# 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).

## References

- Bertschinger, N.; Olbrich, E.; Ay, N.; Jost, J. Autonomy: An information theoretic perspective. Biosystems
**2008**, 91, 331–345. [Google Scholar] [CrossRef] [PubMed] - Boden, M.A. Autonomy: What is it? Biosystems
**2008**, 91, 305–308. [Google Scholar] [CrossRef] - Albantakis, L. A Tale of Two Animats: What Does It Take to Have Goas? Springer: Cham, Switzerland, 2018; pp. 5–15. [Google Scholar] [CrossRef][Green Version]
- Krakauer, D.; Bertschinger, N.; Olbrich, E.; Flack, J.C.; Ay, N. The information theory of individuality. Theory Biosci.
**2020**, 139, 209–223. [Google Scholar] [CrossRef][Green Version] - Vakhrameev, D.; Aguilera, M.; Barandiaran, X.E.; Bedia, M. Measuring Autonomy for Life-Like AI. In Proceedings of the 2020 Conference on Artificial Life, Montréal, QC, Canada, 13–17 July 2020; MIT Press: Cambridge, MA, USA, 2020; pp. 589–591. [Google Scholar] [CrossRef]
- Maturana, H.R.; Varela, F.J. Autopoiesis and Cognition: The Realization of the Living; Boston Studies in the Philosophy and History of Science; Springer: Dordrecht, The Netherlands, 1980. [Google Scholar]
- Tononi, G. On the Irreducibility of Consciousness and Its Relevance to Free Will; Springer New York: New York, NY, USA, 2013; pp. 147–176. [Google Scholar] [CrossRef]
- Marshall, W.; Kim, H.; Walker, S.I.; Tononi, G.; Albantakis, L. How causal analysis can reveal autonomy in models of biological systems. Philos. Trans. Ser. Math. Phys. Eng. Sci.
**2017**, 375, 20160358. [Google Scholar] [CrossRef] - Aguilera, M.; Di Paolo, E. Integrated Information and Autonomy in the Thermodynamic Limit. arXiv
**2018**, arXiv:1805.00393. [Google Scholar] - Farnsworth, K.D. How Organisms Gained Causal Independence and How It Might Be Quantified. Biology
**2018**, 7, 38. [Google Scholar] [CrossRef][Green Version] - Silver, D.; Huang, A.; Maddison, C.J.; Guez, A.; Sifre, L.; Van Den Driessche, G.; Schrittwieser, J.; Antonoglou, I.; Panneershelvam, V.; Lanctot, M.; et al. Mastering the game of Go with deep neural networks and tree search. Nature
**2016**, 529, 484–489. [Google Scholar] [CrossRef] - Moreno, A.; Etxeberria, A.; Umerez, J. The autonomy of biological individuals and artificial models. BioSystems
**2008**, 91, 309–319. [Google Scholar] [CrossRef] - Moreno, A.; Mossio, M. Biological Autonomy. In History, Philosophy and Theory of the Life Sciences; Springer: Dordrecht, The Netherlands, 2015; Volume 12. [Google Scholar] [CrossRef][Green Version]
- Barandiaran, X.; Ruiz-Mirazo, K. Modelling autonomy: Simulating the essence of life and cognition. BioSystems
**2008**, 91, 295–304. [Google Scholar] [CrossRef] - Hintze, A.; Schossau, J.; Bohm, C. The Evolutionary Buffet Method; Springer: Cham, Switzerland, 2019; pp. 17–36. [Google Scholar] [CrossRef]
- Hintze, A.; Edlund, J.A.; Olson, R.S.; Knoester, D.B.; Schossau, J.; Albantakis, L.; Tehrani-Saleh, A.; Kvam, P.; Sheneman, L.; Goldsby, H.; et al. Markov Brains: A Technical Introduction. arXiv
**2017**, arXiv:1709.05601. [Google Scholar] - Rocha, L.M. Syntactic Autonomy: Why There Is No Autonomy without Symbols and How Self-Organizing Systems Might Evolve Them; Annals of the New York Academy of Sciences; John Wiley & Sons, Ltd.: Hoboken, NJ, USA, 2000; Volume 901, pp. 207–223. [Google Scholar] [CrossRef]
- Bertschinger, N.; Olbrich, E. Information and Closure in Systems Theory. In Proceedings of the 7th German Workshop on Artificial Life, Jena, Germany, 26–28 July 2006. [Google Scholar]
- Kirchhoff, M.; Parr, T.; Palacios, E.; Friston, K.; Kiverstein, J. The Markov blankets of life: Autonomy, active inference and the free energy principle. J. R. Soc. Interface
**2018**, 15, 20170792. [Google Scholar] [CrossRef] [PubMed] - Pearl, J. Causality: Models, Reasoning and Inference; Cambridge University Press: Cambridge, UK, 2000; Volume 29. [Google Scholar]
- Friston, K. Life as we know it. J. R. Soc. Interface
**2013**, 10, 20130475. [Google Scholar] [CrossRef][Green Version] - Bruineberg, J.; Dolega, K.; Dewhurst, J.; Baltieri, M. The Emperor’s New Markov Blankets. 2020. Available online: http://philsciarchive.pitt.edu/18467/1/The%20Emperor%27s%20New%20Markov%20Blankets.pdf (accessed on 15 September 2021).
- Kolchinsky, A.; Wolpert, D.H. Semantic information, autonomous agency and non-equilibrium statistical physics. Interface Focus
**2018**, 8, 20180041. [Google Scholar] [CrossRef] [PubMed] - Hagberg, A.A.; Schult, D.A.; Swart, P.J. Exploring Network Structure, Dynamics, and Function using NetworkX. In Proceedings of the 7th Python in Science Conference, Pasadena, CA, USA, 19–24 August 2008; pp. 11–15. [Google Scholar]
- Freeman, L.C. A Set of Measures of Centrality Based on Betweenness. Sociometry
**1977**, 40, 35. [Google Scholar] [CrossRef] - Luo, J.; Magee, C.L. Detecting Evolving Patterns of Self-Organizing Networks by Flow Hierarchy Measurement. Complexity
**2011**, 16, 53–61. [Google Scholar] [CrossRef][Green Version] - Fischer, D.; Mostaghim, S.; Albantakis, L. How swarm size during evolution impacts the behavior, generalizability, and brain complexity of animats performing a spatial navigation task. In Proceedings of the Genetic and Evolutionary Computation Conference on—GECCO 18, Kyoto, Japan, 15–19 July 2018; pp. 77–84. [Google Scholar] [CrossRef][Green Version]
- Walker, S.I.; Davies, P.C.W. The algorithmic origins of life. J. R. Soc. Interface R. Soc.
**2013**, 10, 20120869. [Google Scholar] [CrossRef] - Edlund, J.A.; Chaumont, N.; Hintze, A.; Koch, C.; Tononi, G.; Adami, C. Integrated information increases with fitness in the evolution of animats. PLoS Comput. Biol.
**2011**, 7, e1002236. [Google Scholar] [CrossRef][Green Version] - Albantakis, L.; Hintze, A.; Koch, C.; Adami, C.; Tononi, G. Evolution of Integrated Causal Structures in Animats Exposed to Environments of Increasing Complexity. PLoS Comput. Biol.
**2014**, 10, e1003966. [Google Scholar] [CrossRef] - Beer, R.D.; Williams, P.L. Information processing and dynamics in minimally cognitive agents. Cogn. Sci.
**2015**, 39, 1–38. [Google Scholar] [CrossRef] [PubMed] - Salge, C.; Glackin, C.; Polani, D. Empowerment—An Introduction. arXiv
**2013**, arXiv:cs.AI/1310.1863. [Google Scholar] - Bialek, W.; Nemenman, I.; Tishby, N. Predictability, complexity, and learning. Neural. Comput.
**2001**, 13, 2409–2463. [Google Scholar] [CrossRef] [PubMed] - Schwartz-Ziv, R.; Tishby, N. Opening the Black Box of Deep Neural Networks via Information. arXiv
**2017**, arXiv:1703.00810. [Google Scholar] - Marstaller, L.; Hintze, A.; Adami, C. The evolution of representation in simple cognitive networks. Neural. Comput.
**2013**, 25, 2079–2107. [Google Scholar] [CrossRef] [PubMed][Green Version] - Williams, P.L.; Beer, R.D. Generalized Measures of Information Transfer. arXiv
**2011**, arXiv:1102.1507. [Google Scholar] - Mediano, P.A.; Seth, A.K.; Barrett, A.B. Measuring integrated information: Comparison of candidate measures in theory and simulation. Entropy
**2019**, 21, 17. [Google Scholar] [CrossRef][Green Version] - Krakauer, D.C.; Zanotto, P. Viral individuality and limitations of the life concept. In Protocells: Bridging Nonliving and Living Matter; MIT Press: Cambridge, MA, USA, 2009. [Google Scholar]
- Krakauer, D.; Bertschinger, N.; Olbrich, E.; Ay, N.; Flack, J.C. The Information Theory of Individuality. arXiv
**2014**, arXiv:1412.2447. [Google Scholar] [CrossRef][Green Version] - Schreiber, T. Measuring information transfer. Phys. Rev. Lett.
**2000**, 85, 461–464. [Google Scholar] [CrossRef][Green Version] - Chang, A.Y.C.; Biehl, M.; Yu, Y.; Kanai, R. Information Closure Theory of Consciousness. Front. Psychol.
**2020**, 11, 1504. [Google Scholar] [CrossRef] - Kanwal, M.; Grochow, J.; Ay, N. Comparing Information-Theoretic Measures of Complexity in Boltzmann Machines. Entropy
**2017**, 19, 310. [Google Scholar] [CrossRef] - Oizumi, M.; Tsuchiya, N.; Amari, S.I. A unified framework for information integration based on information geometry. Proc. Natl. Acad. Sci. USA
**2015**, 113, 14817–14822. [Google Scholar] [CrossRef][Green Version] - Tegmark, M. Improved Measures of Integrated Information. PLoS Comput. Biol.
**2016**, 12, e1005123. [Google Scholar] [CrossRef] - Tononi, G.; Sporns, O. Measuring information integration. BMC Neurosci.
**2003**, 4, 1–20. [Google Scholar] [CrossRef] [PubMed][Green Version] - Balduzzi, D.; Tononi, G. Integrated information in discrete dynamical systems: Motivation and theoretical framework. PLoS Comput. Biol.
**2008**, 4, e1000091. [Google Scholar] [CrossRef][Green Version] - Oizumi, M.; Albantakis, L.; Tononi, G. From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0. PLoS Comput. Biol.
**2014**, 10, e1003588. [Google Scholar] [CrossRef] [PubMed][Green Version] - Tononi, G.; Boly, M.; Massimini, M.; Koch, C. Integrated information theory: From consciousness to its physical substrate. Nat. Rev. Neurosci.
**2016**, 17, 450–461. [Google Scholar] [CrossRef] [PubMed] - Barbosa, L.S.; Marshall, W.; Albantakis, L.; Tononi, G. Mechanism Integrated Information. Entropy
**2021**, 23, 362. [Google Scholar] [CrossRef] - McGill, W. Multivariate information transmission. Trans. Ire Prof. Group Inf. Theory
**1954**, 4, 93–111. [Google Scholar] [CrossRef] - Watanabe, S. Information Theoretical Analysis of Multivariate Correlation. IBM J. Res. Dev.
**1960**, 4, 66–82. [Google Scholar] [CrossRef] - Tononi, G.; Sporns, O.; Edelman, G.M. A measure for brain complexity: Relating functional segregation and integration in the nervous system. Proc. Natl. Acad. Sci. USA
**1994**, 91, 5033–5037. [Google Scholar] [CrossRef][Green Version] - Olbrich, E.; Bertschinger, N.; Ay, N.; Jost, J. How should complexity scale with system size? Eur. Phys. J.
**2008**, 63, 407–415. [Google Scholar] [CrossRef][Green Version] - Timme, N.; Alford, W.; Flecker, B.; Beggs, J.M. Synergy, redundancy, and multivariate information measures: An experimentalist’s perspective. J. Comput. Neurosci.
**2014**, 36, 119–140. [Google Scholar] [CrossRef] - Williams, P.L.; Beer, R.D. Nonnegative Decomposition of Multivariate Information. arXiv
**2010**, arXiv:1004.2515. [Google Scholar] - Harder, M.; Salge, C.; Polani, D. Bivariate measure of redundant information. Phys. Rev. -Stat. Nonlinear Soft Matter Phys.
**2013**, 87, 012130. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bertschinger, N.; Rauh, J.; Olbrich, E.; Jost, J.; Ay, N. Quantifying Unique Information. Entropy
**2014**, 16, 2161–2183. [Google Scholar] [CrossRef][Green Version] - Chicharro, D. Quantifying Multivariate Redundancy with Maximum Entropy Decompositions of Mutual Information. arXiv
**2017**, arXiv:1708.03845. [Google Scholar] - Kolchinsky, A. A novel Approach to Multivariate Redundancy and Synergy. arXiv
**2019**, arXiv:1908.08642. [Google Scholar] - Tax, T.; Mediano, P.; Shanahan, M.; Tax, T.M.; Mediano, P.A.; Shanahan, M. The Partial Information Decomposition of Generative Neural Network Models. Entropy
**2017**, 19, 474. [Google Scholar] [CrossRef] - Yu, S.; Wickstrøm, K.; Jenssen, R.; Principe, J.C. Understanding Convolutional Neural Network Training with Information Theory. arXiv
**2018**, arXiv:1804.06537. [Google Scholar] - Mediano, P.A.M.; Rosas, F.; Carhart-Harris, R.L.; Seth, A.K.; Barrett, A.B. Beyond Integrated Information: A Taxonomy of Information Dynamics Phenomena. arXiv
**2019**, arXiv:1909.02297. [Google Scholar] - Ay, N.; Polani, D. Information Flows in Causal Networks. Adv. Complex Syst.
**2008**, 11, 17–41. [Google Scholar] [CrossRef][Green Version] - Hoel, E.P.; Albantakis, L.; Tononi, G. Quantifying causal emergence shows that macro can beat micro. Proc. Natl. Acad. Sci. USA
**2013**, 110, 19790–19795. [Google Scholar] [CrossRef][Green Version] - Cover, T.M.; Thomas, J.A. Elements of Information Theory; Wiley-Interscience: Hoboken, NJ, USA, 2006. [Google Scholar]
- Tononi, G. Integrated information theory. Scholarpedia
**2015**, 10, 4164. [Google Scholar] [CrossRef] - Albantakis, L.; Tononi, G. Causal Composition: Structural Differences among Dynamically Equivalent Systems. Entropy
**2019**, 21, 989. [Google Scholar] [CrossRef][Green Version] - Albantakis, L.; Tononi, G. The Intrinsic Cause-Effect Power of Discrete Dynamical Systems—From Elementary Cellular Automata to Adapting Animats. Entropy
**2015**, 17, 5472–5502. [Google Scholar] [CrossRef][Green Version] - Mayner, W.G.; Marshall, W.; Albantakis, L.; Findlay, G.; Marchman, R.; Tononi, G. PyPhi: A toolbox for integrated information theory. PLoS Comput. Biol.
**2018**, 14, e1006343. [Google Scholar] [CrossRef] - Albantakis, L.; Marshall, W.; Hoel, E.; Tononi, G. What caused what? A quantitative account of actual causation using dynamical causal networks. Entropy
**2019**, 21, 459. [Google Scholar] [CrossRef][Green Version] - Korb, K.B.; Nyberg, E.P.; Hope, L. A new causal power theory. In Causality in the Sciences; Oxford University Press: Oxford, UK, 2011. [Google Scholar] [CrossRef]
- Juel, B.E.; Comolatti, R.; Tononi, G.; Albantakis, L. When is an action caused from within? Quantifying the causal chain leading to actions in simulated agents. arXiv
**2019**, arXiv:1904.02995. [Google Scholar] - Shapley, L.S. Contributions to the Theory of Games, Chapter A Value for n-person Games; Princeton University Press: Princeton, NJ, USA, 1953. [Google Scholar]
- Albantakis, L. Integrated information theory. In Beyond Neural Correlates of Consciousness; Overgaard, M., Mogensen, J., Kirkeby-Hinrup, A., Eds.; Routledge: London, UK, 2020; pp. 87–103. [Google Scholar] [CrossRef]
- Strogatz, S.H.; Dichter, M. Nonlinear Dynamics and Chaos, 2nd ed.; SET with Student Solutions Manual; Studies in Nonlinearity; Avalon Publishing: New York, NY, USA, 2016. [Google Scholar]
- Adamatzky, A.; Martinez, G.J. On generative morphological diversity of elementary cellular automata. Kybernetes
**2010**, 39, 72–82. [Google Scholar] [CrossRef][Green Version] - Lempel, A.; Ziv, J. On the Complexity of Finite Sequences. IEEE Trans. Inf. Theory
**1976**, 22, 75–81. [Google Scholar] [CrossRef] - Zenil, H.; Villarreal-Zapata, E. Asymptotic Behaviour and Ratios of Complexity in Cellular Automata. arXiv
**2013**, arXiv:1304.2816. [Google Scholar] - Gauvrit, N.; Zenil, H.; Tegnér, J. The Information-theoretic and Algorithmic Approach to Human, Animal and Artificial Cognition. In Representation and Reality in Humans, Other Living Organisms and Intelligent Machines; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Zenil, H. Compression-based investigation of the dynamical properties of cellular automata and other systems. arXiv
**2009**, arXiv:0910.4042. [Google Scholar] [CrossRef] - Nilsen, A.S.; Juel, B.E.; Marshall, W.; Storm, J.F. Evaluating Approximations and Heuristic Measures of Integrated Information. Entropy
**2019**, 21, 525. [Google Scholar] [CrossRef] [PubMed][Green Version] - Casali, A.G.; Gosseries, O.; Rosanova, M.; Boly, M.; Sarasso, S.; Casali, K.R.; Casarotto, S.; Bruno, M.A.; Laureys, S.; Tononi, G.; et al. A theoretically based index of consciousness independent of sensory processing and behavior. Sci. Transl. Med.
**2013**, 5, 198ra105. [Google Scholar] [CrossRef] - Bohm, C.; Hintze, A. MABE (Modular Agent Based Evolver): A framework for digital evolution research. In Proceedings of the 14th European Conference on Artificial Life ECAL, Lyon, France, 4–8 September 2017; MIT Press: Cambridge, MA, USA, 2017; pp. 76–83. [Google Scholar] [CrossRef][Green Version]
- Olson, R.S.; Hintze, A.; Dyer, F.C.; Knoester, D.B.; Adami, C. Predator confusion is sufficient to evolve swarming behaviour. J. R. Soc. Interface
**2013**, 10. [Google Scholar] [CrossRef] [PubMed][Green Version] - Fischer, D.; Mostaghim, S.; Albantakis, L. How cognitive and environmental constraints influence the reliability of simulated animats in groups. PLoS ONE
**2020**, 15, e0228879. [Google Scholar] [CrossRef] [PubMed] - Boden, M.A. Autonomy and artificiality. In The Philosophy of Artificial Life; Oxford University Press: Oxford, UK, 1996; pp. 95–108. [Google Scholar]
- Varela, F.; Maturana, H.; Uribe, R. Autopoiesis: The organization of living systems, its characterization and a model. Biosystems
**1974**, 5, 187–196. [Google Scholar] [CrossRef] - Varela, F.J. Principles of Biological Autonomy; North Holland: Amsterdam, The Netherlands, 1979. [Google Scholar]
- Letelier, J.C.; Soto-Andrade, J.; Guíñez Abarzúa, F.; Cornish-Bowden, A.; Luz Cárdenas, M. Organizational invariance and metabolic closure: Analysis in terms of (M,R) systems. J. Theor. Biol.
**2006**, 238, 949–961. [Google Scholar] [CrossRef] - Clark, A. How to Knit Your Own Markov Blanket. In Philosophy and Predictive Processing; Metzinger, T.K., Wiese, W., Eds.; MIND Group: Frankfurt, Germany, 2017. [Google Scholar]
- Rovelli, C. Agency in Physics. arXiv
**2020**, arXiv:2007.05300. [Google Scholar] - Waade, P.T.; Olesen, C.L.; Ito, M.M.; Mathys, C. Consciousness Fluctuates with Surprise: An empirical pre-study for the synthesis of the Free Energy Principle and Integrated Information Theory. PsyArXiv
**2020**. [Google Scholar] [CrossRef] - Friston, K.J.; Wiese, W.; Hobson, J.A. Sentience and the origins of consciousness: From cartesian duality to Markovian monism. Entropy
**2020**, 22, 516. [Google Scholar] [CrossRef] - Safron, A. An Integrated World Modeling Theory (IWMT) of Consciousness: Combining Integrated Information and Global Neuronal Workspace Theories With the Free Energy Principle and Active Inference Framework; Toward Solving the Hard Problem and Characterizing Agentic Causation. Front. Artif. Intell.
**2020**, 3, 30. [Google Scholar] [CrossRef] [PubMed] - Albantakis, L. Review of Sentience and the Origins of Consciousness: From Cartesian Duality to Markovian Monism. 2020. Available online: https://www.consciousnessrealist.com/sentience-and-the-origins-of-consciousness/ (accessed on 15 September 2021).
- Shalizi, C.; Crutchfield, J. Computational mechanics: Pattern and prediction, structure and simplicity. J. Stat. Phys.
**2001**, 104, 817–879. [Google Scholar] [CrossRef] - Marshall, W.; Gomez-Ramirez, J.; Tononi, G. Integrated Information and State Differentiation. Front. Psychol.
**2016**, 7, 926. [Google Scholar] [CrossRef][Green Version] - Lizier, J.; Prokopenko, M.; Zomaya, A. A framework for the local information dynamics of distributed computation in complex systems. In Guided Self-Organization: Inception; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Lizier, J.T. JIDT: An Information-Theoretic Toolkit for Studying the Dynamics of Complex Systems. Front. Robot. AI
**2014**, 1, 37. [Google Scholar] [CrossRef][Green Version] - Shalizi, C.R.; Haslinger, R.; Rouquier, J.B.; Klinkner, K.L.; Moore, C. Automatic filters for the detection of coherent structure in spatiotemporal systems. Phys. Rev.
**2006**, 73, 036104. [Google Scholar] [CrossRef] [PubMed][Green Version] - Biehl, M.; Ikegami, T.; Polani, D. Towards information based spatiotemporal patterns as a foundation for agent representation in dynamical systems. In Proceedings of the Artificial Life Conference 2016, Cancun, Mexico, 4–6 July 2016. [Google Scholar] [CrossRef][Green Version]
- Biehl, M.; Polani, D. Action and perception for spatiotemporal patterns. arXiv
**2017**, arXiv:1706.03576. [Google Scholar] - Hintze, A.; Kirkpatrick, D.; Adami, C. The structure of evolved representations across different substrates for artificial intelligence. arXiv
**2018**, arXiv:1804.01660. [Google Scholar] - Chicharro, D.; Ledberg, A.; Robins, J.; J, T.; Corbetta, M. When two become one: The limits of causality analysis of brain dynamics. PLoS ONE
**2012**, 7, e32466. [Google Scholar] [CrossRef][Green Version] - Rohde, M.; Stewart, J. Ascriptional and ‘genuine’ autonomy. Biosystems
**2008**, 91, 424–433. [Google Scholar] [CrossRef] - Albantakis, L. The Greek Cave: Why a Little Bit of Causal Structure Is Necessary... Even for Functionalist, 2020. Available online: https://www.consciousnessrealist.com/greek-cave/ (accessed on 15 September 2021).
- Doerig, A.; Schurger, A.; Hess, K.; Herzog, M.H. The unfolding argument: Why IIT and other causal structure theories cannot explain consciousness. Conscious. Cogn.
**2019**, 72, 49–59. [Google Scholar] [CrossRef] - Dale, R.; Spivey, M.J. From apples and oranges to symbolic dynamics: A framework for conciliating notions of cognitive representation. J. Exp. Theor. Artif. Intell.
**2005**, 17, 317–342. [Google Scholar] [CrossRef]

**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