Skip to main content

Defining and simulating open-ended novelty: requirements, guidelines, and challenges


The open-endedness of a system is often defined as a continual production of novelty. Here we pin down this concept more fully by defining several types of novelty that a system may exhibit, classified as variation, innovation, and emergence. We then provide a meta-model for including levels of structure in a system’s model. From there, we define an architecture suitable for building simulations of open-ended novelty-generating systems and discuss how previously proposed systems fit into this framework. We discuss the design principles applicable to those systems and close with some challenges for the community.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. A review of the definitions found in the literature is provided in “Definitions of Open-Endedness

  2. Note, however, that the last definition may not be exclusive. In particular, one can define open-ended evolution as an unbounded evolutionary process and simultaneously consider that open-ended evolution is a definition of life.

  3. Though proponents of group selection, who claim that natural selection can act on populations and not just on individuals (Wilson 1997), would disagree.

  4. We circumvent the question of whether or in what sense “downward causation” exists in reality by focusing on models where we introduce it as existing.

  5. By “perception” we mean the ability of the system to sense (by whatever means) aspects of its environment, allowing it to act and react; it would not necessarily need to be a living system.

  6. It may also have no effect at all, as, for example, is the case of a neutral mutation.

  7. With our meta-model model, it is not “turtles all the way down”!

  8. This does not imply or require that the set of \(m_i\) entities constituting any specific level-\(i+1\) individual is fixed or static for the lifetime of that individual. It is a specific characteristic of biological individuals that they continuously turn over their constituent lower level components, while retaining their systemic coherence and individuality.

  9. One can still argue that, on a larger horizon (namely on the level of the entire universe), although the universe is bounded, the potential for novelty is unlimited, be it generated by variation, innovation or emergence. Following this idea, one could consider that the universe is effectively open-ended.

  10. As opposed to statistical simulations, such as “draw an infinite number of reals from (0, 1), with an exponential distribution”, where innovation in the form of seeing new numbers is certain. Our argument here applies to any physical simulation, not just to computational ones.

  11. A two scale model is technically “multiscale”, but can perhaps be simulated in some cases.

  12. Fitness in biological systems is defined as differential reproductive success. But in nature fitness is assessed retrospectively, through natural selection.

  13. This is another possible difference between our computer simulation-motivated meta-model and biology. This argument in a biological context would imply that innovation in living systems is impossible without group selection. This would be a highly contentious claim.

  14. This is different from external modification, or “patching”, running code.

  15. The variation is technically pseudorandom, being generated by a deterministic algorithm. This process is itself a lower level system in the model. Details of this level can significantly affect GA behavior.

  16. Note that open-ended GAs could be implemented providing the bitstring length and the number of genes it encodes can evolve. In this case innovation events could be possible as observed, e.g., in Knibbe et al. (2007).

  17. The “core” in “coreworld” derives from the earlier CoreWar programming game (Dewdney 1987), and invokes the typical linear, random access, memory configuration originally associated with early magnetic core hardware.

  18. Strictly, Coreworld does not incorporate a memory “allocation” shortcut per se at all.


  • Adami C, Brown CT (1994) Evolutionary learning in the 2D artificial life system avida. In: Artificial life IV: proceedings of the 4th international workshop on the synthesis and simulation of living systems, pp 377–381

  • Amar P, Legent G, Thellier M, Ripoll C, Bernot G, Nystrom T, Saier M, Norris V (2008) A stochastic automaton shows how enzyme assemblies may contribute to metabolic efficiency. BMC Syst Biol 2:27

    Article  PubMed  PubMed Central  Google Scholar 

  • Andrews PS, Polack FAC, Sampson AT, Stepney S, Timmis J (2010) The CoSMoS process, version 0.1: a process for the modelling and simulation of complex systems. Technical Report YCS-2010-453, Department of Computer Science, University of York

  • Andrews PS, Stepney S, Hoverd T, Polack FA, Sampson AT, Timmis J (2011) CoSMoS process, models, and metamodels. In: CoSMoS 2011: proceedings of the 2011 workshop on complex systems modelling and simulation, pp 1–14

  • Andrews PS, Stepney S, Timmis J (2012) Simulation as a scientific instrument. In: Proceedings of the 2012 workshop on complex systems modelling and simulation, Orleans, France, pp 1–10

  • Anderson PW (1972) More is different. Sci 177:393–396

    CAS  Google Scholar 

  • Banzhaf W, Nordin P, Keller R, Francone F (1998) Genetic programming: an introduction. Morgan Kaufmann, San Francisco, CA, USA

    Book  Google Scholar 

  • Banzhaf W, Yamamoto L (2015) Artificial chemistries. MIT Press, Cambridge, MA, USA

    Google Scholar 

  • Baptista T, Costa E (2013) Step evolution: improving the performance of open-ended evolution simulations. In: IEEE symposium on artificial life, Singapore 2013, pp 52–59. IEEE

  • Barricelli NA (1962) Numerical testing of evolution theories: part i theoretical introduction and basic tests. Acta Biotheor 16(1–2):69–98

    Article  Google Scholar 

  • Barrick JE, Lenski RE (2013) Genome dynamics during experimental evolution. Nat Rev Genet 14:827–839

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  • Barrick JE, Yu DS, Yoon SH, Jeong H, Oh TK, Schneider D, Lenski RE, Kim JF (2009) Genome evolution and adaptation in a long-term experiment with Escherichia coli. Nature 461:1243–1247

    CAS  Article  PubMed  Google Scholar 

  • Batut B, Knibbe C, Marais G, Daubin V (2014) Reductive genome evolution at both ends of bacterial population size spectrum. Nat Rev Microbiol 12(12):841–850

    CAS  Article  PubMed  Google Scholar 

  • Baugh D (2015) Implementing von Neumann’s architecture for machine self reproduction within the tierra artificial life platform to investigate evolvable genotype-phenotype mappings. PhD, Dublin City University. School of Electronic Engineering

  • Bedau MA (1991) Can biological teleology be naturalized? J Philos 88:647–655

    Article  Google Scholar 

  • Bedau MA (1996) The nature of life. In: Boden M (ed) The philosophy of artificial life. Oxford University Press, Oxford, UK, pp 332–357

    Google Scholar 

  • Bedau MA (1999) Can unrealistic computer models illuminate theoretical biology? In: Proceedings of the 1999 genetic and evolutionary computation conference, workshop companion, pp 20–23

  • Bedau MA, Packard NH (1992) Measurement of evolutionary activity, teleology, and life. In: Langton C, Taylor C, Farme D, Rasmussen S (eds) Artificial life II. Addison-Wesley, Reading, MA, USA, pp 431–461

    Google Scholar 

  • Bedau MA, Snyder E, Packard NH (1998) A classification of long-term evolutionary dynamics. In: ALife IV, MIT Press, Cambridge, MA, USA, pp 228–237

  • Bedau MA, McCaskill JS, Packard NH, Rasmussen S, Adami C, Green DG, Ikegami T, Kaneko K, Ray TS (2000) Open problems in artificial life. Artif Life 6(4):363–376

    CAS  Article  PubMed  Google Scholar 

  • Bentley PJ (2003) Evolving fractal gene regulatory networks for robot control. In: ECAL 2003, vol 2801 of LNCS. Springer, Berlin, Germany, pp 753–762

    Google Scholar 

  • Berlekamp ER, Conway JH, Guy RK (1982) Winning ways for your mathematical plays, volume 2: games in particular. Academic Press, New York, NY, USA

  • Bianco R, Nolfi S (2004) Toward open-ended evolutionary robotics: evolving elementary robotic units able to self-assemble and self-reproduce. Connect Sci 16(4):227–248

    Article  Google Scholar 

  • Blount ZD, Borland CZ, Lenski RE (2008) Historical contingency and the evolution of a key innovation in an experimental population of Escherichia coli. Proc Natl Acad Sci USA 105:7899–7906

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  • Buss L (1987) The evolution of individuality. Princeton University Press, Princeton, NJ, USA

    Google Scholar 

  • Channon A (2001) Passing the ALife test: activity statistics classify evolution in Geb as unbounded. In: ECAL ’01: proceedings of the 6th European conference on artificial life. Springer, Berlin, Germany, pp 417–426

  • Channon A (2003) Improving and still passing the ALife test: component-normalised activity statistics classify evolution in Geb as unbounded. In: Proceedings of artificial life VIII. MIT Press, Cambridge, MA, USA, pp 173–181

    Google Scholar 

  • Channon AD, Damper RI (2000) Towards the evolutionary emergence of increasingly complex advantageous behaviours. Int J Syst Sci 31(7):843–860

    Article  Google Scholar 

  • Chomsky N (1956) Three models for the description of language. IRE Trans Inf Theory 2:113–124

    Article  Google Scholar 

  • Church A (1936) An unsolvable problem of elementary number theory. Am J Math 58:345–363

    Article  Google Scholar 

  • Cook SA (1971) On the complexity of theorem-proving procedures. In: Proceedings ACM symposium on theory of computing, pp 151–158

  • Craver C (2007) A field guide to levels. In: Explaining the brain: mechanisms and the Mosaic Unity of Neuroscience, chapter 5. Clarendon Press, Oxford

  • Dewdney AK (1987) Computer recreations: a program called mice nibbles its way to victory at the first core wars tournament. Sci Am 256(1):8–11

    Article  Google Scholar 

  • Dittrich P, Ziegler J, Banzhaf W (2001) Artificial chemistries: a review. Artif Life 7:225–275

    CAS  Article  PubMed  Google Scholar 

  • Droop A, Hickinbotham S (2012) A quantitative measure of non-neutral evolutionary activity for systems that exhibit intrinsic fitness. Artificial Life XIII, pp 45–52

  • Edmonds B (1998) Meta-genetic programming: co-evolving the operators of variation. CPM Report 98–32, Centre for Policy Modelling, Manchester Metropolitan University, UK, Aytoun St., Manchester, M1 3GH, UK

  • Ellis GFR (2011) Top-down causation and emergence: some comments on mechanisms. R Soc Interf Focus 6(1):1–15

    Google Scholar 

  • Fernández JD, Lobo D, Martn GM, Doursat R, Vico FJ (2012) Emergent diversity in an open-ended evolving virtual community. Artif Life 18(2):199–222

    Article  PubMed  Google Scholar 

  • Fernando C, Kampis G, Szathmáry E (2011) Evolvability of natural and artificial systems. Proc Comput Sci 7:73–76

    Article  Google Scholar 

  • Foster JA (2001) Computational genetics: evolutionary computation. Nat Rev Genet 2:428–436

    CAS  Article  PubMed  Google Scholar 

  • Frigg R, Hartmann S (2012) Models in science. In: Zalta EN (ed) The Stanford encyclopedia of philosophy. Fall 2012 edition

  • Gardner M (1970) Mathematical games: the fantastic combinations of John Conway’s new solitaire game “life”. Sci Am 223(4):120–123

  • Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman, San Francisco, CA, USA

    Google Scholar 

  • Gold EM (1967) Language identification in the limit. Inform Contr 10:447–474

    Article  Google Scholar 

  • Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading, MA, USA

    Google Scholar 

  • Gotts NM (2009) Ramifying feedback networks, cross-scale interactions, and emergent quasi individuals in Conway’s Game of Life. Artif Life 15(3):351–375

    Article  PubMed  Google Scholar 

  • Hartmanis J, Stearns RE (1965) On the computational complexity of algorithms. Trans Am Math Soc 117:285–306

    Article  Google Scholar 

  • Harvey I (1992) Species adaptation genetic algorithms: a basis for a continuing SAGA. In: Toward a practice of autonomous systems: proceedings of the 1st European conference on artificial life, pp 346–354

  • Hasegawa T (2015) On the evolution of genotype-phenotype mapping: exploring viability in the Avida articial life system. PhD, Dublin City University, School of Electronic Engineering

  • Heylighen F (2012) Brain in a vat cannot break out. J Conscious Stud

  • Hickinbotham S, Clark E, Nellis A, Stepney S, Clarke T, Young P (2016) Maximising the adjacent possible in automata chemistries. Artif Life 22(1):49–75

    Article  PubMed  Google Scholar 

  • Hickinbotham S, Clark E, Stepney S, Clarke T, Nellis A, Pay M, Young P. Specification of the stringmol chemical programming language version 0.2. Technical report, Technical Report YCS-2010-458, University of York

  • Hilbert D (1901) Mathematical problems. Archiv der Mathematik und Physik, 3rd series, vol 1, pp 44–65, 213–237

  • Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann Arbor, MI, USA

    Google Scholar 

  • Horsman C, Stepney S, Wagner RC, Kendon V (2014) When does a physical system compute? Proc R Soc A 470(2169):20140182

    Article  PubMed  PubMed Central  Google Scholar 

  • Hoverd T, Stepney S (2011) Energy as a driver of diversity in open-ended evolution. In: ECAL 2011, Paris, France, August 2011. MIT Press, Cambridge, MA, USA, pp 356–363

    Google Scholar 

  • Humphreys P (2004) Extending ourselves: computational science, empiricism, and scientific method. Oxford University Press, Oxford, UK

  • Huneman P (2012) Determinism, predictability and open-ended evolution: lessons from computational emergence. Synthese 185(2):195–214

    Article  Google Scholar 

  • Hutton TJ (2002) Evolvable self-replicating molecules in an artificial chemistry. Artif Life 8(4):341–356

    Article  PubMed  Google Scholar 

  • Kaneko K (1994) Chaos as a source of complexity and diversity in evolution. Artif Life 1(1/2):163–178

  • Kantschik W, Dittrich P, Brameier M, Banzhaf W (1999) Meta-evolution in graph GP. In: Poli R, Nordin P, Langdon W, Fogarty T (eds) Genetic programming: proceedings EuroGP 1999. Springer, Berlin, Germany, pp 15–28

    Chapter  Google Scholar 

  • Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum, New York

    Google Scholar 

  • Kimura M (1984) The neutral theory of molecular evolution. Cambridge University Press, Cambridge, UK

  • Kleppe A, Warmer J, Bast W (2003) MDA explained: the model driven architecture: practice and promise. Addison-Wesley, Reading, MA, USA

  • Knibbe C, Coulon A, Mazet O, Fayard J-M, Beslon G (2007) A long-term evolutionary pressure on the amount of noncoding DNA. Mol Biol Evolut 24(10):2344–2353

    CAS  Article  Google Scholar 

  • Koestler A (1970) Beyond atomism and holism: the concept of the holon. Perspect Biol Med 13(2):131–154

    Article  Google Scholar 

  • Koza J (1992) Genetic programming. MIT Press, Cambridge, MA, USA

  • Lan D (2006) Hierarchy, complexity, society. In: Pumain D (ed) Hierarchy in natural and social sciences. Springer, Berlin, Germany, pp 81–119

  • Lehman J, Stanley KO (2011) Abandoning objectives: evolution through the search for novelty alone. Evol Comput 19(2):189–223

    Article  PubMed  Google Scholar 

  • Lehman J, Stanley KO (2012) Beyond open-endedness: quantifying impressiveness. ALIFE XIII, pp 75–82

  • Maley C (1999) Four steps toward open-ended evolution. In: Proceedings of the genetic and evolutionary computation conference (GECCO-1999), vol 2, pp 1336

  • Markovitch O, Sorek D, Lui LT, Lancet D, Krasnogor N (2012) Is there an optimal level of open-endedness in prebiotic evolution? Orig Life Evolut Biosph 42(5):469–474

    Article  Google Scholar 

  • Maynard Smith J (1988) Evolutionary progress and levels of selection. In: Nitecki M (ed) Evolutionary progress. University of Chicago Press, Chicago, IL, USA, pp 219–230

    Google Scholar 

  • Maynard Smith J, Szathmáry E (1995) The major transitions in evolution. Oxford University Press

  • McCutcheon JP, Moran NA (2012) Extreme genome reduction in symbiotic bacteria. Nat Rev Microbiol 10(1):13–26

    CAS  Google Scholar 

  • McMullin B (2012) Architectures for self-reproduction: abstractions, realisations and a research program. In: Adami C, Bryson DM, Ofria C, Pennock RT (eds) Artificial life XIII. MIT Press, Cambridge, MA, USA, pp 83–90

    Google Scholar 

  • Medernach D, Kowaliw T, Ryan C, Doursat R (2013) Long-term evolutionary dynamics in heterogeneous cellular automata. In: Proceedings of the 15th annual conference on Genetic and evolutionary computation (GECCO’13), pp 231–238. ACM

  • Morris I (2010) Why the west rules-for now: the patterns of history and what they reveal about the future. Profile Books, London, UK

  • Nehaniv CL, Hewitt J, Christianson B, Wernick P. What software evolution and biological evolution don’t have in common. In: 2nd international IEEE workshop on software evolvability (SE’06), pp 58–65. IEEE

  • Odling-Smee F, Laland K, Feldman M (2003) Niche construction: the neglected process in evolution. Princeton University Press, Princeton, NJ, USA

  • Pargellis AN (2001) Digital life behavior in the amoeba world. Artif Life 7(1):63–75

    CAS  Article  PubMed  Google Scholar 

  • Pfante O, Bertschinger N, Obrich E, Ay N, Jost J (2014) Comparison between different methods of level identification. Adv Complex Syst 17:1450007–1–1450007-21

    Article  Google Scholar 

  • Plucain J, Hindré T, Le Gac M, Tenaillon O, Cruveiller S, Médigue C, Leiby N, Harcombe WR, Marx CJ, Lenski RE, Schneider D (2014) Epistasis and allele specificity in the emergence of a stable polymorphism in Escherichia coli. Science 343:1366–1369

    CAS  Article  PubMed  Google Scholar 

  • Popper K (1982) The open universe: an argument for indeterminism. Hutchinson, London, UK

  • Post EL (1944) Recursively enumerable sets of positive integers and their decision problems. Bull Am Math Soc 50:284–316

    Article  Google Scholar 

  • Rasmussen S, Chen L, Deamer D, Krakauer DC, Packard NH, Stadler PF, Bedau MA (2004) Transitions from nonliving to living matter. Science 303(5660):963–965

    CAS  Article  PubMed  Google Scholar 

  • Rasmussen S, Knudsen C, Feldberg R, Hindsholm M (1990) The coreworld: emergence and evolution of cooperative structures in a computational chemistry. Physica 42D:111–134

    Google Scholar 

  • Ray TS (1992) An approach to the synthesis of life. In: Langton CG, Taylor C, Farmer JD, Rasmussen S (eds) Artifical life II. Addison-Wesley, Reading, MA, USA, pp 371–408

  • Ray TS (1992) Evolution, ecology and optimization of digital organisms. Working paper 92–08-042, Santa Fe

  • Rendell P (2002) Turing universality of the game of life. In: Adamatzky A (ed) Collision-based computing. Springer, Berlin, Germany

  • Renner G, Ekárt A (2003) Genetic algorithms in computer aided design. Comput Aided Design 35(8):709–726

    Article  Google Scholar 

  • Rensch B (1959) Evolution above the species level. Methuen, London

    Google Scholar 

  • Reynolds CW (1987) Flocks, herds and schools: a distributed behavioral model. ACM SIGGRAPH Comput Graph 21(4):25–34

    Article  Google Scholar 

  • Rogers H (1987) Theory of recursive functions and effective computability. MIT Press, Cambridge, MA, USA

  • Rosen R (1991) Life itself: a comprehensive inquiry into the nature, origin, and fabrication of life. Columbia University Press, New York, NY, USA

  • Ruiz-Mirazo K, Moreno A (2012) Autonomy in evolution: from minimal to complex life. Synthese 185(1):21–52

    Article  Google Scholar 

  • Ruiz-Mirazo K, Pereto J, Moreno A (2004) A universal definition of life: autonomy and open-ended evolution. Origins Life Evolut Biosph 34(3):323–346

    CAS  Article  Google Scholar 

  • Ruiz-Mirazo K, Umerez J, Moreno A (2008) Enabling conditions for ‘open-ended evolution’. Biol Philos 23(1):67–85

    Article  Google Scholar 

  • Schrödinger E (1944) What is life? The physical aspect of the living cell. Cambridge University Press, Cambridge, UK

  • Schulman R, Yurke B, Winfree E (2012) Robust self-replication of combinatorial information via crystal growth and scission. PNAS 109(17):6405–6410

    CAS  Article  PubMed  PubMed Central  Google Scholar 

  • Sipper M, Sanchez E, Mange D, Tomassini M, Pérez-Uribe A, Stauffer A (1997) A phylogenetic, ontogenetic, and epigenetic view of bio-inspired hardware systems. IEEE Trans Evolut Comput 1(1):83–97

    Article  Google Scholar 

  • Sipper M, Sanchez E, Mange D, Tomassini M, Pérez-Uribe A, Stauffer A (1998) An introduction to bioinspired machines. In: Bio-inspired computing machines towards novel computational architectures. Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, pp 1–12

  • Skusa A, Bedau MA (2002) Towards a comparison of evolutionary creativity in biological and cultural evolution. In: ALife VIII. MIT Press, Cambridge, MA, USA, pp 233–242

  • Soare RI (1987) Recursively enumerable sets and degrees: a study of computable functions and computably generated sets. MIT Press

  • Soros LB, Stanley KO (2014) Identifying necessary conditions for open-ended evolution through the artificial life world of Chromaria. In: Artificial life 14: international conference on the synthesis and simulation of living systems, vol 14, pp 793–800

  • Spector L (2010) Towards practical autoconstructive evolution: self-evolution of problem-solving genetic programming systems. In: Riolo R, McConaghy T, Vladislavleva E (eds) Genetic programming theory and practice VIII, volume 8 of genetic and evolutionary computation, chapter 2, pp 17–33. Springer, Ann Arbor, USA, 20–22 May 2010

  • Spector L, Robinson A (2002) Genetic programming and autoconstructive evolution with the push programming language. Genet Program Evol Mach 3:7–40

    Article  Google Scholar 

  • Standish RK (2003) Open-ended artificial evolution. Int J Comput Intell Appl 3(2):167–175

    Article  Google Scholar 

  • Stepney S (2012) A pattern language for scientific simulations. In: Proceedings of the 2012 workshop on complex systems modelling and simulation, Orleans, France, pp 77–103

  • Stepney S, Alden K, Paul JLB, Andrews S, Droop A, Ghetiu T, Hoverd T, Polack FAC, Read M, Ritson CG, Sampson AT, Timmis J, Welch PH, Winfield AFT (2016) Engineering simulations as scientific instruments. Springer (in preparation), Berlin, Germany

  • Stepney S, Andrews PS (2015) CoSMoS special issue editorial. Nat Comput 14:1–6

    Article  Google Scholar 

  • Stepney S, Hoverd T (2011) Reflecting on open-ended evolution. In: ECAL ’11: proceedings of the 11th European conference on artificial life. MIT Press, Cambridge, MA, USA, pp 781–788

  • Suppes P (1960) A comparison of the meaning and uses of models in mathematics and the empirical sciences. Synthese 12:287–301

    Article  Google Scholar 

  • Szathmáry E (2015) Toward major evolutionary transitions theory 2.0. Proc Natl Acad Sci 112(33):10104–10111

    Article  PubMed  PubMed Central  Google Scholar 

  • Taylor T (1999) From artificial evolution to artificial life. PhD thesis, The University of Edinburgh

  • Turing A (1936–1937) On computable numbers, with an application to the Entscheidungsproblem. Proc London Math Soc Ser 2 42:230–265

  • Valiant L (1984) A theory of the learnable. Commun ACM 27:1134–1135

    Article  Google Scholar 

  • Waddington C (2008) Paradigm for an evolutionary process. Biol Theory 3:258–266

    Article  Google Scholar 

  • Weisberg M (2013) Simulation and similarity: using models to understand the world. Oxford University Press, Oxford, UK

  • Wilson D (1997) Biological communities as functionally organized units. Ecology 78:2018–2024

    Article  Google Scholar 

  • Wimsatt W (1987) False models as means to truer theories. In: Nitecki N, Hoffman A (eds) Neutral models in biology. Oxford University Press, New York, pp 23–55

  • Wimsatt W (1994) The ontology of complex systems: levels, perspectives, and causal thickets. Can J Philos 20(Suppl):207–274

  • Winsberg E (2010) Science in the age of computer simulation. Chicago University Press, Chicago, IL, USA

Download references


We thank the anonymous referees for their insightful comments, which have helped to improve this paper from an earlier version. The authors acknowledge funding from diverse agencies: W. Banzhaf from NSERC under Discovery Grant RGPIN 283304-2012, B. Baumgaertner and J. A. Foster from the BEACON Center for Evolution in Action and from IBEST, G. Beslon and S. Stepney from the European Commission 7th Framework Programme (FPFP7-ICT-2013.9.6 FET Proactive: Evolving Living Technologies) EvoEvo project (ICT-610427), V.V. de Melo from Brazilian Government CNPq (Universal) grant (486950/2013-1) and Brazilian Government CAPES (Science without Borders Program) grant (12180-13-0), L. Spector from the National Science Foundation under Grant Nos. 1129139 and 1331283. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the CAPES/CNPq (Brazilian Government), European Commission (EU), NSERC (Canada), or NSF (USA). The authors are grateful to Memorial University for providing infrastructure for our workshop and to Overleaf for providing an excellent collaborative tool for writing a LaTeX document.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Wolfgang Banzhaf.

Additional information

This article forms part of a special issue of Theory in Biosciences in commemoration of Olaf Breidbach.

Vinicius Veloso de Melo: On leave from Institute of Science and Technology (ICT), Federal University of São Paulo (UNIFESP), São José dos Campos, SP, Brazil.



On levels

Levels in philosophy

There are at least two different approaches one could take to characterizing the notion of level. On the one hand, philosophers like Wimsatt have provided a prototype of levels (as opposed to providing definitions) (Wimsatt 1994). Under this treatment, levels are distinguished by a cluster of rankable features: objects at different levels will have different sizes; objects at different levels stand in composition relations to one another; objects at the same level are governed by the same laws and the forces at play have similar magnitudes; objects at the same level are reliable detectors of one another because they stand in regular and predictable relations with one another; objects at the same level are investigated with the same set of techniques with respect to similar disciplinary perspectives. The advantage of providing a prototype treatment of levels is that some examples of levels may lack one or more of the aforementioned features.

On the other hand, philosophers like Craver have approached the issue by providing a taxonomy of the different senses of level (Craver 2007). This approach highlights the similarities and differences between the senses of level and can provide clarity where there are often misleading associations between them. For example, Craver distinguishes between four different senses of levels of composition: mereology, aggregativity, mere material/spatial containment, and mechanism:

  1. 1.

    Levels of mereology are formed by part–whole relations so that the collection of parts are at a lower level than the object that the parts make up. On the mereological conception of levels, complex things are regarded as wholes, but this does not emphasize whether the part–whole relation is constituted by material or spatial containment, or some other feature. It also does not specify what relations hold between the parts.

  2. 2.

    Levels of aggregativity, on the other hand, specify that the properties of items at a higher level are the simple sums of the properties of the items at the lower level. For example, the mass of a pile of sand is the sum of the masses of the individual grains.

  3. 3.

    Levels of mere material/spatial containment are permissive conception of composition. In this sense of level, an entity at a higher level is constituted by pieces. For example, to model climate we might divide the atmosphere into cubic-kilometer pieces. Pieces are to be contrasted with components. If we divide the human body into cubic-centimeter pieces, we would have a haphazard collection of things that do not have clear contributions to the workings of the human body.

  4. 4.

    Dividing the human body into components, however, involves the identification of how the part is relevant to the behavior of the whole. This is the defining feature of levels of mechanisms. On this conception of levels, components at a lower level are organized together to form components at a higher level, such that the behaviors of the components are relevant to the behavior of the whole.

An important aspect of thinking about mechanistic levels is that they are defined within a hierarchically organized mechanism; levels are not defined by objective kinds that are independently identifiable from a mechanism. To ask whether a given molecule and cell are at different levels makes sense only in the context of a given mechanism, which in turn makes sense only in a given model: the actual levels are model dependent. So, there is no unique answer to whether two items are at the same level; they are if they are both components in the same mechanism without being components of each other. However, under a different decomposition hierarchy, those items can be at different levels (for example, if one is a component of the other). Thus, an elephant and a bacterium it carries can be at the same level if they are components of the same mechanism, but can be at different levels if the hierarchical mechanism in question is the elephant and the components that make it up include bacteria. In recent years, level identification has also become an interesting topic of mathematical and information theoretic analysis (Pfante 2014).

On timescales and levels

Levels and timescales, both of dynamics and evolution, may be intimately related, but the relationship might be complicated. In abiotic systems it may be the case that dynamics are faster at lower than at higher levels, but this may be a relatively trivial observation in that the higher levels are simply large-scale patterns of the lower level entities, and the patterns to be observed and said to exist must have a longer timescale. Thus, the large-scale weather system depends for its existence on the Brownian motions of the atmospheric molecules, whereas the process of self-organization generating the pattern requires many times the Brownian timescale to unfold. But here the weather system’s timescale can only be defined in terms of the large-scale spatial pattern. Bénard cells provide another example. In this case, the lower level building blocks are much more stable than the emergent higher level structures but the low-level dynamic is much more rapid than the high-level dynamic.

In biological systems the relationship is more varied, and also more important. Living systems are not just compositions of molecules; they themselves construct most of their composite molecules. It is frequently claimed that the timescale of biological (Darwinian) evolution is slower than that of the individuals in the evolving population, so this is also a case of the higher level (species) having a slower timescale than the lower level (individuals or populations of individuals). It is also the case that the inherent rate of the appearance of new molecules in the genome is much greater than the rate of evolution in the individual, but in this case the evolution is largely suppressed by the repair mechanisms available to the phenotype to ensure the survival of the phenotype level, and hence also the survival of the genotype. As a result, the rate of evolution of the two levels is effectively the same—lower level variation uncorrelated with high-level variation simply dies out.

In human systems, the situation may be even less straightforward. A corporation may be considered to be a higher level entity in that it has a distinct physical existence (buildings, equipment, etc.) as well as a legal existence: incorporation gives it much the same legal status as a human being, and allows it to act in relevant respects as a person. Yet it is composed of a collection of people, who as employees become components of the corporate entity. The timescale of their evolution in terms of their function (for example, the products they make) may be slow relative to the lifetime of an employee, or much faster. Financial entities show an even greater range of evolutionary timescales, from a century or more in the case of savings banks to years or even months in the case of derivatives or closed-end funds. In the case of derivatives, the higher level evolves much faster than the lower level of the instruments being securitized. In human systems, we may even find circular hierarchies when individuals are subject to the constraints of various higher level organizations such as their employer, their church, and the bylaws of the local government, these organizations being in turn subject to the laws and regulations of the national government, but the national government being subject to a president/king/dictator—an individual belonging to the lowest level of the hierarchy. Note that humans have had the ability to build entities—computers—faster than themselves that now drive large parts of, e.g., financial exchanges.

At this point it would seem safe to say that for biological and human systems the relationship between levels and timescales is one that is fundamentally important. However, it is difficult to identify simple generalizations. The problem is one that for the time being must be examined in particular contexts.

On computational limits

Classical theoretical computer science investigates the relationships between objects that can be described or instantiated with algorithms, which are formalizations of Hilbert’s concept of an “effective mathematical procedure” (Hilbert 1901). Several different algorithmic models exist, the most well-known being grammars in formal languages (Post 1944), recursive functions (Church 1936), and finite automata (Turing 1936). Less well known, but relevant to our discussion, are automata that infer functions from examples (Gold 1967; Valiant 1984). The objects described or instantiated by these models include primarily sets of finite (but indefinitely long) strings over finite alphabets, known as formal languages, and functions which map either strings onto strings or natural numbers onto natural numbers.

The fundamental results of theoretical computer science prove that there are hierarchies of sets of objects described or instantiated by abstract models of algorithms, such that variations in the models determine which level of the hierarchy the model describes or instantiates. In our terms, this means that even abstract mathematical models of algorithms define classes of innovations that are achievable by varying the models. Each of these classes of innovations are open-ended in the sense that there is no final or ultimate model; a model can always be extended so as to make a new class of objects accessible.

Consider some examples from automata theory. Turing machines are finite automata with indefinitely extensible memory. The finitude of this model implies that there is a hierarchy of three classes of sets or functions that can be described or instantiated: those that can be computed, those that can be enumerated, and those which are beyond computation. Augmenting the Turing machine model with access to a hypothetical (not actually implementable) external set, known as an oracle, produces an infinite, strict hierarchy of classes beyond these three [known as the Kleene–Post Hierarchy (Rogers 1987; Soare 1987)], and therefore open-ended innovation in our sense. Restricting the Turing machine model by limiting access to memory produces another strict hierarchy which corresponds very closely to a hierarchy of formal languages known as the Chomsky Hierarchy (Chomsky 1956): random access to memory yields the class of languages describable by context-sensitive grammars, access to only the last item remembered yields the context-free grammars, and no access to memory yields the regular languages. Again, expanding a grammar with new types of production rules changes the level of the formal language hierarchy, moving up a well-defined hierarchy.

The branch of theoretical computer science known as computational complexity proves that restricting the number of steps or the amount of memory an automaton can use also produces strict hierarchies of formal languages (Hartmanis and Stearns 1965). Thus, for example, Turing machines that can run for a number of steps bounded by an exponential function of the input size can recognize languages that such machines bounded by a polynomial number of steps cannot. Similarly, automata that can access no more memory locations than some exponential of the input size can recognize languages that such automata with a polynomial bound cannot. Many of the most important problems in computer science involve understanding when different resource bounds or different types of automata (such as deterministic versus nondeterministic) produce genuine innovation, in the sense that one variation can recognize languages that the other cannot. For example, the famous P versus NP problem asks whether adding nondeterminism to polynomially bounded Turing machines is an innovation or not. If it is, then the hundreds of critically important practical problems in NP cannot be solved by algorithms that run in a reasonable amount of time (Garey and Johnson 1979; Cook 1971; Karp 1972). Unfortunately, we do not know the answers to most of these questions.

In summary, there are three lessons to learn from theoretical computer science that are relevant to this paper. First, enumeration can indeed provide innovation, since algorithms exist to enumerate some sets that no algorithm can fully describe. But this fact requires the abstraction of what an algorithm can do “in the limit”, including the possibility that the algorithm never halts. Which brings us to the second lesson: computational complexity theory does indeed provide a context in which one can prove that certain variations in finite computations, such as allowing exponentially more memory access, generate innovations. Unfortunately, this theory breaks down precisely where it becomes useful: with reasonable limits on the number of steps or the amount of memory that a computation requires. Therefore, theoretical computer science is a useful framework in which to study open-ended innovation, but only in an abstract, mathematical sense. In particular, it does not currently directly explain open-ended innovation in physical systems such as computer simulations or biology.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Banzhaf, W., Baumgaertner, B., Beslon, G. et al. Defining and simulating open-ended novelty: requirements, guidelines, and challenges. Theory Biosci. 135, 131–161 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Modelling and simulation
  • Open-ended evolution
  • Novelty
  • Innovation
  • Major transitions
  • Emergence