Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Assessing planetary complexity and potential agnostic biosignatures using epsilon machines


We present a new approach to exoplanet characterization using techniques from complexity science, with potential applications to biosignature detection. This agnostic method makes use of the temporal variability of light reflected or emitted from a planet. We use a technique known as epsilon machine reconstruction to compute the statistical complexity, a measure of the minimal model size for time series data. We demonstrate that statistical complexity is an effective measure of the complexity of planetary features. Increasing levels of qualitative planetary complexity correlate with increases in statistical complexity and Shannon entropy, demonstrating that our approach can identify planets with the richest dynamics. We also compare Earth time series with Jupiter data, and find that for the three wavelengths considered Earth’s average complexity and entropy rate are approximately 50% and 43% higher than Jupiter’s, respectively. The majority of schemes for the detection of extraterrestrial life rely upon biochemical signatures and planetary context. However, it is increasingly recognized that extraterrestrial life could be very different from life on Earth. Under the hypothesis that there is a correlation between the presence of a biosphere and observable planetary complexity, our technique offers an agnostic and quantitative method for the measurement thereof.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Complexity–entropy diagram for Earth time series.
Fig. 2: Complexity–entropy diagram for Earth and Jupiter time series.

Similar content being viewed by others

Data availability

Source data for all time series as well as the data points in the figures in the main text are provided with this paper.

Code availability

C++ code for the EMR process used in this study can be accessed here:


  1. Kiang, N. Y. et al. Exoplanet biosignatures: at the dawn of a new era of planetary observations. Astrobiology 18, 619–629 (2018).

    Article  ADS  Google Scholar 

  2. Meadows, V. S. et al. Exoplanet biosignatures: understanding oxygen as a biosignature in the context of its environment. Astrobiology 18, 630–662 (2018).

    Article  ADS  Google Scholar 

  3. Bartlett, S. & Wong, M. L. Defining lyfe in the universe: from three privileged functions to four pillars. Life 10, 42 (2020).

    Article  Google Scholar 

  4. Dorn, E. D., Nealson, K. H. & Adami, C. Monomer abundance distribution patterns as a universal biosignature: examples from terrestrial and digital life. J. Mol. Evol. 72, 283–295 (2011).

    Article  ADS  Google Scholar 

  5. Guttenberg, N., Chen, H., Mochizuki, T. & Cleaves, H. J. Classification of the biogenicity of complex organic mixtures for the detection of extraterrestrial life. Life 11, 234 (2021).

    Article  Google Scholar 

  6. Johnson, S. S., Anslyn, E. V., Graham, H. V., Mahaffy, P. R. & Ellington, A. D. Fingerprinting non-terran biosignatures. Astrobiology 18, 915–922 (2018).

    Article  ADS  Google Scholar 

  7. Krissansen-Totton, J., Bergsman, D. S. & Catling, D. C. On detecting biospheres from chemical thermodynamic disequilibrium in planetary atmospheres. Astrobiology 16, 39–67 (2016).

    Article  ADS  Google Scholar 

  8. Krissansen-Totton, J., Olson, S. & Catling, D. C. Disequilibrium biosignatures over earth history and implications for detecting exoplanet life. Sci. Adv. 4, eaao5747 (2018).

    Article  ADS  Google Scholar 

  9. Marshall, S. M. et al. Identifying molecules as biosignatures with assembly theory and mass spectrometry. Nat. Commun. 12, 3033 (2021).

    Article  ADS  Google Scholar 

  10. Solé, R. V. & Munteanu, A. The large-scale organization of chemical reaction networks in astrophysics. Europhys. Lett. 68, 170 (2004).

    Article  ADS  Google Scholar 

  11. Walker, S. I. et al. Exoplanet biosignatures: future directions. Astrobiology 18, 779–824 (2018).

    Article  ADS  Google Scholar 

  12. Marshak, A. et al. Earth observations from DSCOVR EPIC instrument. Bull. Am. Meteorol. Soc. 99, 1829–1850 (2018).

    Article  ADS  Google Scholar 

  13. Fan, S. et al. Earth as an exoplanet: a two-dimensional alien map. Astrophys. J. Lett. 882, L1 (2019).

    Article  ADS  Google Scholar 

  14. Gu, L. et al. Earth as a proxy exoplanet: deconstructing and reconstructing spectrophotometric light curves. Astron. J. 161, 122 (2021).

    Article  ADS  Google Scholar 

  15. Jiang, J. H. et al. Using Deep Space Climate Observatory measurements to study the earth as an exoplanet. Astron. J. 156, 26 (2018).

    Article  ADS  Google Scholar 

  16. Baluška, F. & Levin, M. On having no head: cognition throughout biological systems. Front. Psychol. 7, 902 (2016).

    Article  Google Scholar 

  17. Davies, P. C. & Walker, S. I. The hidden simplicity of biology. Rep. Prog. Phys. 79, 102601 (2016).

    Article  ADS  Google Scholar 

  18. Farnsworth, K. D., Nelson, J. & Gershenson, C. Living is information processing: from molecules to global systems. Acta Biotheor. 61, 203–222 (2013).

    Article  Google Scholar 

  19. Tkačik, G. & Bialek, W. Information processing in living systems. Annu. Rev. Condens. Matter Phys. 7, 89–117 (2016).

    Article  ADS  Google Scholar 

  20. Walker, S. I., Kim, H. & Davies, P. C. The informational architecture of the cell. Phil. Trans. R. Soc. A 374, 20150057 (2016).

    Article  ADS  Google Scholar 

  21. Witzany, G. What is life? Front. Astron. Space Sci. 7, 7 (2020).

    Article  ADS  Google Scholar 

  22. Brodu, N. & Crutchfield, J. P. Discovering causal structure with reproducing-kernel Hilbert space ϵ-machines. Preprint at (2020).

  23. Marzen, S. & Crutchfield, J. P. Informational and causal architecture of continuous-time renewal processes. J. Stat. Phys. 168, 109–127 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Sinapayen, L. & Ikegami, T. Online fitting of computational cost to environmental complexity: predictive coding with the ε-network. Artif. Life Conf. Proc. 14, 380–387 (2017).

    Google Scholar 

  25. Shannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948).

    Article  MathSciNet  MATH  Google Scholar 

  26. Crutchfield, J. P. Between order and chaos. Nat. Phys. 8, 17–24 (2012).

    Article  Google Scholar 

  27. Feldman, D. P., McTague, C. S. & Crutchfield, J. P. The organization of intrinsic computation: complexity–entropy diagrams and the diversity of natural information processing. Chaos 18, 043106 (2008).

    Article  ADS  MathSciNet  Google Scholar 

  28. Bak, P. How Nature Works: the Science of Self-Organized Criticality (Springer, 2013).

  29. Kauffman, S. At Home in the Universe: the Search for the Laws of Self-Organization and Complexity (Oxford Univ. Press, 1996).

  30. Langton, C. G. Computation at the edge of chaos: phase transitions and emergent computation. Physica D 42, 12–37 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  31. Catling, D. C. et al. Exoplanet biosignatures: a framework for their assessment. Astrobiology 18, 709–738 (2018).

    Article  ADS  Google Scholar 

  32. Pierrehumbert, R. T. & Hammond, M. Atmospheric circulation of tide-locked exoplanets. Annu. Rev. Fluid Mech. 51, 275–303 (2019).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Smith, E. & Morowitz, H. J. The Origin and Nature of Life on Earth: the Emergence of the Fourth Geosphere (Cambridge Univ. Press, 2016).

  34. Chopra, A. & Lineweaver, C. H. The case for a Gaian bottleneck: the biology of habitability. Astrobiology 16, 7–22 (2016).

    Article  ADS  Google Scholar 

  35. Lenardic, A. & Seales, J. Habitability: a process versus a state variable framework with observational tests and theoretical implications. Int. J. Astrobiol. 20, 125–132 (2021).

    Article  ADS  Google Scholar 

  36. Lenton, T. M. et al. Selection for Gaia across multiple scales. Trends Ecol. Evol. 33, 633–645 (2018).

    Article  Google Scholar 

  37. Nicholson, A. E., Wilkinson, D. M., Williams, H. T. & Lenton, T. M. Gaian bottlenecks and planetary habitability maintained by evolving model biospheres: the ExoGaia model. Mon. Not. R. Astron. Soc. 477, 727–740 (2018).

    Article  ADS  Google Scholar 

  38. Kasting, J. F. & Siefert, J. L. Life and the evolution of Earth’s atmosphere. Science 296, 1066–1068 (2002).

    Article  ADS  Google Scholar 

  39. Dyke, J., Gans, F. & Kleidon, A. Towards understanding how surface life can affect interior geological processes: a non-equilibrium thermodynamics approach. Earth Syst. Dyn. 2, 139–160 (2011).

    Article  ADS  Google Scholar 

  40. Li, K.-F., Pahlevan, K., Kirschvink, J. L. & Yung, Y. L. Atmospheric pressure as a natural climate regulator for a terrestrial planet with a biosphere. Proc. Natl Acad. Sci. USA. 106, 9576–9579 (2009).

    Article  ADS  Google Scholar 

  41. Harvey, I. The circular logic of Gaia: fragility and fallacies, regulation and proofs. Eur. Conf. Artif. Life Proc. 13, 90–97 (2015).

    Google Scholar 

  42. Wood, A. J., Ackland, G. J., Dyke, J. G., Williams, H. T. & Lenton, T. M. Daisyworld: a review. Rev. Geophys. 46, RG1001 (2008).

    Article  ADS  Google Scholar 

  43. West, G. B. Scale: the Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies (Penguin, 2017).

  44. Galas, D. J., Sakhanenko, N. A., Skupin, A. & Ignac, T. Describing the complexity of systems: multivariable ‘set complexity’ and the information basis of systems biology. J. Comput. Biol. 21, 118–140 (2014).

    Article  MathSciNet  Google Scholar 

  45. Marzen, S. E. & Crutchfield, J. P. Structure and randomness of continuous-time, discrete-event processes. J. Stat. Phys. 169, 303–315 (2017).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Bertello, G., Arduin, P.-J., Boschetti, F. & Weatherley, D. Application of computational mechanics to the analysis of seismic time-series via numerical optimisation. New Gener. Comput. 27, 1–23 (2008).

    Article  MATH  Google Scholar 

  47. Marzen, S. Intrinsic computation of a Monod–Wyman–Changeux molecule. Entropy 20, 599 (2018).

    Article  ADS  Google Scholar 

  48. Muñoz, R. N. et al. General anesthesia reduces complexity and temporal asymmetry of the informational structures derived from neural recordings in Drosophila. Phys. Rev. Res. 2, 023219 (2020).

    Article  Google Scholar 

  49. Park, J. B., Lee, J. W., Yang, J.-S., Jo, H.-H. & Moon, H.-T. Complexity analysis of the stock market. Physica A 379, 179–187 (2007).

    Article  ADS  Google Scholar 

  50. Varn, D. P. & Crutchfield, J. P. Chaotic crystallography: how the physics of information reveals structural order in materials. Curr. Opin. Chem. Eng. 7, 47–56 (2015).

    Article  Google Scholar 

  51. Adami, C., Ofria, C. & Collier, T. C. Evolution of biological complexity. Proc. Natl Acad. Sci. USA 97, 4463–4468 (2000).

    Article  ADS  Google Scholar 

  52. Lineweaver, C. H., Davies, P. C. & Ruse, M. Complexity and the Arrow of Time (Cambridge Univ. Press, 2013).

  53. Smith, J. M. & Szathmary, E. The Major Transitions in Evolution (Oxford Univ. Press, 1997).

  54. Adami, C. What is complexity? BioEssays 24, 1085–1094 (2002).

    Article  Google Scholar 

  55. Gell-Mann, M. & Lloyd, S. Information measures, effective complexity, and total information. Complexity 2, 44–52 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  56. Wolpert, D. H. & Macready, W. Using self-dissimilarity to quantify complexity. Complexity 12, 77–85 (2007).

    Article  MathSciNet  Google Scholar 

  57. Chaitin, G. J. Information, Randomness & Incompleteness: Papers on Algorithmic Information Theory Vol. 8 (World Scientific, 1990).

  58. Kolmogorov, A. N. On tables of random numbers. Sankhyā A 25, 369–376 (1963).

    MathSciNet  MATH  Google Scholar 

  59. Packard, N. H., Crutchfield, J. P., Farmer, J. D. & Shaw, R. S. Geometry from a time series. Phys. Rev. Lett. 45, 712 (1980).

    Article  ADS  Google Scholar 

  60. Shalizi, C. R. & Crutchfield, J. P. Computational mechanics: pattern and prediction, structure and simplicity. J. Stat. Phys. 104, 817–879 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  61. Crutchfield, J., Farmer, J., Packard, N. & Shaw, R. Chaos. Sci. Am. 225, 46–57 (1986).

    Article  Google Scholar 

  62. Brodu, N. Reconstruction of epsilon-machines in predictive frameworks and decisional states. Adv. Complex Syst. 14, 761–794 (2011).

    Article  MathSciNet  Google Scholar 

  63. Lane, N. Oxygen: the Molecule that Made the World (Oxford Univ. Press, 2002).

  64. Olejarz, J., Iwasa, Y., Knoll, A. H. & Nowak, M. A. The Great Oxygenation Event as a consequence of ecological dynamics modulated by planetary change. Nat. Commun. 12, 3985 (2021).

    Article  ADS  Google Scholar 

Download references


We acknowledge partial funding support from the NASA Exoplanet Research Program NNH18ZDA001N-2XRP. A portion of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA (80NM0018D0004). Y.L.Y. was supported in part by an NAI Virtual Planetary Laboratory grant from the University of Washington. We thank the members of the Caltech GPS ‘Astrobiothermoevoinfo’ reading group for the various inspiring discussions that have helped catalyse ideas such as those presented here. We also thank T. Ewald at Caltech for valuable help with the processing of Jupiter data from Cassini. Finally, S.B. thanks S. Bullock for being his guide into the world of complexity.

Author information

Authors and Affiliations



S.B. conceived of the idea of using EMR to analyse planetary complexity and the hypothesized correlation between planetary complexity and the presence of life. He performed the complexity analysis, produced the figures and wrote the manuscript. J.L. provided the Jupiter Cassini data. L.G. and S.F. produced the synthetic Earth and recomposed datasets. L.S. assisted with the complexity analysis, results interpretation, literature review and manuscript editing. V.N. assisted with results interpretation and manuscript editing. J.H.J., D.C. and Y.L.Y. provided essential guidance, assistance with data provision, results interpretation and manuscript editing.

Corresponding author

Correspondence to Stuart Bartlett.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Astronomy thanks Cole Mathis and the other, anonymous, reviewer(s) for their contribution to the peer review of this work

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Epsilon machine example.

A simple epsilon machine with a set of discrete states (blue circles), a start state (double circle on the left), a set of state-to-state transitions (orange arrows), and a set of measurement observables for each transition (green numbers). Each transition has a certain probability of occurrence (not shown in the diagram). The Shannon entropy measures the intrinsic randomness of the model (technically the state probability-weighted information content of the transition probability distributions). The statistical complexity measures the information content of the state space of the model. Notation follows that of26.

Extended Data Fig. 2 Time series processing.

Examples showing the pre-processing of time series in preparation for EMR. a) Original, unaltered time series from the 10 reflectance channels, b) Original time series after normalization and discretization, c) Synthetic time series generated by replacing all pixels with a characteristic desert spectrum, d) Normalized and discretized versions of the series in (c).

Extended Data Fig. 3 Earth, Jupiter time series.

Time series for Earth (blue, 443nm wavelength) and Jupiter (red, 450.9nm wavelength) used for the complexity comparison.

Extended Data Fig. 4 Earth, Jupiter complexity comparison.

Statistical complexities of Earth and Jupiter time series in three wavelength bands for a range of kernel width parameter values.

Supplementary information

Supplementary Information

Supplementary Figs. 1–5, Tables 1 and 2 and Sections 1–4.

Source data

Source Data Fig. 1

Data points for Fig. 1.

Source Data Fig. 2

Data points for Fig. 2.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bartlett, S., Li, J., Gu, L. et al. Assessing planetary complexity and potential agnostic biosignatures using epsilon machines. Nat Astron 6, 387–392 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing