Original: A LITTLE TAXONOMY OF OPEN - ENDEDNESS
Revised: A LITTLE TAXONOMY OF OPEN - ENDEDNESS

Original: Anonymous authors

Original: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: This paper aims to provide a partial taxonomy of the ways that the term openendedness is used in Artificial Life, the ways that open-endedness is used outside of Artificial Life, open-endedness referred to via other terms.
Revised: This paper aims to provide a partial taxonomy of the ways that the term openendedness is used in Artificial Life, the ways that open-endedness is used outside of Artificial Life, open-endedness referred to via other terms.

Original: The  definitions for open-endednessfall in the conceptual or measuring categories. The conceptual categories are rich, while the measuring categories—and the actual measurements—tend to be simplistic.
Revised: The conceptual definitions for open-endedness are rich, while the operational definitions—and the actual measurements—tend to be simplistic.

Original: Related concepts that we describe include meta-learning, compositionality (in language),  creativity
Revised: Related concepts that we describe include meta-learning, compositionality (in language), and creativity

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: What is “open-endedness”?
Revised: What is “open-endedness”?

Original: Bedau et al.
Revised: Bedau et al.

Original: (2000)’s open problems in Artificial Life (ALife) refers to “open-ended evolution”, implying that open-endedness is a feature of evolutionary processes—or at least that it is the open-ended kind of evolutionary process that ALife is interested in.
Revised: (2000)’s open problems in Artificial Life (ALife) refers to “open-ended evolution”, implying that open-endedness is a feature of evolutionary processes—or at least that it is the open-ended kind of evolutionary process that ALife is interested in.

Original: Packard et al.
Revised: Packard et al.

Original: (2019a;b) describes how “open-endedness” over time becomes an important idea in its own right: although biology is believed to be open-ended and “creative production of ongoing novelty” tends to be called open-ended evolution, open-endedness is not necessarily biological or evolutionary.
Revised: (2019a;b) describes how “open-endedness” over time becomes an important idea in its own right: although biology is believed to be open-ended and “creative production of ongoing novelty” tends to be called open-ended evolution, open-endedness is not necessarily biological or evolutionary.

Original: Stepney (2021); Stanley (2019) describes open-endedness as a major goal and grand challenge in
Revised: Stepney (2021); Stanley (2019) describes open-endedness as a major goal and grand challenge in

Original: ALife.
Revised: ALife.

Original: Two general types of definitions for open-endedness currently exist in ALife.
Revised: Two distinct types of definitions for open-endedness currently exist in ALife: the conceptual and the operational.

Original: The conceptual type describes open-endednesson a spectrum (Banzhaf et al., 2016; Taylor, 2019); the measuring type describes open-endedness according to how it could be measured (Packard et al., 2019a;b; Hintze, 2019).
Revised: The conceptual definitions in some instances describe relative levels of open-endedness (Banzhaf et al., 2016; Taylor, 2019), and in other instances place open-endedness in categories depending on the context in which open-endedness is considered (Packard et al., 2019a;b; Hintze, 2019).

Original: Packard et al.
Revised: Packard et al.

Original: (2019a) notes that the term “open-endedness” is subjective and that it is best to identify which definitions of open-endedness—possibly multiple at once—is being used before diving into any discussion; it provides a partial list of definitions to start with.
Revised: (2019a) notes that the term “open-endedness” is subjective and that it is best to identify which definitions of open-endedness—possibly multiple at once—is being used before diving into any discussion; it provides a partial list of definitions to start with.

Original: This plurality of definitions for open-endedness is not only in ALife, but also in robotics, deep learning, computational creativity, and so on.
Revised: This plurality of definitions for open-endedness is not only in ALife, but also in robotics, deep learning, computational creativity, and so on.

Revised: The operational definitions are commonly described in the evaluation of open-ended systems; these operationalizations often refer to the same concepts of novelty, diversity, and complexity without further discussion of conceptual frameworks of open-endedness.

Original: We aim to provide a partial taxonomy of the ways that open-endedness is used in ALife, the ways that open-endedness is used outside of ALife, open-endedness referred to via other terms, as well as related but different concepts.
Revised: We aim to provide a partial taxonomy of the ways that open-endedness is used in ALife, the ways that open-endedness is used outside of ALife, open-endedness referred to via other terms, as well as related but different concepts.

Original: We are not attempting to provide a genealogy or a history.
Revised: We are not attempting to provide a genealogy or a history.

Original: Most of the papers cited are found via a forward-and-backward bibliography walk from Taylor (2019).
Revised: Most of the papers cited are found via a forward-and-backward bibliography walk from Taylor (2019).

Original: The intended contribution of this paper is to help researchers answer questions such as the following:
Revised: The intended contribution of this paper is to help researchers answer questions such as the following:

Original: • Is my goal open-endedness for open-endedness’ sake?
Revised: • Is my goal open-endedness for open-endedness’ sake?

Original: Which definition of openendedness?
Revised: Which definition of openendedness?

Original: Can this be implemented given certain computational resources?
Revised: Can this be implemented given certain computational resources?

Original: If the openendedness of an implementation is contingent, is it measurable?
Revised: If the openendedness of an implementation is contingent, is it measurable?

Original: How good of an approximation are my best measurements?
Revised: How good of an approximation are my best measurements?

Original: • Is my goal some other characteristics that having open-endedness would imply?
Revised: • Is my goal the implications of open-endedness? Does the implication logically or otherwise provably hold, or is just assumed by convention?

Original: Which definition of open-endedness actually implies those other characteristics?
Revised: What definition of open-endedness is required? Can the “implied characteristics” be directly measured?

Original: Can this be implemented given certain computational resources?
Revised: Can this be implemented given my computational resources?

Original: If the open-endedness of an implementation is contingent, is it measurable?
Revised: If the open-endedness of an implementation is contingent, is it measurable?

Original: Does the approximation of open-endedness using my best measurements imply those other desired characteristics?
Revised: Does the approximation of open-endedness using my best measurements imply those other desired characteristics?

Original: • Is my goal  one or more of the proxy characteristics used to measure open-endedness, novelty/diversity or complexity?
Revised: • Is my goal actually one or more of the proxy characteristics used to measure openendedness – novelty, diversity, complexity, or some other characteristic?

Original: Would the metrics used to measure novelty itself be the same as the metrics used to measure open-endedness-by-way-of-ongoing-novelty?
Revised: Would the metrics used to measure novelty itself be the same as the metrics used to measure open-endednessby-way-of-ongoing-novelty?

Revised: Why not directly measure novelty, diversity, complexity, etc.?

Original:  D EFINITIONS OF OPEN - ENDEDNESS 2. O PEN - ENDEDNESS IN ARTIFICIAL LIFE : CONCEPTUAL
Revised:  D EFINITIONS OF OPEN - ENDEDNESS 2. O PEN - ENDEDNESS IN ARTIFICIAL LIFE : CONCEPTUAL

Original: See Figure 1 for a diagram of the conceptual definitions discussed in the following paragraphs.
Revised: See Figure 1 for a diagram of the conceptual definitions discussed in the following paragraphs.

Original: Open-ended processes are ones that are ongoing and do not have a specific objective.
Revised: Open-ended processes are ones that are ongoing and do not have a specific objective.

Original: This ongoing change should be unbounded rather than asymptotic.
Revised: This ongoing change should be unbounded rather than asymptotic.

Original: This open-endedness applies to the phase of a dynamical system before it converges, but does not require the dynamical system to never converge.
Revised: This open-endedness applies to the phase of a dynamical system before it converges, but does not require the dynamical system to never converge.

Original: (Banzhaf et al., 2016; Taylor et al., 2016)
Revised: (Banzhaf et al., 2016; Taylor et al., 2016)

Original: Without hypothesizing a theoretical dynamical system that never converges, it is obvious that such a system is ultimately bounded by resources in the physical or simulated world.
Revised: Without hypothesizing a theoretical dynamical system that never converges, it is obvious that such a system is ultimately bounded by resources in the physical or simulated world.

Original: (Ray, 1991)
Revised: (Ray, 1991)

Original: When considering simulated open-ended processes, it should not matter if the processes start simple and grow into an environment that allows for complex processes, or if the processes and the environment co-evolve.
Revised: When considering simulated open-ended processes, it should not matter if the processes start simple and grow into an environment that allows for complex processes, or if the processes and the environment co-evolve.

Original: However, the environment should be complex enough to allow for the contained processes to be open-ended.
Revised: However, the environment should be complex enough to allow for the contained processes to be open-ended.

Original: (Taylor, 2015).
Revised: (Taylor, 2015).

Original: In practice, simulations are ultimately limited by the programming language used.
Revised: In practice, simulations are ultimately limited by the programming language used.

Original: Bedau et al.
Revised: Bedau et al.

Original: (1998) describes three classes of evolutionary processes: Class 1.
Revised: (1998) describes three classes of evolutionary processes: Class 1.

Original: no adaptations; Class 2.
Revised: no adaptations; Class 2.

Original: adaptations with bounded diversity (the number of atomic innovations in the system); Class 3.
Revised: adaptations with bounded diversity (the number of atomic innovations in the system); Class 3.

Original: adaptations with unbounded diversity.
Revised: adaptations with unbounded diversity.

Original: Class 2 corresponds to Banzhaf et al.
Revised: Class 2 corresponds to Banzhaf et al.

Original: (2016)’s variation, and
Revised: (2016)’s variation, and

Original: Class 3 corresponds to Banzhaf et al.
Revised: Class 3 corresponds to Banzhaf et al.

Original: (2016)’s innovation and emergence.
Revised: (2016)’s innovation and emergence.

Original: The diversity of a system could simply be thought of as the novelty of each entity in relation to the rest of the entities in the system aggregated in some fashion.
Revised: The diversity of a system could simply be thought of as the novelty of each entity in relation to the rest of the entities in the system aggregated in some fashion.

Original: Banzhaf et al.
Revised: Banzhaf et al.

Original: (2016) finds that various definitions of open-endedness in the literature boil down to novelty and complexity.
Revised: (2016) finds that various definitions of open-endedness in the literature boil down to novelty and complexity.

Original: Further, they find that novelty is a necessary but not sufficient requirement, while complexity is a sufficient but not necessary requirement.
Revised: Further, they find that novelty is a necessary but not sufficient requirement, while complexity is a sufficient but not necessary requirement.

Original: They extend the novelty requirement by adding the concept of the model and the meta-model, which results in three levels of novelty in relation to the most recent state of the model and meta-model: novelty within the model (variation), novelty that changes the model (innovation), and novelty that changes the meta-model (emergence).
Revised: They extend the novelty requirement by adding the concept of the model and the meta-model, which results in three levels of novelty in relation to the most recent state of the model and meta-model: novelty within the model (variation), novelty that changes the model (innovation), and novelty that changes the meta-model (emergence).

Original: An open-ended system is one that can continually generate innovation and emergence events.
Revised: An open-ended system is one that can continually generate innovation and emergence events.

Original: Taylor (2019) adapts the model in Banzhaf et al.
Revised: Taylor (2019) adapts the model in Banzhaf et al.

Original: and also describes three levels of novelty, but in relation to the initialization state of the model and meta-model: novelty within the model (exploratory), novelty that changes the model (expansive), and novelty that changes the meta-model (transformational).
Revised: and also describes three levels of novelty, but in relation to the initialization state of the model and meta-model: novelty within the model (exploratory), novelty that changes the model (expansive), and novelty that changes the meta-model (transformational).

Original: All three levels indicate some kind of open-endedness in this system, and the boundary between the levels can be fuzzy.
Revised: All three levels indicate some kind of open-endedness in this system, and the boundary between the levels can be fuzzy.

Original: The examples, given in a simulated creature evolution domain, are: from short limbs to long limbs (exploratory), from no limbs to limbs (expansive), and from running legs to flying wings (transformational).
Revised: The examples, given in a simulated creature evolution domain, are: from short limbs to long limbs (exploratory), from no limbs to limbs (expansive), and from running legs to flying wings (transformational).

Original: Additionally, the three levels may be evaluated in different contexts: the novelty of an event may be different in the genotype state space compared the the novelty of the same event in the phenotype state space.
Revised: Additionally, the three levels may be evaluated in different contexts: the novelty of an event may be different in the genotype state space compared the the novelty of the same event in the phenotype state space.

Original: The question of whether openendedness is an evaluation of process or outcome can be answered by the plurality of state spaces: the outcome is merely the mapping of the process in another state space.
Revised: The question of whether openendedness is an evaluation of process or outcome can be answered by the plurality of state spaces: the outcome is merely the mapping of the process in another state space.

Original: This also helps absorb some definitions of open-endedness into the categories discussed here.
Revised: This also helps absorb some definitions of open-endedness into the categories discussed here.

Original: For example, Channon (2019) uses three classifications: unbounded diversity of components but bounded adaptive success; bounded diversity of components but unbounded adaptive success; unbounded diversity of components and unbounded adaptive success.
Revised: For example, Channon (2019) uses three classifications: unbounded diversity of components but bounded adaptive success; bounded diversity of components but unbounded adaptive success; unbounded diversity of components and unbounded adaptive success.

Original: The components (often the genome) and adaptive success (behaviors resulting from phenotypes) can be interpreted using the plurality of state spaces.
Revised: The components (often the genome) and adaptive success (behaviors resulting from phenotypes) can be interpreted using the plurality of state spaces.

Original: Stepney (2021), also adapting Banzhaf et al.
Revised: Stepney (2021), also adapting Banzhaf et al.

Original: and taking inspiration from Kleppe et al.
Revised: and taking inspiration from Kleppe et al.

Original: (2003), suggests having levels beyond emergence—novelty that changes the (meta-)meta-meta model—as long as it’s possible to identify such meta-ness.
Revised: (2003), suggests having levels beyond emergence—novelty that changes the (meta-)meta-meta model—as long as it’s possible to identify such meta-ness.

Original: They suggest using a discontinuity in novelty value (or derivatives thereof) to identify the boundary between change to a model and change to a metamodel, making an analogy to phase change in physics.
Revised: They suggest using a discontinuity in novelty value (or derivatives thereof) to identify the boundary between change to a model and change to a metamodel, making an analogy to phase change in physics.

Original: Sommerer & Mignonneau (2000); Sayama (2019); Meyerson et al.
Revised: Sommerer & Mignonneau (2000); Sayama (2019); Meyerson et al.

Original: (2022) similarly propose the ideas “phase transition”, “cardinality leap”, and “miracle jump”, respectively.
Revised: (2022) similarly propose the ideas “phase transition”, “cardinality leap”, and “miracle jump”, respectively.

Original: Taylor et al.
Revised: Taylor et al.

Original: ; Packard et al.
Revised: ; Packard et al.

Original: (2019a) report a set of categories for open-ended evolution (the
Revised: (2019a) report a set of categories for open-ended evolution (the

Original: York Categories), the result of a workshop in York, UK on this topic.
Revised: York Categories), the result of a workshop in York, UK on this topic.

Original: According to their survey, definitions of open-endedness are usually some combinations of these categories, which fall in two meta-categories: the ongoing generation of novelty and the ongoing generation of complexity.
Revised: According to their survey, definitions of open-endedness are usually some combinations of these categories, which fall in two meta-categories: the ongoing generation of novelty and the ongoing generation of complexity.

Original: In their survey, definitions can have labels from both categories.
Revised: In their survey, definitions can have labels from both categories.

Original: The novelty aspect includes: (a) ongoing generation of new adaptations (within entities); (b) ongoing generation of new kinds of entities; (c) ongoing major transitions (Szathm ´ ary & Smith, 1995); (d) evolution of evolvability (Pattee & Sayama, 2019).
Revised: The novelty aspect includes: (a) ongoing generation of new adaptations (within entities); (b) ongoing generation of new kinds of entities; (c) ongoing major transitions (Szathm ´ ary & Smith, 1995); (d) evolution of evolvability (Pattee & Sayama, 2019).

Original: The first two types of novelty-based open-endedness (a, b) could map to the first two levels in Banzhaf et al.
Revised: The first two types of novelty-based open-endedness (a, b) could map to the first two levels in Banzhaf et al.

Original: ; Taylor (2019); Stepney (2021).
Revised: ; Taylor (2019); Stepney (2021).

Original: The third (c) can be interpreted as aggregating models and meta-models and valuing the both the whole and its internal layers and transitions all together.
Revised: The third (c) can be interpreted as aggregating models and meta-models and valuing the both the whole and its internal layers and transitions all together.

Original: The fourth (d) refers to evolution of the programmed evolution process, but can also be interpreted conceptually that there is always the possibility of wrapping existing aggregates and designating novelty one layer above that—depending on how one reads the “programmed evolution process”.
Revised: The fourth (d) refers to evolution of the programmed evolution process, but can also be interpreted conceptually that there is always the possibility of wrapping existing aggregates and designating novelty one layer above that—depending on how one reads the “programmed evolution process”.

Original: The complexity aspect includes: (a) ongoing growth of entity complexity and (b) ongoing growth of interaction complexity.
Revised: The complexity aspect includes: (a) ongoing growth of entity complexity and (b) ongoing growth of interaction complexity.

Original: One could also extend this to include the ongoing growth of complexity of the aggregate in dynamic hierarchies, in terms of additional levels of sub-entities in the aggregate as wells as more complex intra-entity interactions.
Revised: One could also extend this to include the ongoing growth of complexity of the aggregate in dynamic hierarchies, in terms of additional levels of sub-entities in the aggregate as wells as more complex intra-entity interactions.

Original: Packard et al.
Revised: Packard et al.

Original: (2019b) reports a revised set of categories (the Tokyo Categories), the result of a workshop in Tokyo, Japan on this topic.
Revised: (2019b) reports a revised set of categories (the Tokyo Categories), the result of a workshop in Tokyo, Japan on this topic.

Original: Those are the ongoing generation of: (a) interesting new kinds of entities and interactions, (b) evolution of evolvability, (c) major transitions, and (d) semantic evolution (Ikegami et al., 2019).
Revised: Those are the ongoing generation of: (a) interesting new kinds of entities and interactions, (b) evolution of evolvability, (c) major transitions, and (d) semantic evolution (Ikegami et al., 2019).

Original: The first three categories are supposed to subsume the ones in the York Categories.
Revised: The first three categories are supposed to subsume the ones in the York Categories.

Original: (d) refers to the shifting semantic information in a system, and has been identified from analyzing human-produced evolution processes.
Revised: (d) refers to the shifting semantic information in a system, and has been identified from analyzing human-produced evolution processes.

Original: However, taking Kockelman (2020)’s interpretation of Saussurean meaning, that the meaning of a sign is “constituted by the relation (of combination or substitution) between relations (qua signifier-signified pairings)”, (d) could be restated as “ongoing change in relation complexity”, which is not far removed from the “ongoing growth in relation complexity” used in the York Categories.
Revised: However, taking Kockelman (2020)’s interpretation of Saussurean meaning, that the meaning of a sign is “constituted by the relation (of combination or substitution) between relations (qua signifier-signified pairings)”, (d) could be restated as “ongoing change in relation complexity”, which is not far removed from the “ongoing growth in relation complexity” used in the York Categories.

Original: With (a), the authors account for the implication in a general understanding of open-endedness for interestingness rather than just novelty and complexity, while acknowledging that interestingness is a vague term that needs to be further specified in future work.
Revised: With (a), the authors account for the implication in a general understanding of open-endedness for interestingness rather than just novelty and complexity, while acknowledging that interestingness is a vague term that needs to be further specified in future work.

Original: It makes sense to collapse several categories from the York Categories,because in practice it is difficult to determine the line between the adaptation of an entity and a new kind of entity, etc. Conceptual treatment of open-endedness appears to come either in terms of mutually-exclusive categories (where individual evolutionary systems can straddle multiple categories) or levels (where each subsequent model level indicates some additional meta-quality compared to the previous level). Within the mutually-exclusive categories, each conceptual category could be associated with nov-
Revised: It makes sense to collapse several categories from the York Categories,

Original: novelty
Revised: novelty

Original: • Euclidean k-nearest neighbor in feature space  • count of immediate neighbors in a discrete feature space  • count of new adaptive innovations divided by the count of all current adaptive innovations • human eye
Revised: • Euclidean k-nearest neighbor in feature space (Lehman et al., 2008) • count of immediate neighbors in a discrete feature space (Cully & Demiris, 2017) • count of new adaptive innovations divided by the count of all current adaptive innovations (Bedau et al., 1998)

Original: diversity
Revised: diversity

Original: • Euclidean k-nearest neighbors in feature space  • Shannon complexity of the distribution of component size in the system  • total count of persisting components in the system  • total count of types of components in the system  • pair-wise Levenstein distance  • human eye 
Revised: • Euclidean k-nearest neighbors in feature space (Lehman et al., 2008; Lehman & Stanley, 2011b; Pugh et al., 2016; Mouret & Clune, 2015; Brant & Stanley, 2017) • Shannon complexity of the distribution of component size in the system (Bedau et al., 1998) • total count of persisting components in the system (Bedau et al., 1998) • total count of types of components in the system (Bedau et al., 1998) • pair-wise Levenstein distance (Hintze, 2019) • human eye (Hintze, 2019)

Original: complexity
Revised: complexity

Original: • grammar-based compression (approximation of Kolmogorov complexity) • sliding-window compression (approximation of Kolmogorov complexity) • jpeg compression  • (requires approximation)
Revised: • grammar-based compression (Mondol & Brown, 2021) • sliding-window compression (Hintze, 2019) • jpeg compression (Earle et al., 2021) • (requires approximation)

Original: Kolmogorov complexity—modification: 0 if sequence is random  • (requires approximation) statistical complexity  • Shannon complexity  • the count of atomic functions used to compose a more complex function (where the complex function has been generated by the program to start with)  • human eye 
Revised: Kolmogorov complexity—modification: 0 if sequence is random (Standish, 2003) • (requires approximation) statistical complexity (Stepney, 2021) • Shannon complexity (Standish, 2003) • the count of atomic functions used to compose a more complex function (where the complex function has been generated by the program to start with) (Wang et al., 2019) • human eye (Hintze, 2019)

Revised: Conceptual treatment of open-endedness appears to come either in terms of mutually-exclusive categories (where individual evolutionary systems can straddle multiple categories) or levels (where each subsequent model level indicates some additional meta-quality compared to the previous level).

Revised: Within the mutually-exclusive categories, each conceptual category could be associated with novelty, diversity, complexity, or some combination thereof; novelty, diversity, and complexity are in turn fuzzy terms that map to “new-ness” of entity characteristics and entity relations.

Revised: See Figure 1.

Original: 2. O PEN - ENDEDNESS IN ARTIFICIAL LIFE : MEASUREMENTS
Revised: 2. O PEN - ENDEDNESS IN ARTIFICIAL LIFE : MEASUREMENTS

Original: See Table 1 for a list of algorithms used to calculate novelty, diversity, and complexity.
Revised: See Table 1 for a list of algorithms used to calculate novelty, diversity, and complexity.

Original: In the papers reviewed, projects that involve actual calculations turn out to be one of the three York Categories: new entity adaptations, new kinds of entities, and entity complexity.
Revised: In the papers reviewed, projects that involve actual calculations turn out to be one of the three York Categories: new entity adaptations, new kinds of entities, and entity complexity.

Original: Bedau et al.
Revised: Bedau et al.

Original: (1998) suggests that diversity could be measured by a) Shannon complexity (Shannon & Weaver, 1949) of the distribution of sizes of components in the system, b) the count of persisting components in the system, c) the count of types of components in the system, or d) the count of new adaptive innovations divided by the count of all current adaptive innovations.
Revised: (1998) suggests that diversity could be measured by a) Shannon complexity (Shannon & Weaver, 1949) of the distribution of sizes of components in the system, b) the count of persisting components in the system, c) the count of types of components in the system, or d) the count of new adaptive innovations divided by the count of all current adaptive innovations.

Original: Using the diversity of the system at different points of time, calculated using the second metric, one could then calculate the adaptive evolutionary activity statistics, or ongoing diversity, of the system.
Revised: Using the diversity of the system at different points of time, calculated using the second metric, one could then calculate the adaptive evolutionary activity statistics, or ongoing diversity, of the system.

Original: The weakness of the system is that what is considered a component in the system has to be decided before hand, in order to count their activity.
Revised: The weakness of the system is that what is considered a component in the system has to be decided before hand, in order to count their activity.

Original: This weakness may be resolved by having an automated algorithm that takes as input the observation of a system and outputs deciding criteria for what should be considered a component in that system.
Revised: This weakness may be resolved by having an automated algorithm that takes as input the observation of a system and outputs deciding criteria for what should be considered a component in that system.

Original: Standish (2003) highlights the context-dependence of calculating complexity (e.g., whether it is based on genotypes or phenotypes) and proposes to measure complexity in general terms by using
Revised: Standish (2003) highlights the context-dependence of calculating complexity (e.g., whether it is based on genotypes or phenotypes) and proposes to measure complexity in general terms by using

Original: Shannon’s information complexity (Shannon & Weaver, 1949) or Kolmogorov’s algorithmic complexity (Kolmogorov, 1965).
Revised: Shannon’s information complexity (Shannon & Weaver, 1949) or Kolmogorov’s algorithmic complexity (Kolmogorov, 1965).

Original: One problem with the latter is that random sequences have maximum
Revised: One problem with the latter is that random sequences have maximum

Original: Kolmogorov complexity, which is contrary to the human intuition that random sequences are not interesting, therefore they do not feel complex.
Revised: Kolmogorov complexity, which is contrary to the human intuition that random sequences are not interesting, therefore they do not feel complex.

Original: To solve this problem, Standish (2003) changes the complexity of random sequences to 0, while keeping other parts of the Kolmogorov complexity the same.
Revised: To solve this problem, Standish (2003) changes the complexity of random sequences to 0, while keeping other parts of the Kolmogorov complexity the same.

Original: This is a simpler version of statistical complexity in Crutchfield (1994).
Revised: This is a simpler version of statistical complexity in Crutchfield (1994).

Original: Also consider the humble cockroach, which is said to be able to adapt to many more environmental events than humans can; consider the thought experiment of monkeys on typewriters having a miniscule but non-zero chance of generating Shakespeare’s poetry.
Revised: Also consider the humble cockroach, which is said to be able to adapt to many more environmental events than humans can; consider the thought experiment of monkeys on typewriters having a miniscule but non-zero chance of generating Shakespeare’s poetry.

Original: Both are examples of a seemingly simple descriptive/genotypical space that project into a seemingly richer behavioral/phenotypical space.
Revised: Both are examples of a seemingly simple descriptive/genotypical space that project into a seemingly richer behavioral/phenotypical space.

Original: If the Shakespearean poetry is possible from the random typing, should random typing really have a complexity of 0?
Revised: If the Shakespearean poetry is possible from the random typing, should random typing really have a complexity of 0?

Original: Perhaps complexity could be divided into potential complexity and structural complexity.
Revised: Perhaps complexity could be divided into potential complexity and structural complexity.

Original: Lehman & Stanley (2012) discusses impressiveness in open-ended evolutionary systems.
Revised: Lehman & Stanley (2012) discusses impressiveness in open-ended evolutionary systems.

Original: Impressiveness is closely related to interestingness and beauty; interestingness decreases over time, while impressiveness and beauty do not.
Revised: Impressiveness is closely related to interestingness and beauty; interestingness decreases over time, while impressiveness and beauty do not.

Original: They consider interestingness and beauty orthogonal to compressibility and make an analogy to P/NP, where an impressive product would be “easier to appreciate than to create”, correlating to an NP-complete problem in complexity theory.
Revised: They consider interestingness and beauty orthogonal to compressibility and make an analogy to P/NP, where an impressive product would be “easier to appreciate than to create”, correlating to an NP-complete problem in complexity theory.

Original: To measure the open-endedness of their system, the authors use the criterion of novelty, operationalized as sparsity.
Revised: To measure the open-endedness of their system, the authors use the criterion of novelty, operationalized as sparsity.

Original: More specifically, they calculate the novelty of an individual as the average Euclidean distance to the k-nearest neighbor of where that individual is located in the n-dimensional space of relevant features, also known as the behavioral characterization, or solution descriptors.
Revised: More specifically, they calculate the novelty of an individual as the average Euclidean distance to the k-nearest neighbor of where that individual is located in the n-dimensional space of relevant features, also known as the behavioral characterization, or solution descriptors.

Original: (Lehman et al., 2008)
Revised: (Lehman et al., 2008)

Original: In their experiment, they use the following eight features: brightness, BZip2 compression, wavelet compression, color variety, x-axis symmetry, y-axis symmetry, and choppiness.
Revised: In their experiment, they use the following eight features: brightness, BZip2 compression, wavelet compression, color variety, x-axis symmetry, y-axis symmetry, and choppiness.

Original: Several quality diversity algorithms Lehman & Stanley (2011b); Pugh et al.
Revised: Several quality diversity algorithms Lehman & Stanley (2011b); Pugh et al.

Original: ; Mouret & Clune (2015) also use sparsity in the n-dimensional feature space to determine diversity.
Revised: ; Mouret & Clune (2015) also use sparsity in the n-dimensional feature space to determine diversity.

Original: Brant & Stanley (2017) evaluates the minimum criterion coevolution algorithm on diversity, again via sparsity in the feature space where a relevant feature is the point in each agent’s trajectory at a given time step.
Revised: Brant & Stanley (2017) evaluates the minimum criterion coevolution algorithm on diversity, again via sparsity in the feature space where a relevant feature is the point in each agent’s trajectory at a given time step.

Original: Another quality diversity algorithm using the MAP-Elite multi-dimensional grid, Cully & Demiris (2017), calculates novelty using a simplification of sparsity: the count of the filled cells surrounding the location of the individual.
Revised: Another quality diversity algorithm using the MAP-Elite multi-dimensional grid, Cully & Demiris (2017), calculates novelty using a simplification of sparsity: the count of the filled cells surrounding the location of the individual.

Original: That is, the more filled cells there are, the less novel it is, which sounds a lot like a gravitational search algorithm Rashedi et al.
Revised: That is, the more filled cells there are, the less novel it is, which sounds a lot like a gravitational search algorithm Rashedi et al.

Original: (2009) without attraction forces.
Revised: (2009) without attraction forces.

Original: Hintze (2019)’s open-endedness includes diversity and complexity.
Revised: Hintze (2019)’s open-endedness includes diversity and complexity.

Original: They measure diversity with pairwise Levenstein distance (Levenshtein et al., 1966) and to measure complexity with Kolmogorov complexity (Kolmogorov, 1965) by proxy of compression (Ziv & Lempel, 1978).
Revised: They measure diversity with pair-wise Levenstein distance (Levenshtein et al., 1966) and to measure complexity with Kolmogorov complexity (Kolmogorov, 1965) by proxy of compression (Ziv & Lempel, 1978).

Original: Levenstein’s distance is a rough approximation of diversity as well as a rough approximation of novelty, so one could interpret this model as, again, reducing to novelty and complexity.
Revised: Levenstein’s distance is a rough approximation of diversity as well as a rough approximation of novelty, so one could interpret this model as, again, reducing to novelty and complexity.

Original: While they use sliding-window compression to approximate Kolmogorov complexity, some kind of grammar-based compression (Mondol & Brown, 2021) might be a better approximator of Kolmogorov complexity.
Revised: While they use sliding-window compression to approximate Kolmogorov complexity, some kind of grammar-based compression (Mondol & Brown, 2021) might be a better approximator of Kolmogorov complexity.

Original: Stepney (2021) complains that Kolmogorov complexity, even when accurately calculated, would be a measure of randomness rather than complexity in the general sense, and that it would be better to use something like statistical complexity to measure complexity.
Revised: Stepney (2021) complains that Kolmogorov complexity, even when accurately calculated, would be a measure of randomness rather than complexity in the general sense, and that it would be better to use something like statistical complexity to measure complexity.

Original: (Stepney, 2021)
Revised: (Stepney, 2021)

Original: Statistical complexity is similar to Standish (2003)’s modified Kolmogorov complexity, but more specifically states that Kolmogorov complexity ( L ) ≈ statistical complexity ( L ) + Shannon complexity ( L ) for a string L .
Revised: Statistical complexity is similar to Standish (2003)’s modified Kolmogorov complexity, but more specifically states that Kolmogorov complexity ( L ) ≈ statistical complexity ( L )+ Shannon complexity ( L ) for a string L .

Original: (Crutchfield, 1994)
Revised: (Crutchfield, 1994)

Original:  O PEN - ENDEDNESS IN OTHER FIELDS , BY OTHER NAMES
Revised:  O PEN - ENDEDNESS IN OTHER FIELDS , BY OTHER NAMES

Original: Lehman et al.
Revised: Lehman et al.

Original: ; Lehman & Stanley (2011a) come from the point of view of ALife and apply novelty to solve maze path-finding problems.
Revised: ; Lehman & Stanley (2011a) come from the point of view of ALife and apply novelty to solve maze path-finding problems.

Original: For the authors, complexity comes necessarily after a continuous novelty search, because there are a limited number of simple novel behaviors in this domain; and the complexity allows the path-finding problem to be solved.
Revised: For the authors, complexity comes necessarily after a continuous novelty search, because there are a limited number of simple novel behaviors in this domain; and the complexity allows the path-finding problem to be solved.

Original: They measure novelty using sparsity.
Revised: They measure novelty using sparsity.

Original: Earle et al.
Revised: Earle et al.

Original: (2021) uses jpeg compression as a proxy for complexity in the domain of simulated artificial life in video games.
Revised: (2021) uses jpeg compression as a proxy for complexity in the domain of simulated artificial life in video games.

Original: Wang et al.
Revised: Wang et al.

Original: (2019)that uses co-evolved environments and agents to solve terrain walker problems.
Revised: (2019) uses co-evolved environments and agents to solve terrain walker problems.

Original: They observe that the after some evolution, the agents are able to walk relatively gracefully (compared to algorithms) in more complex domains.
Revised: They observe that the after some evolution, the agents are able to walk relatively gracefully (compared to algorithms) in more complex domains.

Original: In an evolved environment like this, the count of smaller modular functions used to compose a certain terrain could be a good proxy for Kolmogorov complexity, although it is unclear if this is what the authors used.
Revised: In an evolved environment like this, the count of smaller modular functions used to compose a certain terrain could be a good proxy for Kolmogorov complexity, although it is unclear if this is what the authors used.

Original: In deep learning, approaches such as meta-learning Finn et al.
Revised: In deep learning, approaches such as meta-learning Finn et al.

Original: (2017) and one-shot learning Vinyals et al.
Revised: (2017) and one-shot learning Vinyals et al.

Original: (2016) appear to have comparable goals to open-endedness of being able to embody changes to the meta-model in a system without explicitly modeling the meta-model, although the term openendedness is not used, and the criteria for success are very different.
Revised: (2016) appear to have comparable goals to open-endedness of being able to embody changes to the meta-model in a system without explicitly modeling the meta-model, although the term openendedness is not used, and the criteria for success are very different.

Original: It might be said that the deep learning varieties are more focused on the product of their systems, while open-endedness as a research goal is an idea that focuses on the correctness of the architecture and process.
Revised: It might be said that the deep learning varieties are more focused on the product of their systems, while open-endedness as a research goal is an idea that focuses on the correctness of the architecture and process.

Original: Boden (2004)’s criteria for human creativity include surprise, novelty, and usefulness; couple this with Schmidhuber (2008)’s theory that usefulness and beauty correlate to compressibility, and the fact that compression is often used to estimate complexity, we arrive at a loose equivalence between creativity and open-endedness.
Revised: Boden (2004)’s criteria for human creativity include surprise, novelty, and usefulness; couple this with Schmidhuber (2008)’s theory that usefulness and beauty correlate to compressibility, and the fact that compression is often used to estimate complexity, we arrive at a loose equivalence between creativity and open-endedness.

Original: Similar to Bedau et al.
Revised: Similar to Bedau et al.

Original: ; Banzhaf et al.
Revised: ; Banzhaf et al.

Original: ; Taylor (2019),
Revised: ; Taylor (2019),

Original: Boden (2004)’s creativity has a triad of increasing meta-creativity: exploratory, combinatorial, and transformative creativity.
Revised: Boden (2004)’s creativity has a triad of increasing meta-creativity: exploratory, combinatorial, and transformative creativity.

Original: The claims of open-endedness are weaker than that of Boden (2004)’s human creativity, and requires one of novelty, diversity, and complexity, while having the contingency for the other qualities.
Revised: The claims of open-endedness are weaker than that of Boden (2004)’s human creativity, and requires one of novelty, diversity, and complexity, while having the contingency for the other qualities.

Original: Cellular automata (Von Neumann et al., 1966) such as the Game of Life correspond to exploratory or expansive open-endedness.
Revised: Cellular automata (Von Neumann et al., 1966) such as the Game of Life correspond to exploratory or expansive open-endedness.

Original: Relatedly, Mange et al.
Revised: Relatedly, Mange et al.

Original: (2004) implements Lindenmayer systems (L-systems) which are basically the expansion of context-free grammars.
Revised: (2004) implements Lindenmayer systems (L-systems) which are basically the expansion of context-free grammars.

Original: L-systems might exhibit exploratory open-endedness.
Revised: L-systems might exhibit exploratory open-endedness.

Original: A well-specified grammar (such the grammar in a natural language) could have compositionality, which is a desired goal in emergent language systems (Lowe et al., 2019).
Revised: A well-specified grammar (such the grammar in a natural language) could have compositionality, which is a desired goal in emergent language systems (Lowe et al., 2019).

Original: The idea of compositionality in emergent language systems is quite similar to openendedness in evolutionary systems: both are initially identified as co-existing with natural systems (language and biology, respectively) and deemed important, if not essential to those natural systems.
Revised: The idea of compositionality in emergent language systems is quite similar to openendedness in evolutionary systems: both are initially identified as co-existing with natural systems (language and biology, respectively) and deemed important, if not essential to those natural systems.

Original: The desires to research compositionality and open-endedness are both sometimes motivated by the idea that perhaps they would, in the simulated systems, lead to desirable properties from the natural systems.
Revised: The desires to research compositionality and open-endedness are both sometimes motivated by the idea that perhaps they would, in the simulated systems, lead to desirable properties from the natural systems.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: This paper presented a taxonomy of the ways that “open-endedness” is used in artificial life/openended evolution.
Revised: This paper presented a taxonomy of the ways that “open-endedness” is used in artificial life/openended evolution.

Original: We also make comparisons to similar concepts used in other areas.
Revised: We also make comparisons to similar concepts used in other areas.

Original: Some of those other areas, such as compositionality in emergent language research, seem to follow similar disjunction between the conceptual definitions, measuring definitions, and the actual implementation of measurements.
Revised: Some of those other areas, such as compositionality in emergent language research, seem to follow similar disjunction between the conceptual definitions, operational definitions, and the actual implementation of measurements.

Original: While it does not make an extensive survey, it is hoped that the taxonomy itself is relatively complete and provides illumination for future research
Revised: While it does not make an extensive survey, it is hoped that the taxonomy itself is relatively complete and provides illumination for future research.