# Quantifying the Autonomy of Structurally Diverse Automata: A Comparison of Candidate Measures

## Abstract

**:**

## 1. Introduction

## 2. Quantitative Measures Related to Agency, Autonomy, and Intelligence

#### 2.1. Structural and Graph-Theoretical Measures

#### 2.2. Information Theoretical Measures

#### 2.3. Causal Measures

#### 2.4. Dynamical Measures

## 3. Evolution Simulation

#### 3.1. Markov Brains (MBs)

#### 3.2. “PathFollow” Environment

#### 3.3. Data Analysis

## 4. Results

#### 4.1. Evolved Network Structures

#### 4.2. Information Theoretical Analysis

#### 4.3. Causal Analysis

#### 4.4. Dynamical Analysis

## 5. Discussion

#### 5.1. Scope and Limitations

#### 5.2. Related Work

#### 5.3. Memory and Autonomy

#### 5.4. Correlation, Causation, and Internal Structure

#### 5.5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

**Figure A2.**Additional evaluated quantities. (

**A**) Structural measures: cH denotes the number of connected hidden units; len_LWCC the length of the largest weakly connected component. If len_LWCC is smaller than the number of connected units (Figure 4) the MB is constituted of two or more independent modules. (

**B**) Information-theoretical measures: shown are the system entropy H (Equation (2)), ${I}_{pred}$ of the hidden and motor units, without the sensors, and ${C}_{TSE}$ (Equation (10)). (

**C**) Dynamical measures: shown are the normalized Lempel-Ziv complexity (nLZ) reshaped along the time axis, applied to the MBs’ recorded activity (nLZ_time) and the MBs’ transients upon perturbation with fixed sensors (nLZ_tr_time), as well as the maximum transient lengths upon perturbation (maxTL).

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**Figure 1.**Simulated evolution experiment. (

**A**) Example connectome of a Markov Brain (MB) evolved in condition A2 (fitness = 0.92, completion = 1.0, generation = 150,000). The MB has four connected sensors (red), four hidden units (green), and three motor units (blue). Evolutionary optimization determines both the input-output function of each individual node (here binary and deterministic) and the MB connectivity. (

**B**) One of the four paths used in the “PathFollow” environment. Green: start location; yellow: left turn symbols; orange: right turn symbols; and red: goal.

**Figure 2.**Fitness evolution and distribution across task conditions. (

**A**) Fitness evolution across number of generations. Shaded area indicates 95% confidence interval. (

**B**) Distribution of fitness values (left) and percentage of path completion (right) in the final generation. Black triangles indicate mean. Perfect completion was achieved by 50/50 MBs in NA, 31/50 in A2, and 16/50 in A4.

**Figure 3.**Example connectomes of two A2 MBs with perfect completion, but feed-forward or fully recurrent connectivity, respectively. (

**A**) MB with only feed-forward connections between units, although nodes B and C have self-loops. Thus, the length of the LSCC is one for this MB. (

**B**) MB with recurrent connections between all hidden units and largest possible LSCC length of four hidden units.

**Figure 4.**Structural analysis. (

**A**) Stacked histogram of the LSCC length for the three task conditions. While most MBs in the NA condition are feed-forward (len_LSCC = 1), both feed-forward and recurrent architectures evolved in all three task conditions. (

**B**) Distributions of the number of connected nodes (cN), average degree centrality, average betweenness centrality, and flow hierarchy are shown across task conditions and color-coded according to the length of their LSCC. MBs evolved in A2 and A4 were larger than those in NA by approximately two nodes. The other graph-theoretical measures show little difference between task conditions. As the flow hierarchy depends on cyclical connectivity, lower values correspond to MBs with larger LSCCs. Please note that throughout, axis labels correspond to variable names assigned to the various measures in the accompanying autonomy toolbox.

**Figure 5.**Information-theoretical analysis. The complimentary measures of autonomy proposed in [1], ${A}_{4}$ and ${I}_{pred}$, as well as ${I}_{SMMI}$ identify significant differences across task conditions (top row). By contrast, the information closure measures, $NTI{C}_{4}$ and ${J}_{t}$ (here “IC”) (bottom row) do not differ much between conditions. The multi-information ($MI$) is higher for A4, than the other two conditions, with higher values for MBs with len_LSCC $>1$.

**Figure 6.**Causal analysis. The top row shows the causal version of the autonomy measures proposed in [1], ${\widehat{A}}_{4}$ and $EI({V}_{t},{V}_{t-1})$, as well as $\langle \sum \phi \rangle $ evaluated for the whole MB including sensors and motors. Note however, that here ${\widehat{A}}_{4}$ (“A_4c”) and $EI({V}_{t},{V}_{t-1})$ are based on a maximum entropy distribution of input states rather than the marginal observed distribution proposed in [1]. For all three measures, the NA condition had lower values than A2 and A4. The bottom row shows $\langle {\overline{\alpha}}_{c}(O\prec M)\rangle $, the relative contribution of the hidden units (O) to the actual causes of the agent’s motor states (“alpha_ratio_hidden” in the figure), together with $\langle {\mathsf{\Phi}}^{max}\rangle $ and ${\langle \sum \phi \rangle}_{MC}$ values of the major complex (the maximally integrated subset of hidden units). $\langle {\overline{\alpha}}_{c}(O\prec M)\rangle $ values vary substantially within task condition rather than across conditions, which indicates a large variety of behavioral strategies within each task condition. While condition A2 on average has higher values of $\langle {\mathsf{\Phi}}^{max}\rangle $ and ${\langle \sum \phi \rangle}_{MC}$ than NA and A4, these IIT measures are zero by definition for MBs with len_LSCC $<2$ and, in general, depend strongly on implementation.

**Figure 7.**Dynamical analysis. The first panel shows the number of unique transients per task condition while performing the task. The middle two panels show the normalized Lempel-Ziv complexity of the MBs’ recorded activity (nLZ_space) and the MBs’ transients upon perturbation into all possible initial states for fixed sensor inputs (nLZ_tr_space). Notably, the ordering of nLZ for recorded activity patterns (nLZ_space) across conditions is reversed under perturbation (nLZ_tr_space). Average transient length (avTL) is larger for A2 and A4 than NA.

**Figure 8.**Example networks with different amounts of autonomy. (

**A**) The scatter plot of $\langle {\overline{\alpha}}_{c}(O\prec M)\rangle $ (alpha_ratio_hidden) against ${\widehat{A}}_{4}^{S}$, color-coded by the amount of $\langle {\mathsf{\Phi}}^{max}\rangle $ compares three causal measures of autonomy that represent agency, self-determination, and causal closure, respectively. (

**B**) Connectome of A2 MB with high values for three orthogonal measures of autonomy, $\langle {\overline{\alpha}}_{c}(O\prec M)\rangle $, ${\widehat{A}}_{4}^{S}$, and $\langle {\mathsf{\Phi}}^{max}\rangle $. (

**C**) Connectome of NA MB with low ${\widehat{A}}_{4}^{S}$ and $\langle {\mathsf{\Phi}}^{max}\rangle =0$, but high $\langle {\overline{\alpha}}_{c}(O\prec M)\rangle $. (

**D**) Connectome of NA MB with low $\langle {\overline{\alpha}}_{c}(O\prec M)\rangle $, but intermediate ${\widehat{A}}_{4}^{S}$ and $\langle {\mathsf{\Phi}}^{max}\rangle $.

**Table 1.**Agents were evolved under three task conditions. The table highlights the differences between conditions. All other parameters remained the same.

Condition | NA | A2 | A4 |
---|---|---|---|

Number of generations | 50 k | 150 k | 150 k |

Number of turn symbols | 2 | 2 | 4 |

Random turn symbols | No | Yes | Yes |

Number of evaluations per generation | 1 | 10 | 10 |

Number of available sensors | 4 | 4 | 5 |

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Albantakis, L.
Quantifying the Autonomy of Structurally Diverse Automata: A Comparison of Candidate Measures. *Entropy* **2021**, *23*, 1415.
https://doi.org/10.3390/e23111415

**AMA Style**

Albantakis L.
Quantifying the Autonomy of Structurally Diverse Automata: A Comparison of Candidate Measures. *Entropy*. 2021; 23(11):1415.
https://doi.org/10.3390/e23111415

**Chicago/Turabian Style**

Albantakis, Larissa.
2021. "Quantifying the Autonomy of Structurally Diverse Automata: A Comparison of Candidate Measures" *Entropy* 23, no. 11: 1415.
https://doi.org/10.3390/e23111415