non-zero density) to extinct phase (zero density). A good survey about Life and perturbation can found in [21].

In particular, in this present article, our topic of interest is dynamics of Life and Life-like games under \(\alpha \)-asynchronous perturbation. Fatès and Michel [15, 16] have identified a phase transition of Life from an ‘inactive sparse’ phase to ‘labyrinth’ phase for changing value of \(\alpha \) where each cell is updated with \(\alpha \) probability. For evidence, Fig. 1 depicts the dynamics of Life after transient (1000 time steps) steps for (a) synchronous, i.e. \(\alpha = 1.0\); (b) \(\alpha = 0.9\); (c) \(\alpha = 0.2\); (d) \(\alpha = 0.1\) environment. Note that, Life shows labyrinth phase for \(\alpha = 0.1, 0.2\) in Fig. 1. Following the literature, the current study explores the dynamics of low-density (Eight Life, High Life, Honey Life, Pedestrian Life, Flock Life, LowDeath Life etc.) and high-density (Dry Life, Drigh Life etc.) Life - like games under \(\alpha \)-asynchronous perturbation. Hereafter, we compare Life - like games with Life under the effect of \(\alpha \)-asynchronous perturbation. Basically, we are asking the fundamental question again and again – what is there so special about the Life rules? Moreover, this study displays the gallery of special extended-Life games (identified by Torre and Mártin [13]) under \(\alpha \)-asynchronous perturbation. Finally, the study concludes with some random games with different peculiar phase transition dynamics. In this scenario, the next section introduces the Life and \(\alpha \)-asynchronous perturbation.

2 Life, \(\alpha \)-Asynchronism and Experimental Setup

Traditionally, Life evolves in a regular subset of \(\mathbb {Z}^{2}\) with state set S = {0, 1} (on other words, S = {dead, alive}). In this experimental study, we consider configuration of finite squares with \(N \times N\) cells under periodic boundary condition, i.e. \(\mathbb {Z}/N\mathbb {Z}\). Life shows dramatic change in dynamics for other (specifically, open) boundary condition [9]. Life follows Moore neighbourhood dependency, i.e. self and eight nearest neighbours. Following are the local transition rule of Life.

  • Birth rule: A dead cell with exactly three live neighbours evolves to live state. Here, we denote by B3 where B represents birth; and

  • Survival rule: A live cell with exactly two or three live neighbours will remain alive. We denote by S23 where S depicts survival.

To sum up, Life rule can be written as B3/S23. In general, Bp/Sq is traditionally used for the naming of these 2-D outer-totalistic games where p and q are the subsets that can contains digits from 0 to 8 to represent the number of live neighbours. Note that, we follow this naming approach to represent Life-like games and extended Life games.

Traditionally, like other synchronous systems Life also assume a global clock that forces the cells to get updated simultaneously. However, the assumption of global clock is not very natural. In this study, for Life and Life-like games we follow \(\alpha \)-asynchronous updating scheme [11] where each cell is updated with probability \(\alpha \) at each time step, on other words, each cell is left unchanged with probability \(1 - \alpha \) at each time step. Note that, \(\alpha = 1.0\) depicts the traditional synchronous dynamics. On the other hand, for \(\alpha = 0.01\), the system shows almost same dynamics as fully asynchronous perturbationFootnote 1.

Fig. 2.
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Space-time diagrams of low density Life-like games after transient time steps.

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Next, to understand the effect of \(\alpha \)-asynchronous perturbation, we consider the following qualitative and quantitative experimental setup.

  • Firstly, in the qualitative experiment, we evolve Life and Life-like games starting with \(50 \times 50\) (\(N \times N\)) random initial configuration of fixed density (no of alive cell). During the evolution of the system, we need to observe the space-time digram of Life and Life-like games. In this experimental approach, we can able to provide a visual comparison between Life and Life-like games under \(\alpha \)-asynchronous perturbation.

  • Secondly, quantitative experiment follows the well-known approach of [16, 27]. Here, in the quantitative (formal) experiment, we observe the density (say, \(d_x\)) of configuration (say, x), i.e. \(d_x = \frac{x_{alive}}{\vert x \vert }\) where \(x_{alive}\) counts the number of alive cell in the configuration x; and \(\vert x \vert \) is the size of the configuration space (in this study, \(50 \times 50\)). In this quantitative experiment, we evolve the Life or Life-like games for a transient time period (say, \(t_{transient}\)) starting from an initial configuration with density \(d_{ini}\). Here, in this study, \(t_{transient} = 1000\) time steps. Next, we calculate the average density of Life or Life-like games for a sampling period of time (say, \(t_{sampling}\)). We denote this average density by \(d_{avg}\). In this study, \(t_{sampling} = 100\) time steps. To sum up, \(d_{avg}(d_{ini},\alpha )\) depicts the steady-state density under the proposed \(\alpha \)-asynchronous perturbation environment. In general, we consider \(d_{ini} = 0.5\).

Following this experimental setup, we next explore – (a) low density Life-like games, like, Eight Life, Pedestrian Life, Flock Life [14, 18, 19, 26]; (b) high density Life-like games, like, Dry Life, Drigh Life [14, 19, 26]; (c) extended-Life games capable to generate natural patterns like, ‘labyrinths’, ‘island with borders’, ‘solid-liquid mixture’ [13]; (d) random games capable to show different kind of phase transitions; under \(\alpha \)-asynchronous perturbation.

3 Life-Like Games and Perturbation

3.1 Low-Density Life-Like Games and Perturbation

In this section, we explore the effect of \(\alpha \)-asynchronism perturbation on low density Life-like games, specifically – Flock Life (B3/S12), \(2 \times 2\) Life (B36/S125), High Life (B36/S23), Pedestrian Life (B38/S23), Eight Life (B3/S238), Honey Life (B38/S238), LowDeath Life (B368/S238).

Let us first focus on the qualitative experimental results. Here, Fig. 2 depicts the space-time diagrams after transient time steps for Flock Life, Honey Life, \(2 \times 2\) Life, Eight Life and High Life starting with \(\alpha = 1.0\) (synchronous) to \(\alpha = 0.1\). According to Fig. 2, Flock Life and \(2 \times 2\) Life show a solid resistance against perturbation, i.e. inactive sparse phase remains same with the changing value of \(\alpha \in [0,1]\). For quantitative evidence, see the steady-state density profile as a function of \(\alpha \) perturbation in Fig. 3.

On the other hand, Honey Life, Eight Life, and Pedestrian Life depict a phase transition from inactive sparse phase to labyrinth phase for changing value of \(\alpha \). For evidence, Fig. 2 depicts the inactive sparse phase for \(\alpha = 1.0(syn), 0.9, 0.5\) and labyrinth phase for vary highFootnote 2 perturbation rate (\(\alpha = 0.1\)) considering Honey Life and Eight Life. Note that, Life also depicts the same dynamics. Recall that, Life also shows a phase transition from inactive sparse phase to labyrinth phase for changing value of \(\alpha \), see Fig. 1. Specifically, according to [15, 16], Life displays a second-order phase transition which belongs to the directed percolation universality class.

In a different dynamics, High Life and LowDeath Life show a phase transition from inactive sparse phase to active dense phase for increasing rate of perturbation, for evidence, see the space-time diagram of High Life in Fig. 2. Figure 3 depicts the quantitative experimental results for all the low-density Life-like games where High Life, Pedestrian Life, Eight Life, Honey Life, LowDeath Life show the interesting phase transition dynamics.

Fig. 3.
figure 3

The plot depicts the density parameter profile as a function of alpha parameter for lower density Life-like games.

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3.2 High-Density Life-Like Games and Perturbation

Here, we discuss the effect of the \(\alpha \)-asynchronous perturbation on high-density Life-like games, specifically – Dry Life (B37/S23), Drigh Life (B367/S23), B356/S23, B356/S238, B3568/S23, B3568/S238, B3578/S23, B3578/S237, B3578 /S238.

Fig. 4.
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The plot depicts the density parameter profile as a function of alpha parameter for higher density Life-like games.

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As a qualitative result, Fig. 5 shows the samples of space-time diagrams after transient time steps for Dry Life (B37/S23), Drigh Life (B367/S23), B356/S238 and B3578/S238 starting from \(\alpha = 1.0\) (synchronous) to \(\alpha = 0.1\). On the other hand, Fig. 4 depicts the quantitative experimental steady-state density profiles as a function of alpha perturbation for all high-density Life-like games. According to Fig. 4, all the high-density Life-like games show the phase transition dynamics with the effect of alpha-perturbation. Therefore, phase transition is a general dynamics for these games with the effect of noise.

Here, only Dry Life (B37/S23) shows (out of high-density Life-like games) a phase transition from inactive-sparse phase to labyrinth phase for changing value of \(\alpha \) perturbation, for evidence see Fig. 5. Recall that, Life also shows the similar phase change towards labyrinth phase with the effect of \(\alpha \) perturbation [15, 16]. However, the rest high-density Life-like games depict a phase transition from inactive sparse phase to active dense phase for changing value of \(\alpha \) perturbation, see Fig. 5 for evidence. Therefore, only Dry Life follows the dynamics of Life. In this context, note that, Life also shows phase transition from inactive sparse phase to active dense phase with the effect of probabilistic loos of information perturbation [27].

3.3 Extended Life Games and Perturbation

In a early work, Torre and Mártin [13] have explored all 1296 extendedFootnote 3 Life games and have identified (some of) their capability of generating natural patterns like ‘labyrinths’, ‘island with borders’, ‘ferromagnetic domains’, ‘solid-liquid mixture’. In this section, we specifically explore those special (capable to generate natural patterns) extended Life games under \(\alpha \)-asynchronous environment.

Fig. 5.
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Space-time diagrams of high density Life-like games after transient time steps.

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According to Torre and Mártin [13], games like B3/S1234, B345678/S56, B12/S1234 show labyrinths structure under traditional synchronous environment. These rules show solid resistance against the perturbation. For evidence, Fig. 6 depicts the dynamics of B12/S1234 under \(\alpha \)-asynchronous environment. On the other hand, B123/S1234 shows labyrinths pattern for high perturbation rate, \(\alpha = 0.1, 0.5\), see Fig. 6. However, the game B123/S1234 shows dynamics like ferromagnetic domains under synchronous environment. Note that, games B4/S1234 and B1234567/S56 also show the same dynamics (like B123/S1234). Under synchronous environment, game B234567/S5678 shows phases in combat where different patterns are generated in few region of the lattice starting with small initial density, hereafter, those pattern grows until the complete coverage of the lattice. For \(\alpha \)-asynchronous environment, one can also observe the phases in combat, however, finally, the game converges and shows pattern like ‘large building with some light-on windows during late night’, see Fig. 6 for evidence. Here, B345678/S5678 and B34567/S567 also depicts the same dynamics. According to [13], game B45678/S5678 shows islands with active borders starting with initial density \(d_{ini} = 0.5\) under synchronous environment. However, the size of islands decrease with the decreasing value \(\alpha \). Moreover, B45678/S5678 also depicts ‘large building with some light-on windows during late night’ pattern for high rate of perturbation, for \(\alpha = 0.1\) see Fig. 6. Here also, game B45/S456 shows the similar dynamics. Similarly, under the synchronous environment, B45678/S4567 shows pattern like ‘islands with overlapping lake inside’ starting with initial density \(\alpha = 0.3\). This game shows resistance against small perturbation, for \(\alpha = 0.9\) see Fig. 6. However, for high perturbation rate (i.e. \(\alpha = 0.1\)), game B45678/S4567 shows the similar ‘large building with some light-on windows during late night’ dynamics. To sum up, Fig. 6 displays the beautiful gallery of these rules with the effect of perturbation. Note that, this part of study on extended Life games only focuses on the qualitative experiment.

Fig. 6.
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Space-time diagrams of extended Life games after transient time steps.

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4 Discussion

To sum up, this first experiment on Life-like cellular automata with the effect of \(\alpha \)-asynchronous perturbation shows following peculiar behaviour –

  • Low density Flock Life and \(2 \times 2\) Life show a solid resistance against perturbation.

  • Low density games like Honey Life, Eight Life and Pedestrian Life show similarity in dynamics with Life. With the effect of perturbation, these games show phase transition towards labyrinth patterns.

  • Low density games like High Life and LowDeath Life also depict phase transition dynamics, however, for high rate of perturbation, these games show active-dense pattern.

  • All high density Life-like games show phase transition dynamics with the effect of perturbation. However, only Dry Life follows similar dynamics with Life, i.e. labyrinth patterns.

  • Extended Life games with labyrinth patterns also show solid resistance against perturbation. However, most of natural patterns of extended Life games show stability with the effect of \(\alpha \)-asynchronous perturbation.

Fig. 7.
figure 7

The plot depicts the density parameter profile as a function of alpha parameter for random games.

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According to all these evidence, we can say that phase transition is a general trend for Life-like games under \(\alpha \)-asynchronous environment. For more evidence, in a first experiment, we explore 1000 random outer-totalistic games and 137 (out of 1000) games show different kind of phase transition dynamics with the effect of perturbation. Figure 7 displays some of the quantitative experimental results associated with phase transition dynamics. Figure 7(a) shows games which show almost zero density (inactive-sparse phase) for traditional synchronous environment and depicts active dense phase (\(d_{avg} > 0.2\)) after a certain perturbation rate. Here, game B457/S13567 shows very early (low perturbation rate) phase transition for critical value \(\alpha _c \sim 0.9\), on the other hand, game B458/S2567 depicts very late (\(\alpha _c \sim 0.2\)) phase transition. Some games like B456/S1345, B468/S4568 show phase transition for \(\alpha _c \sim 0.5\), see Fig. 7(a) for all these various behaviour.

In other interesting dynamics, game like B3458/S1, B258/S6, B248/S8, B26/ S56 show a phase transition from active-dense phase to extinct phase (zero-density), see Fig. 7(b). Here also, B3458/S1 shows early phase transition (\(\alpha _c \sim 0.9\)) and games like B258/S6, B248/S8, B26/S56 depict late phase transition (\(\alpha _c \sim 0.2\)). In this context, note that, during the evolution of stochastic Life rules, Monetti and Albano [23, 24] have also observed first-order irreversible phase transition from steady state (non-zero density) to extinct phase (zero density). Lastly, games like B356/S3578, B3468/S367 show two phase transitions – firstly, form active-dense to almost zero density; secondly, from almost zero density to again active-dense for very high rate of perturbation, see Fig. 7(b) for all these peculiar dynamics. Therefore, phase transition dynamics with the effect of perturbation is also visible for many Bp/Sq 2-D outer-totalistic rules (along with Life). However, the connection between a random game (the set p and q) and phase transition is still open to us.