Compression is Comprehension, and the Unreasonable Effectiveness of Digital Computation in the Natural World (1904.10258v3)

Abstract: Chaitin's work, in its depth and breadth, encompasses many areas of scientific and philosophical interest. It helped establish the accepted mathematical concept of randomness, which in turn is the basis of tools that I have developed to justify and quantify what I think is clear evidence of the algorithmic nature of the world. To illustrate the concept I will establish novel upper bounds of algorithmic randomness for elementary cellular automata. I will discuss how the practice of science consists in conceiving a model that starts from certain initial values, running a computable instantiation, and awaiting a result in order to determine where the system may be in a future state--in a shorter time than the time taken by the actual unfolding of the phenomenon in question. If a model does not comply with all or some of these requirements it is traditionally considered useless or even unscientific, so the more precise and faster the better. A model is thus better if it can explain more with less, which is at the core of Chaitin's "compression is comprehension". I will pursue these questions related to the random versus possibly algorithmic nature of the world in two directions, drawing heavily on the work of Chaitin. I will also discuss how the algorithmic approach is related to the success of science at producing models of the world, allowing computer simulations to better understand it and make more accurate predictions and interventions.

• The paper demonstrates how algorithmic compression, rooted in Chaitin’s theory, reveals the computational structure underlying natural processes.
• It introduces Algorithmic Information Dynamics and employs CTM and BDM to empirically bound and measure complexity in deterministic systems.
• The findings offer practical insights for modeling biological systems and advancing AI by reprogramming discrete computational processes.

Compression is Comprehension: The Unreasonable Effectiveness of Digital Computation in the Natural World

Hector Zenil's paper, "Compression is Comprehension: The Unreasonable Effectiveness of Digital Computation in the Natural World," leverages the foundational work of Greg Chaitin to argue for and demonstrate the algorithmic nature of the universe. The paper, stemming from a talk at the University of Barcelona, interweaves theoretical insights with empirical investigations, primarily focusing on the concept of algorithmic randomness and its relation to natural phenomena.

Theoretical Foundations

Central to this work is the influential concept of algorithmic probability, particularly Chaitin's Omega ($\Omega$) number, which encapsulates the likelihood of a random program halting. This idea stems from Chaitin's exploration into the nature of randomness using Turing machines and Kolmogorov complexity. Zenil further builds on Chaitin's premise that the universe can be seen as comprehensible through compression—explaining more with less—by examining systems such as elementary cellular automata (ECA).

Cellular Automata and Algorithmic Complexity

Zenil empirically explores the bounds of algorithmic randomness in ECAs to invoke a deep understanding of how simple rules can generate complex behaviors. By establishing upper bounds of algorithmic randomness, Zenil underscores that even the simplest deterministic systems, when viewed through the lens of Chaitin's theory, can display a vast spectrum of behavior from highly regular to seemingly random.

Practical Implications of Compression

The paper examines how scientific modeling itself leverages algorithmic principles. The essence of creating effective models involves compressing observations into the simplest possible form, a key principle in both theoretical computer science and practical science.

Algorithmic Information Dynamics

One of the novel aspects Zenil introduces is Algorithmic Information Dynamics (AID). This method characterizes the evolution of complexity over time, particularly useful in studying dynamic systems. By focusing on perturbations and their effects, AID can uncover underlying generating mechanisms in both artificial and natural processes.

Empirical Methods: CTM and BDM

Zenil elaborates on two empirical techniques central to the analyses:

1. Coding Theorem Method (CTM): This approach leverages the relationship between the frequency of a string produced by a random program and its algorithmic complexity.
2. Block Decomposition Method (BDM): This method involves decomposing complex structures into smaller, analyzable components to approximate their algorithmic complexity.

Application to Biological Systems

Zenil asserts the utility of these methods in fields beyond theoretical computation, extending into biological systems. By interpreting biological processes as computational, discrete mechanisms akin to cellular automata, Zenil opens avenues for reprogramming biological entities at a molecular level, mirroring computational reprogramming techniques.

Relevance to Intelligence and AI

An intriguing extension of Zenil's work touches on human cognition and AI. He explores how algorithmic information theory might explain peaks in human cognitive capabilities and how these principles can inform the development of artificial intelligence that better mirrors human reasoning, particularly in causal inference and symbol manipulation.

Implications for Future Research

Zenil's exploration consolidates foundational theories in algorithmic information theory with practical insights, suggesting a framework for future research across scientific disciplines. The implications are twofold:

1. Theoretical Understanding: Reinforcing the view of the universe as fundamentally algorithmic, providing a coherent explanation of the ordered and comprehensible nature of physical laws.
2. Practical Applications: Enabling advancements in fields such as biology, AI, and complex systems through methods that bridge empirical observations and computational models.

Conclusion

Zenil's contribution, rooted deeply in Chaitin's and Turing's pioneering work, elegantly argues for the inherent simplicity underpinning apparent complexity in the natural world. His empirical validations and methodological innovations collectively assert that comprehension indeed lies in compression, thus promoting an algorithmic paradigm in both understanding and manipulating natural phenomena.