Complex systems research is becoming ever more important in both the natural and social sciences. It is commonly implied that there is such a thing as a complex system, different examples of which are studied across many disciplines. However, there is no concise definition of a complex system, let alone a definition on which all scientists agree. We review various attempts to characterize a complex system, and consider a core set of features that are widely associated with complex systems in (...) the literature and by those in the field. We argue that some of these features are neither necessary nor sufficient for complexity, and that some of them are too vague or confused to be of any analytical use. In order to bring mathematical rigour to the issue we then review some standard measures of complexity from the scientific literature, and offer a taxonomy for them, before arguing that the one that best captures the qualitative notion of the order produced by complex systems is that of the Statistical Complexity. Finally, we offer our own list of necessary conditions as a characterization of complexity. These conditions are qualitative and may not be jointly sufficient for complexity. We close with some suggestions for future work. (shrink)

According to the dominant computational approach in cognitive science, cognitive agents are digital computers; according to the alternative approach, they are dynamical systems. This target article attempts to articulate and support the dynamical hypothesis. The dynamical hypothesis has two major components: the nature hypothesis (cognitive agents are dynamical systems) and the knowledge hypothesis (cognitive agents can be understood dynamically). A wide range of objections to this hypothesis can be rebutted. The conclusion is that cognitive systems may well be dynamical systems, (...) and only sustained empirical research in cognitive science will determine the extent to which that is true. (shrink)

We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system's dynamics). We review the ideas of computational mechanics, (...) an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the ``causal states'' of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables. (shrink)

The functional complexity, or the number of functions, of organisms hasfigured prominently in certain theoretical and empirical work inevolutionary biology. Large-scale trends in functional complexity andcorrelations between functional complexity and other variables, such assize, have been proposed. However, the notion of number of functions hasalso been operationally intractable, in that no method has been developedfor counting functions in an organism in a systematic and reliable way.Thus, studies have had to rely on the largely unsupported assumption thatnumber of functions can be (...) measured indirectly, by using number ofmorphological, physiological, and behavioral parts as a proxy. Here, amodel is developed that supports this assumption. Specifically, the modelpredicts that few parts will have many functions overlapping in them, andtherefore the variance in number of functions per part will be low. If so,then number of parts is expected to be well correlated with number offunctions, and we can use part counts as proxies for function counts incomparative studies of organisms, even when part counts are low. Alsodiscussed briefly is a strategy for identifying certain kinds of parts inorganisms in a systematic way. (shrink)

This paper considers undecidability in the imitation game, the so-called Turing Test. In the Turing Test, a human, a machine, and an interrogator are the players of the game. In our model of the Turing Test, the machine and the interrogator are formalized as Turing machines, allowing us to derive several impossibility results concerning the capabilities of the interrogator. The key issue is that the validity of the Turing test is not attributed to the capability of human or machine, but (...) rather to the capability of the interrogator. In particular, it is shown that no Turing machine can be a perfect interrogator. We also discuss meta-imitation game and imitation game with analog interfaces where both the imitator and the interrogator are mimicked by continuous dynamical systems. (shrink)

The target article sketched and explored a mechanism (action-oriented predictive processing) most plausibly associated with core forms of cortical processing. In assessing the attractions and pitfalls of the proposal we should keep that element distinct from larger, though interlocking, issues concerning the nature of adaptive organization in general.

This article reviews the seven “visions” of evolution proposed by Depew and Weber , concluding that each posited relationship between natural selection and self-organization has suited different aims and approaches. In the second section of the article, we show that these seven viewpoints may be collapsed into three fundamentally different ones: natural selection drives evolution; self-organization drives evolution; and natural selection and self-organization are complementary aspects of the evolutionary process. We then argue that these three approaches are not mutually exclusive, (...) since each may apply to different stages of development of different systems. What emerges from our discussion is a more encompassing view: that self-organization proposes what natural selection disposes. (shrink)

The philosophy of blockchain technology is concerned, among other things, with blockchain ontology, how it might be characterised, how it is being created, implemented, and adopted, how it operates in the world, and how it evolves over time. This paper concentrates on whether Bitcoin/blockchain can be considered a complex system and, if so, whether it is a chaotic one. Beyond mere academic curiosity, a positive response would raise concerns about the likelihood of Bitcoin/blockchain entering a 2010-Flash-Crash-type of chaotic regime, with (...) catastrophic consequences for financial systems based on it. The paper starts by highlighting the relevant details of the Bitcoin/blockchain ecosystem formed by the blockchain itself, bitcoin end users, capital gains seekers, miners, full nodes maintainers, and developers, and their interactions. Then the Information Theory of Complex Systems is briefly discussed for later use. Finally, the blockchain is investigated with the help of Crutchfield's Statistical Complexity measure. The low non-null statistical complexity value obtained suggests that the blockchain may be considered algorithmically complicated but hardly a complex system and unlikely to enter a chaotic regime. (shrink)

There has been considerable debate in the literature about the relative merits of information processing versus dynamical approaches to understanding cognitive processes. In this article, we explore the relationship between these two styles of explanation using a model agent evolved to solve a relational categorization task. Specifically, we separately analyze the operation of this agent using the mathematical tools of information theory and dynamical systems theory. Information-theoretic analysis reveals how task-relevant information flows through the system to be combined into a (...) categorization decision. Dynamical analysis reveals the key geometrical and temporal interrelationships underlying the categorization decision. Finally, we propose a framework for directly relating these two different styles of explanation and discuss the possible implications of our analysis for some of the ongoing debates in cognitive science. (shrink)

The cybernetic definition of a living individual proposed previously (Korzeniewski, 2001) is very abstract and therefore describes the essence of life in a very formal and general way. In the present article this definition is reformulated in order to determine clearly the relation between life in general and a living individual in particular, and it is further explained and defended. Next, the cybernetic definition of a living individual is confronted with the real world. It is demonstrated that numerous restrictions imposed (...) on the cybernetic definition of life by physical reality imply a number of particular properties of life that characterize present life on Earth, namely: (1) a living individual must be a dissipative structure (and therefore a low-entropy thermodynamic system out of the state of equilibrium); (2) spontaneously-originated life must be based on organic compounds; (3) evolutionarily stable self-dependent, free-living individuals must have some minimal level of complexity of structure and function; (4) a living individual must have a record of identity separated from an executive machinery; (5) the identity of living individuals must mutate and may evolve; (6) living individuals may collect and accumulate information in subsequent generations over very long periods of time; (7) the degree of complexity of a living individual reflects the degree of complexity of its environment (ecological niche) and (8) living individuals are capable of supple adaptation to varying environmental conditions. Thus, the cybernetic definition of a living individual, when confronted with the real physical world, generates most of the general properties of the present life on Earth. (shrink)

A wide range of systems appear to perform computation: what common features do they share? I consider three examples, a digital computer, a neural network and an analogue route finding system based on soap-bubbles. The common feature of these systems is that they have autonomous dynamics — their states will change over time without additional external influence. We can take advantage of these dynamics if we understand them well enough to map a problem we want to solve onto them. Programming (...) consists of arranging the starting state of a system so that the effects of the system''s dynamics on some of its variables corresponds to the effects of the equations which describe the problem to be solved on their variables. The measured dynamics of a system, and hence the computation it may be performing, depend on the variables of the system we choose to attend to. Although we cannot determine which are the appropriate variables to measure in a system whose computation basis is unknown to us I go on to discuss how grammatical classifications of computational tasks and symbolic machine reconstruction techniques may allow us to rule out some measurements of a system from contributing to computation of particular tasks. Finally I suggest that these arguments and techniques imply that symbolic descriptions of the computation underlying cognition should be stochastic and that symbols in these descriptions may not be atomic but may have contents in alternative descriptions. (shrink)