Software in the natural world: A computational approach to hierarchical emergence (2402.09090v2)

Abstract: Understanding the functional architecture of complex systems is crucial to illuminate their inner workings and enable effective methods for their prediction and control. Recent advances have introduced tools to characterise emergent macroscopic levels; however, while these approaches are successful in identifying when emergence takes place, they are limited in the extent they can determine how it does. Here we address this limitation by developing a computational approach to emergence, which characterises macroscopic processes in terms of their computational capabilities. Concretely, we articulate a view on emergence based on how software works, which is rooted on a mathematical formalism that articulates how macroscopic processes can express self-contained informational, interventional, and computational properties. This framework establishes a hierarchy of nested self-contained processes that determines what computations take place at what level, which in turn delineates the functional architecture of a complex system. This approach is illustrated on paradigmatic models from the statistical physics and computational neuroscience literature, which are shown to exhibit macroscopic processes that are akin to software in human-engineered systems. Overall, this framework enables a deeper understanding of the multi-level structure of complex systems, revealing specific ways in which they can be efficiently simulated, predicted, and controlled.

• The paper introduces a novel framework that models macroscopic emergent processes as nested, self-contained computational layers.
• It employs information-theoretic and automata principles to establish causal and computational closure, enabling accurate predictions of system behavior.
• Applications to models like cellular automata and neural simulations demonstrate the framework's potential for advancing real-world system analysis.

Essay on "Software in the Natural World: A Computational Approach to Hierarchical Emergence"

The paper "Software in the Natural World: A Computational Approach to Hierarchical Emergence" introduces a novel computational framework aimed at characterizing emergent macroscopic processes in complex systems through the lens of software-like processes. This scholarly work addresses a critical limitation in current emergence studies: while existing methods may identify when emergence occurs, they often fail to elucidate how emergent properties arise. By proposing an approach that integrates information-theoretic methods with principles from theoretical computer science, particularly automata theory and computational mechanics, this research offers a refined perspective on how emergence operates.

The authors contend that emergent macroscopic processes can be construed as collections of nested, self-contained informational, interventional, and computational properties. This framework captures these processes' essence by considering their capacity to perform computations independently of the underlying physical substrate. Furthermore, it reveals the structure of complex systems as hierarchical lattices of computational processes, thus providing insightful apparatus for simulating, predicting, and controlling complex system dynamics.

Summary of Key Points and Results

1. Causal and Computational Closure: The paper defines "causally closed" macroscopic processes as those whose ϵ-machines (capturing the process at a specific level of detail) are equivalent to their υ-machines (capturing the process's causation from finer-grained states). This equivalence implies that all relevant macroscopic distinctions are accessible without lower-level details. Computational closure similarly captures this autonomy by ensuring macro-level descriptions retain deterministic transitions after coarse-graining states and inputs.
2. Information-Theoretic Foundation: Information closeness is employed to ensure macro-level processes can predict themselves without additional microscopic insights. This property is shown to be equivalent to causal closure under certain conditions, providing a robust basis for discerning truly emergent behaviors.
3. Hierarchy and Multi-Level Structures: The paper identifies that computationally closed systems offer a compelling hierarchical organization. These can be seen as lattices where different levels correspond to progressively coarsened but information-sufficient descriptions, akin to renormalization concepts in physics.
4. Application and Case Studies: The framework is illustrated through various case studies, including models like cellular automata, diffusion processes (Ehrenfest model), Ising models with Glauber dynamics, random walks on modular networks, and simulated neural processes. For instance, the analysis of the Ehrenfest model underlines that its macroscopic dynamics (such as particle distributions) can be understood and controlled independently of microscopic configurations, demonstratively running "software" at an emergent level.
5. Implications and Future Directions: The framework's general applicability opens avenues for developing efficient algorithms to identify emergent behaviors in real-world systems through lumpability and coarse-graining strategies. This capability offers promising prospects for theoretical advancements and practical applications in fields that require understanding complex, multi-level dynamics, such as neuroscience, sociology, and ecosystem management.

In sum, this paper provides a rigorous mathematical structure for analyzing emergent processes, promoting a computational view of emergence that parallels human-engineered software systems. Going forward, this conceptual framework could significantly influence our ability to model and manage complex systems, facilitating advancements in both scientific understanding and technological innovation. Future research may focus on refining these techniques for application in non-stationary environments or systems characterized by continuous-variable dynamics, thus broadening the scope of emergent phenomena that can be quantitatively apprehended.