The Intrinsic Cause-Effect Power of Discrete Dynamical Systems—From Elementary Cellular Automata to Adapting Animats

Larissa Albantakis,Giulio Tononi

doi:10.3390/e17085472

Abstract

Current approaches to characterize the complexity of dynamical systems usually rely on state-space trajectories. In this article instead we focus on causal structure, treating discrete dynamical systems as directed causal graphs—systems of elements implementing local update functions. This allows us to characterize the system’s intrinsic cause-effect structure by applying the mathematical and conceptual tools developed within the framework of integrated information theory (IIT). In particular, we assess the number of irreducible mechanisms (concepts) and the total amount of integrated conceptual information Φ specified by a system. We analyze: (i) elementary cellular automata (ECA); and (ii) small, adaptive logic-gate networks (“animats”), similar to ECA in structure but evolving by interacting with an environment. We show that, in general, an integrated cause-effect structure with many concepts and high Φ is likely to have high dynamical complexity. Importantly, while a dynamical analysis describes what is “happening” in a system from the extrinsic perspective of an observer, the analysis of its cause-effect structure reveals what a system “is” from its own intrinsic perspective, exposing its dynamical and evolutionary potential under many different scenarios.

Highlights

The term “dynamical system” encompasses a vast class of objects and phenomena—any system whose state evolves deterministically with time over a state space according to a fixed rule [1]
Our objectives here are to assess whether and how much certain isolated and adaptive discrete dynamical systems exist from their own intrinsic perspective and to determine how their cause-effect structures relate to their dynamic complexity
One hallmark of living systems is that they typically show a wide range of interesting behaviors, far away from thermodynamic equilibrium (e.g., [40])

Summary

Introduction

The term “dynamical system” encompasses a vast class of objects and phenomena—any system whose state evolves deterministically with time over a state space according to a fixed rule [1]. Even simple state-update rules can give rise to complex spatio-temporal patterns This has been demonstrated extensively using a class of simple, discrete dynamical systems called “cellular automata” (CA) [8,9,10,11,12]. The aim of dynamical systems theory is to understand and characterize a system based on its long-term behavior, by classifying the geometry of its long-term trajectories. In this spirit, cellular automata have been classified according to whether their evolution for most initial states leads to fixed points, periodic cycles, or chaotic patterns associated with strange attractors [8,13]

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Results

Conclusion