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

Abstract

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.

Appendix

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. 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. 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. 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. 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.