Open-Endedness & Levels of Novelty

Updated 21 August 2025

Open-endedness is the capacity for unbounded complexity growth in systems, achieved through continual recombination and exploration of new state spaces.
Levels of novelty—exploratory, expansive, and transformational—distinguish incremental variations from paradigm-shifting innovations in evolving systems.
Statistical signatures like Zipf’s law and algorithmic measures provide practical insights for designing artificial systems that mimic natural open-ended evolution.

Open-endedness is defined as the potential for a system to undergo an indefinite and unbounded increase in complexity, continually generating novel states, behaviors, or artifacts. In both biological and technological contexts, open-endedness encapsulates the capacity for combinatorial innovation, continual exploration of new state spaces, and the generative expansion of possibilities. Closely tied to open-endedness are "levels of novelty," which distinguish the incremental discovery of new variants within a fixed model (exploratory), the expansion into new regions of state or phenotype space (expansive), and the introduction of entirely new conceptual domains or mechanisms that transform the system’s fundamental dynamics (transformational).

1. Foundational Definitions and Formal Postulates

A rigorous foundation for open-endedness is provided by Algorithmic Information Theory (AIT), in which the complexity of a system’s state, denoted $\sigma_t$ , is measured by its Kolmogorov complexity $K(\sigma_t)$ —the length of the shortest program generating the state. Open-ended evolution (OEE) is defined by a set of postulates (Corominas-Murtra et al., 2016):

Open-Endedness (normalized complexity non-decreasing):

$\frac{K(\Sigma(t))}{t} \leq \frac{K(\Sigma(t+1))}{t+1}$

where $\Sigma(t)$ is the sequence of system states up to time $t$ .

Unboundedness: For all $N$ , there exists $t$ such that normalized complexity exceeds $N$ .
Heredity Principle: Evolution tends to minimize the conditional complexity between consecutive states,

$K(\sigma_{t+1}|\sigma_t)$

reflecting copying, re-use, and small-step tinkering.

Open-ended systems are characterized not by convergence to equilibrium but by indefinite complexity growth. Re-use and recursion—allowing combinatorial explosion of state space—are critical generative mechanisms. Practical examples include biological genome expansion, increasing protein domain architectures, and recursive generative grammars in language (Corominas-Murtra et al., 2016, Taylor, 2018).

2. Levels of Novelty and Expansion of State Space

Novelty within open-ended systems is classified into levels—each with clear operational implications and consequences for the evolutionary trajectory (Taylor, 2018, Taylor, 2021, Taylor, 2020):

Novelty Type	Definition	Illustration
Exploratory	Discovering new variants within fixed rules/model	Parameter changes: e.g., varying limb length
Expansive	Expanding into new but structurally similar spaces	New regions, e.g., evolving terrestrial gaits
Transformational	Creating new models/rules—meta-model shift	Evolution of flight, emergence of language

Exploratory novelty refers to recombination or optimization within existing rules—a trajectory through a fixed, yet potentially huge, phenotype space. Expansive novelty arises when constraints or mechanisms are altered so that new types or domains become accessible, but the underlying meta-model is intact. Transformational novelty emerges with the introduction of fundamentally new conceptual mechanisms or domains, so the rules or meta-model of the system themselves become subject to change. For instance, a biological innovation like the origination of cell differentiation transforms organizational possibilities (Taylor, 2018, Taylor, 2021).

These distinctions are operationalized by how evolutionary mechanisms are implemented:

Intrinsic implementations (evolvable, encoded within the system) permit the nature of generation, evaluation, or variation operators to themselves evolve, thus enabling metamodel change and deep expansion of search space.
Extrinsic implementations (hard-wired) constrain the system to exploratory novelty only (Taylor, 2018, Taylor, 2020).

3. Statistical Structure and Zipf’s Law as a Signature of Open-endedness

A recurring empirical and theoretical observation is that complex open-ended systems—biological, technological, and linguistic—exhibit Zipf’s law in their size/frequency distribution: $p(s_i) = \frac{1}{Z} i^{-\gamma},\qquad \gamma\approx 1$ where $p(s_i)$ is the frequency of the $i$ th element (e.g., word, protein domain, technological module) (Corominas-Murtra et al., 2016). The emergence of Zipf’s law is not accidental but follows from principled variational arguments: when conditional complexity between states is minimized (heredity) and the process is unbounded, minimizing Kullback–Leibler divergence between consecutive state distributions yields a recurrence relation that, asymptotically, produces $p(i)\sim 1/i$ .

This statistical footprint is interpreted as evidence that a system possesses the generative mechanism to continually expand state space via recombination, tinkering, and hierarchal composition—core features of open-endedness (Corominas-Murtra et al., 2016).

4. Algorithmic vs. Statistical Information: Conservation and Paradox

A critical insight addresses the paradox in information transmission through open-ended evolution. While standard (Shannon) statistical information—e.g., mutual information $I(X_n\!:\!X_m)$ between states separated by long time intervals—tends toward zero in an open-ended system (due to growing normalization and entropy), algorithmic information (Kolmogorov-based) is preserved at the programmatic/structural level (Corominas-Murtra et al., 2016): $I(\sigma_N:\sigma_n) = K(\sigma_N) - K(\sigma_N|\sigma_n)$

$K(\sigma_{t+1}|\sigma_t) \geq |K(\sigma_{t+1}) - K(\sigma_t)| + O(1)$

Thus, while statistical redundancy decays (entropic information is "lost"), the compressed, generative (suffix-program) information—capturing the actual history of novelty production—remains. The implication is that standard Shannon information theory is incomplete for capturing persistence and gain of information content in open-ended systems; Algorithmic Information Theory offers a better formalization (Corominas-Murtra et al., 2016).

5. Mechanisms and Conditions Underpinning Higher Levels of Novelty

The capacity for expansive or transformational open-endedness rests on several factors (Taylor, 2018, Taylor, 2021):

Multiple Domains and Transdomain Bridges: Biological and technological components may have properties in more than one domain; exaptation (repurposing of features) acts as a "bridge" to new behaviors.
Non-additive Compositionality: When the function of a structure is not the sum of its parts, small configuration changes can trigger emergent properties, introducing novel behaviors unattainable in additive systems.
Hierarchical Systems Building: Persistent, stable composite systems (e.g., multicellularity, technological platforms) allow layered innovation, which is key for transformational novelty (Taylor, 2021).
Intrinsic Evolution of Generative/Evaluative Processes: Systems where mapping from genotype to phenotype, evaluation, and mechanisms of reproduction can evolve are more likely to generate unexplored regions of search space and enable meta-level innovation (Taylor, 2018, Taylor, 2020).

6. Broader Implications: Predictability, Universality, and System Design

The presence of open-endedness has profound consequences for predictability, universality, and the design of artificial or evolved systems:

Predictability Limits: Although statistical regularities (such as size/frequency distributions) may be predictable, the actual identities of emergent novelties are fundamentally unpredictable due to combinatorial explosion and the "no free lunch" constraints (Corominas-Murtra et al., 2016).
Universality: The presence of Zipf’s law and combinatorial innovation across domains (language, proteins, technology) indicates universal principles underpinning open-endedness (Corominas-Murtra et al., 2016, Taylor, 2021).
Design Strategies: To engineer or simulate open-ended systems, it is necessary to (1) embed evolvability at multiple process levels (generation, evaluation, reproduction), (2) encourage modularity and recombination, and (3) establish generative grammars or rule-based systems rich enough to allow state space expansion and bridging across domains (Taylor, 2018, Taylor, 2021).
Measuring Open-endedness: Algorithmic and variational measures (Kolmogorov complexity growth, mutual information at the structural level, and statistical scaling laws) serve as diagnostics for determining whether a system is achieving open-endedness.

7. Conclusion

Open-endedness is the property of a system to achieve indefinitely increasing complexity and continual novelty, governed by a dynamic interplay of recombination, modularity, heredity, and expansion of conceptual or practical state space. The phenomenon is quantitatively linked to statistical distributions such as Zipf’s law and formalized by algorithmic information principles, which differentiate it from purely statistical models of information transmission. Levels of novelty—exploratory, expansive, and transformational—operationalize the richness and scope of open-ended evolution. Realizing open-endedness in artificial systems requires intrinsic evolvability, non-additive composition, domain-bridging components, and mechanisms that facilitate not only local search but also meta-level (transformational) innovation. Theoretical and empirical evidence shows that open-endedness underlies the emergence of complexity and diversity across biological, technological, and linguistic domains, providing a unifying framework for the paper and engineering of evolving systems (Corominas-Murtra et al., 2016, Taylor, 2018, Taylor, 2021, Taylor, 2020).