Open-Ended Evolution: Unbounded Complexity

• Open-Ended Evolution is the process of continuously generating novel, increasingly complex entities without converging to a fixed endpoint.
• It leverages robust reproduction, an unlimited diversity of environments, and mechanisms like cardinality leaps and non-additive composition to drive sustained novelty.
• Research on OEE informs practical design principles in natural, digital, and sociotechnical systems, addressing challenges such as unpredictability and evolving evolvability.

Open-Ended Evolution (OEE) is the term for evolutionary dynamics that generate an essentially unlimited diversity of novel, increasingly complex, and functional entities or interactions over time, without convergence to any immutable attractor or upper bound. In both natural and artificial systems, OEE captures the capacity to continually produce new forms and mechanisms, mirroring the creative productivity exhibited by the biosphere across its ~4-billion-year history (Packard et al., 2019). Research on OEE addresses foundational questions about the conditions, mechanisms, formalization, and experimental design needed to realize truly unbounded evolutionary innovation in silico, in vitro, and in empirical sociotechnical systems.

1. Formal Definitions and Conceptual Foundations

OEE is formally distinguished from bounded or closed evolutionary dynamics by its continual, non-saturating novelty production. The minimal necessary requirement for OEE is that the set of distinct types (species, structures, behaviors, or “traits”) realized up to time t, written T(t), satisfies

$$\lim_{t\to\infty}|T(t)| = \infty \quad \text{and} \quad \forall n\in\mathbb{N},\,\exists t:\,|T(t)| > n$$

or, equivalently, that the rate of new type discovery does not vanish asymptotically (Borg et al., 2022, Packard et al., 2019).

A modern theoretical synthesis divides OEE into three increasingly powerful levels of novelty (Taylor, 2018, Taylor, 2020):

• Exploratory OEE: Continuous discovery of new instances or configurations within a fixed phenotype/metamodel space.
• Expansive OEE: Expansion of the search/model space itself by discovering new building blocks or “concepts” (e.g., new metabolic reactions, higher-order assemblies).
• Transformational OEE: Innovation that alters the generative meta-model or even the domains of physical law or interaction, enabling the emergence of qualitatively new organizational/hierarchical levels (e.g., evolution of vision, major evolutionary transitions) (Taylor, 2018, Taylor, 2021).

Algorithmic information theory frames OEE by requiring unbounded growth in the algorithmic complexity (Kolmogorov complexity) of evolutionary histories:

• Axiom 1 (Open-endedness): $(K(\Sigma(t))/t) \leq (K(\Sigma(t+1))/(t+1))$
• Axiom 2 (Unboundedness): For all $N \in \mathbb{N}$, $\exists t$ such that $K(\Sigma(t))/t > N$
• Axiom 3 (Heredity principle): Evolution minimizes $K(\Sigma(t+1) | \Sigma(t))$ (Corominas-Murtra et al., 2016).

Statistical counterparts (e.g., Shannon entropy growth), however, are insufficient: standard information-theoretic mutual informations vanish over long OEE trajectories unless non-statistical (algorithmic) information is considered (Corominas-Murtra et al., 2016).

Decidability results clarify that strong forms of OEE (unbounded complexity growth under nontrivial complexity measures) require undecidability: the system’s adaptation times and evolving states cannot be computed or predicted in advance (Hernández-Orozco et al., 2016).

2. Fundamental Biological and Artificial Requirements

Tim Taylor articulated five necessary requirements for OEE in both natural and artificial evolutionary systems (Taylor, 2015):

1. Robust Reproductive Individuals: Self-maintenance and error resistance sufficient to ensure individuals can reproduce despite perturbations; e.g., DNA repair, redundancy, and write-protection mechanisms.
2. A Medium Allowing Unlimited Diversity: An evolutionary substrate (“physics” or interaction medium) supporting an effectively unlimited diversity of individuals, structures, and interactions, preferably across multiple scales.
3. Capacity for Producing More Complex Offspring: Mechanisms, such as von Neumann’s architecture or multi-parental processes (horizontal gene transfer, endosymbiosis), that allow offspring with greater organizational or functional complexity than their parents.
4. Mutational Pathways to Other Viable Individuals: Fitness and genotype–phenotype mappings with neutral networks, modularity, redundancy, and degeneracy, enabling evolutionary “stepping stones” across phenotype space to avoid stasis in local optima.
5. Drive for Continued Evolution: Intrinsic or extrinsic mechanisms (co-evolution, shifting ecological/fitness landscapes, resource cycling, niche construction) that maintain perpetual selective pressures and prevent evolutionary stagnation.

These requirements are not independent: robust reproduction provides a population base for exploration; only a rich, open medium allows complexity to increase and new pathways to be continually discovered; the drive for ongoing change ensures the persistent occupation and traversal of the medium (Taylor, 2015).

3. Mechanisms and Formal Models Underlying OEE

Canonical OEE-enabling mechanisms include:

• Cardinality Leaps: The ability to form higher-order entities (multisets, aggregates, composites) causes discontinuous increases (“cardinality leaps”) in the size of the possibility space: for a base set S of countable size, forming multisets yields $\mathbb{N}^{|S|}$ or $2^{\aleph_0}$ for uncountable S, permitting continuous novelty discovery (Sayama, 2018).
• Transdomain Bridges and Non-Additive Composition: Innovations can arise from components bridging multiple domains (mechanical/electrical/chemical) or via non-additive combinatorial assemblies producing emergent behaviors, thus expanding organizational/action space and enabling both expansive and transformational OEE (Taylor, 2018, Taylor, 2021).
• Intrinsic Implementation: Evolutionary processes (generation, evaluation, reproduction) should be realized intrinsically, i.e., emergent and evolvable within the system, rather than as static, extrinsic code. This allows the evolution of evolvability, major transitions, and rule discovery (Taylor, 2018).
• Environmental and Population Complexity: Rich, structured environments (resource gradients, multi-modal “physics,” spatial/ecological complexity) and population mechanisms (co-evolutionary arms races, ecosystem engineering, spatial structure) underpin both OEE and the emergence of new levels of individuality (Packard et al., 2019, Taylor, 2015).
• Hierarchical Constraint and Ratcheting: Hierarchies of constraints (completely conserved, quasi-conserved, conditionally conserved) enable local emergent rules and facilitate the evolution of new rule sets from within, i.e. rule-open-endedness (Adams et al., 2 Jun 2024).

4. Empirical Systems and OEE in Practice

OEE has been studied in a variety of experimental and empirical systems:

• Artificial Chemistries and Digital Evolution: The “Hash Chemistry” model demonstrates linear, unbounded novelty production (new multisets) via a cardinality leap, conditional on selection (deterministic hash-based fitness) (Sayama, 2018). Interactive systems such as Picbreeder.org reveal that human-guided, objective-free evolution in rich genotype–phenotype spaces can spontaneously induce canalization, modularity, and deep evolvability (Huizinga et al., 2017). Predator–prey robotics systems show OEE can arise in embodied contexts when reproduction is conditioned on explicit behavioral successes (Kachler et al., 2023).
• Cultural, Technological, and Web Ecosystems: Human culture and technological systems exhibit OEE via the co-evolution of a technological (artifact/skill repertoire) system and a search space (of needs/goals)—with positive feedback and a balance of stochasticity and selection being crucial for unbounded complexity growth (Winters et al., 6 Aug 2025). Social-tagging websites (e.g., Flickr) display combinatorial OEE proportional to the exponential growth of possible tag combinations and continual meaning-drift of tags (Ikegami et al., 2019).
• Decentralized On-Chain Agents: Spore.fun deploys AI agents with on-chain heritable “genomes” and economic selection, operationalized on blockchains with Trusted Execution Environments. This setup yields observable bursts of lineage diversification and innovation akin to OEE, although subject to environmental (API, platform) limitations (Hu et al., 24 May 2025).

5. Major Transitions, Evolution of Evolvability, and Hierarchy

True OEE is inseparable from the possibility of major evolutionary transitions—events where new levels of individuality emerge through the fusion, fission, and cooperation of pre-existing components (Turney, 2018). Turney’s six conditions for OEE—variation, heredity, selection, fission, fusion, and cooperation—provide a minimal declarative class for algorithms supporting continual increases in organizational complexity, facilitating endless reshaping of the unit of selection across hierarchies. These processes must themselves be evolvable, allowing meta-evolution of the evolutionary rules, operators, and selection regimes (Turney, 2018).

The “evolution of evolvability” is pivotal: mechanisms such as canalization, modularity, and hierarchy (arising in open-ended and divergent, rather than directed, search) enable the selective entrenchment of useful directions for subsequent evolutionary change, thereby accelerating innovation rates and preventing evolutionary stasis (Huizinga et al., 2017).

6. Theoretical Insights, Open Problems, and Future Directions

Research on OEE has yielded several key theoretical and practical insights:

• Information Theory Limits: The statistical (Shannon) signature of OEE, such as Zipf's law, masks a paradox: stepwise mutual information decays, erasing all trace of early historical content, while only algorithmic (non-statistical) information-theoretic measures retain the accumulation of historical structure (Corominas-Murtra et al., 2016).
• Computational Irreducibility and Undecidability: Systems capable of open-ended, strong, unbounded evolutionary complexity must entail undecidable aspects; any computable or static framework imposes strict complexity growth ceilings (Hernández-Orozco et al., 2016).
• Rule-Open-Endedness and Emergent Laws: A current frontier is engineering systems that endogenously invent new update laws (“rules for the rules”), not merely traverse new states, by leveraging emergent, hierarchically interacting, quasi- and conditional-conservation constraints maintained internally rather than externally (Adams et al., 2 Jun 2024).
• Design Principles and Benchmarks: Intrinsic implementation of core evolutionary processes, support for compositional/hierarchical building blocks and higher-order aggregates, dynamic, multi-scale environments, and explicit support for meta-evolution are recognized as necessary design features (Taylor, 2018, Packard et al., 2019). Benchmarking for “sustained evolutionary activity” and “interestingness” remains an open challenge (Packard et al., 2019).
• Control, Predictability, and Safety in OEE Systems: OEE’s creative autonomy brings intrinsic unpredictability and loss of design-time control, raising unique safety challenges (goal misalignment, specification drift, emergence of unanticipated behaviors). Safe OEE research seeks formal frameworks for meta-incentive learning, risk-constrained discovery, and interpretability in open-ended search (Ecoffet et al., 2020).

Open questions concern the generality and sufficiency of various OEE enablers, formalization of what “interesting” or “adaptive” novelty is, development of automated metrics for canalization and major transitions, and the mechanisms by which systems can evolve their own open-endedness (evolved OEE) (Borg et al., 2022).

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Key References:

• (Taylor, 2015) Tim Taylor, "Requirements for Open-Ended Evolution in Natural and Artificial Systems"
• (Taylor, 2018) Tim Taylor, "Evolutionary Innovations and Where to Find Them: Routes to Open-Ended Evolution in Natural and Artificial Systems"
• (Sayama, 2018) Sayama, "Cardinality Leap for Open-Ended Evolution"
• (Huizinga et al., 2017) Huizinga et al., "The Emergence of Canalization and Evolvability in an Open-Ended, Interactive Evolutionary System"
• (Corominas-Murtra et al., 2016) Corominas-Murtra et al., "Zipf's law, unbounded complexity and open-ended evolution"
• (Hernández-Orozco et al., 2016) Hernández-Orozco et al., "Undecidability and Irreducibility Conditions for Open-Ended Evolution and Emergence"
• (Taylor, 2021) Taylor, "Evolutionary Innovation Viewed as Novel Physical Phenomena and Hierarchical Systems Building"
• (Packard et al., 2019) Packard et al., "An Overview of Open-Ended Evolution"
• (Adams et al., 2 Jun 2024) Muirhead et al., "An Open-Ended Approach to Understanding Local, Emergent Conservation Laws in Biological Evolution"
• (Winters et al., 6 Aug 2025) Winters & Charbonneau, "Modelling the emergence of open-ended technological evolution"
• (Borg et al., 2022) Borg et al., "Evolved Open-Endedness in Cultural Evolution"
• (Hu et al., 24 May 2025) Spore.fun case paper, "Sovereign Agent Open-ended Evolution on Blockchain with TEEs"
• (Ecoffet et al., 2020) Ecoffet et al., "Open Questions in Creating Safe Open-ended AI: Tensions Between Control and Creativity"
• (Turney, 2018) Turney, "Conditions for Major Transitions in Biological and Cultural Evolution"