# Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty

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## Abstract

**:**

## 1. Introduction

## 2. Physical Universality and Local Physical Universality

**Definition**

**1.**

**Physical universality.**

**Definition**

**2.**

**DLocal Physical Universality**.

## 3. State-Dependent Dynamical Systems

## 4. Open-Ended Evolution

**Definition**

**3.**

**Unbounded evolution (UE)**: A system U composed of at least two interacting subsystems $\mathcal{X},\mathcal{Y}\subseteq U$ interacting according to an arbitrary function $\mathcal{F}$, exhibits unbounded evolution if there exists a recurrence time ${t}_{r}$ such that the state trajectory ${x}_{1}\stackrel{\mathcal{F}}{\to}{x}_{2}\stackrel{\mathcal{F}}{\to}{x}_{3}\stackrel{\mathcal{F}}{\to}\dots \stackrel{\mathcal{F}}{\to}{x}_{r}$ for ${x}_{n}\in \mathcal{X}$ is non-repeating for ${t}_{r}>{t}_{P}$ and ${t}_{P}$ is the Poincaré recurrence time ${t}_{p}=|{\mathrm{\Sigma}}^{\widehat{\mathcal{X}}}|$ of a finite region $\widehat{\mathcal{X}}$, where $\widehat{\mathcal{X}}$ is an isolated equivalent of $\mathcal{X}$ without the interaction with $\mathcal{Y}$ under $\mathcal{F}$.

**Definition**

**4.**

**Innovation (INN)**: A system U composed of at least two interacting subsystems $\mathcal{X},\mathcal{Y}\subseteq U$ interacting according to an arbitrary function $\mathcal{F}$, exhibits innovation if there exists a recurrence time ${t}_{r}$ such that the state trajectory ${x}_{1}\stackrel{\mathcal{F}}{\to}{x}_{2}\stackrel{\mathcal{F}}{\to}{x}_{3}\stackrel{\mathcal{F}}{\to}\dots \stackrel{\mathcal{F}}{\to}{x}_{r}$ for ${x}_{n}\in \mathcal{X}$ is not contained in the set of all possible state-trajectories for a finite region $\widehat{\mathcal{X}}$ in isolation.

## 5. Methodology

**Resource:**The resource $\mathcal{P}$ is a CA of finite width w evolving according to constant function $f={r}_{P}$, where ${r}_{P}\in {R}_{P}$ and ${R}_{P}\subseteq ECA$ is a subset of ECA rules (e.g., $\mathcal{P}$ is an ECA, sets ${R}_{p}$ used in this study are described below).

**System:**The system $\mathcal{S}$ is a CA of finite width w evolving according a state-dependent map $f={r}_{S}\left(t\right)$, where ${r}_{S}(t+1)=\mathcal{M}({\gamma}_{S}\left(t\right),{\gamma}_{P}\left(t\right),{r}_{S}\left(t\right))$ and ${\gamma}_{S}\left(t\right)$ and ${\gamma}_{P}\left(t\right)$ are the configuration of $\mathcal{S}$ and $\mathcal{P}$ at time t, respectively, and ${r}_{S}\left(t\right)$ is the rule implemented by the system at time t. As with the resource ${r}_{S}\in {R}_{P}$ and ${R}_{P}\subseteq ECA$ is a subset of ECA rules (here the system implements an update rule drawn from the same set of rules governing the resource to ensure both obey the same “laws”).

**Controller:**In the definition of the system, the controller $\mathcal{M}$ is an arbitrary function mapping the rule of the system at time t to that at $t+1$, such that the rules of $\mathcal{S}$ evolve in time in addition to the configurations. We regard the ‘metarule’ $\mathcal{M}$ as the controller for the interaction between $\mathcal{P}$ and $\mathcal{S}$. For the results presented here $\mathcal{M}$ is defined as follows:

#### 5.1. Classifying ECA Rules by Reversibility

## 6. Results

#### 6.1. Likelihood of Open-Ended Evolution

#### 6.2. Topology of the State-Transition Diagram

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Examples of state-transition diagrams for $w=6$ Elementary Cellular Automata (ECA) with periodic boundary conditions. Shown are the rule table and state-transition diagram for ECA rules 110 (

**a**,

**c**), respectively, and 240 (

**b**,

**d**), respectively. Garden of eden states are highlighted in red. Rule 240 has no garden of eden states and is logically reversible. Each loop in the state transition diagram of rule 240 represents a set $\mathrm{\Gamma}$ of configurations that is locally physically universal (see text), whereas this property does not hold for rule 110.

**Figure 2.**A state-dependent cellular automaton composed of two spatially segregated interacting parts as shown in (

**b**): a system ($\mathcal{S}$) is coupled to a resource (program) ($\mathcal{P}$) through a “metarule” $\mathcal{M}$ that controls the interaction, illustrated abstractly in (

**a**). The metarule is a function that maps the states of the systems S and P to a new update rule for S.

**Figure 3.**The relative size of all ${R}_{p}$ sets (as percentages here) for ECA widths $w=3,4,5,6,7$. Different w are denoted by concentric circles, where colors indicate separate ${R}_{p}$ sets.

**Figure 4.**State transition diagrams for ${R}_{p}=1.0$ for varying w. Node size is weighted by out-degree and colors indicate betweenness centrality (high values are warm, low values cool tones). Each connected component is locally physically universal.

**Figure 5.**State transition diagrams for with $w=5$ with varying ${R}_{p}$. Node size is weighted by out-degree and colors indicate betweenness centrality (high values are warm, low values cool tones).

**Figure 6.**Distributions of recurrence times (relative to the Poincaré recurrence time $\frac{{t}_{r}}{{t}_{P}}$, y-axis) of sampled systems $\mathcal{S}$ with state trajectories that are innovative, meaning they are not reproducible by any static-rule ECA according to Definition 4. Trivial state-trajectories consisting of all 1’s or 0’s were removed. Distributions are shown for all ${R}_{p}$ sets (x-axis) of a given CA widths w (panels

**a**–

**e**). The horizontal line represents where ${t}_{r}={t}_{P}$. State-trajectories above this line are classified as open-ended according to Definition 3.

**Figure 7.**Frequency distribution of the in-degree (${k}_{\mathrm{in}}$, (

**a**,

**c**,

**e**)) and out-degree (${k}_{\mathrm{out}}$, (

**b**,

**d**,

**f**)) for state-transition diagrams for varying ${R}_{p}$ for widths $w=5$ (

**a**,

**b**), $w=6$ (

**c**,

**d**) and $w=7$ (

**e**,

**f**). State-transition diagrams with sampled trajectories exhibiting open-ended dynamics are shown in red, while those where no open-ended cases were confirmed are shown on a gray scale.

**Figure 8.**Shortest path (${l}_{S}$) for state-transition diagrams for varying ${R}_{p}$ for widths $w=5$ (

**a**), $w=6$ (

**b**) and $w=7$ (

**c**). State-transition diagrams with sampled trajectories exhibiting open-ended dynamics are shown in red, while those where no open-ended cases were confirmed are shown on a grey scale.

**Table 1.**Statistics of the topology for state-transition diagrams for $\mathcal{S}$ of widths $w=5,6,7$. Bold highlighting indicates state-transition graphs that permit OEE or are that have a largest connected component that is locally universal. Included are: ${N}_{\mathrm{rules}}$, the number of rules in the class ${R}_{p}$; ${N}_{\mathrm{edges}}$ the number of edges in the state-transition graph; ${N}_{\mathrm{CC}}$, the number of connected components of the graph; ${S}_{\mathrm{LCC}}$ the size of the largest connected component (LCC); $\langle {k}_{\mathrm{in}}\rangle $, the mean in-degree of states (same as $\langle {k}_{\mathrm{out}}\rangle $); $\langle {l}_{S}\rangle $, the average shortest path length between directed pairs of states in the LCC; whether or not OEE cases were found (Y or N, respectively), and whether or not the largest connected component is locally physically universal (Y or N, respectively).

${\mathit{R}}_{\mathit{p}}$, w | ${\mathit{N}}_{\mathbf{rules}}$ | ${\mathit{N}}_{\mathbf{edges}}$ | ${\mathit{N}}_{\mathbf{CC}}$ | ${\mathit{S}}_{\mathbf{LCC}}$ | $\langle {\mathit{k}}_{\mathbf{in}}\rangle $ | $\langle {\mathit{l}}_{\mathit{S}}\rangle $ | OEE | ${\mathit{LU}}_{\mathbf{LCC}}$ |
---|---|---|---|---|---|---|---|---|

${R}_{p}=0$, $w=5$ | 2 | 64 | 1 | 100% | 13 | 0.65 | N | N |

${R}_{p}=0.2$, $w=5$ | 8 | 204 | 1 | 100% | 41 | 1.35 | N | N |

${R}_{p}=0.3$, $w=5$ | 30 | 524 | 1 | 100% | 105 | 1.42 | Y | N |

${R}_{p}=0.4$, $w=5$ | 8 | 164 | 1 | 100% | 33 | 0.94 | N | N |

${R}_{p}=0.5$, $w=5$ | 66 | 644 | 1 | 100% | 129 | 1.3 | Y | N |

${R}_{p}=0.7$, $w=5$ | 106 | 624 | 1 | 100% | 125 | 1.32 | Y | N |

${R}_{p}=0.8$, $w=5$ | 20 | 344 | 1 | 100% | 69 | 1.65 | Y | N |

${R}_{p}=1.0$, $w=5$ | 16 | 444 | 2 | 94% | 89 | 1.48 | Y | Y |

${R}_{p}=0$, $w=6$ | 2 | 128 | 1 | 100% | 21 | 0.66 | N | N |

${R}_{p}=0.2$, $w=6$ | 8 | 428 | 1 | 100% | 71 | 1.41 | N | N |

${R}_{p}=0.3$, $w=6$ | 36 | 1368 | 1 | 100% | 228 | 1.52 | N | N |

${R}_{p}=0.4$, $w=6$ | 8 | 420 | 1 | 100% | 70 | 1.72 | N | N |

${R}_{p}=0.5$, $w=6$ | 72 | 1848 | 1 | 100% | 308 | 1.48 | N | N |

${R}_{p}=0.6$, $w=6$ | 72 | 1548 | 1 | 100% | 258 | 1.57 | Y | N |

${R}_{p}=0.7$, $w=6$ | 36 | 1088 | 1 | 100% | 181 | 1.76 | N | N |

${R}_{p}=0.8$, $w=6$ | 8 | 428 | 2 | 97% | 71 | 2.34 | N | N |

${R}_{p}=0.9$, $w=6$ | 8 | 428 | 3 | 84% | 71 | 2.46 | N | Y |

${R}_{p}=1.0$, $w=6$ | 6 | 368 | 8 | 18% | 61 | 1.58 | N | Y |

${R}_{p}=0$, $w=7$ | 2 | 256 | 1 | 100% | 36 | 0.66 | N | N |

${R}_{p}=0.1$, $w=7$ | 8 | 900 | 1 | 100% | 129 | 1.72 | N | N |

${R}_{p}=0.2$, $w=7$ | 28 | 2524 | 1 | 100% | 361 | 2.35 | N | N |

${R}_{p}=0.3$, $w=7$ | 8 | 956 | 1 | 100% | 137 | 1.85 | N | N |

${R}_{p}=0.4$, $w=7$ | 40 | 3280 | 1 | 100% | 469 | 2.14 | N | N |

${R}_{p}=0.5$, $w=7$ | 26 | 2440 | 1 | 100% | 349 | 2.25 | Y | N |

${R}_{p}=0.6$, $w=7$ | 100 | 4960 | 1 | 100% | 709 | 1.87 | Y | N |

${R}_{p}=0.7$, $w=7$ | 20 | 1964 | 2 | 98% | 281 | 2.26 | N | Y |

${R}_{p}=0.8$, $w=7$ | 8 | 900 | 2 | 98% | 129 | 3.11 | Y | N |

${R}_{p}=1.0$, $w=7$ | 16 | 1908 | 2 | 98% | 273 | 2.43 | Y | Y |

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**MDPI and ACS Style**

Adams, A.M.; Berner, A.; Davies, P.C.W.; Walker, S.I.
Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty. *Entropy* **2017**, *19*, 461.
https://doi.org/10.3390/e19090461

**AMA Style**

Adams AM, Berner A, Davies PCW, Walker SI.
Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty. *Entropy*. 2017; 19(9):461.
https://doi.org/10.3390/e19090461

**Chicago/Turabian Style**

Adams, Alyssa M., Angelica Berner, Paul C. W. Davies, and Sara I. Walker.
2017. "Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty" *Entropy* 19, no. 9: 461.
https://doi.org/10.3390/e19090461