Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos (1711.02456v2)

Abstract: In this paper we explore several fundamental relations between formal systems, algorithms, and dynamical systems, focussing on the roles of undecidability, universality, diagonalization, and self-reference in each of these computational frameworks. Some of these interconnections are well-known, while some are clarified in this study as a result of a fine-grained comparison between recursive formal systems, Turing machines, and Cellular Automata (CAs). In particular, we elaborate on the diagonalization argument applied to distributed computation carried out by CAs, illustrating the key elements of G\"odel's proof for CAs. The comparative analysis emphasizes three factors which underlie the capacity to generate undecidable dynamics within the examined computational frameworks: (i) the program-data duality; (ii) the potential to access an infinite computational medium; and (iii) the ability to implement negation. The considered adaptations of G\"odel's proof distinguish between computational universality and undecidability, and show how the diagonalization argument exploits, on several levels, the self-referential basis of undecidability.

• The paper analyzes how self-reference, particularly through diagonalization, underpins the emergence of undecidable dynamics across formal systems (like Gödel's incompleteness), Turing machines (like the Halting Problem), and Cellular Automata (like Rule 110).
• It identifies three key conditions necessary for generating undecidable dynamics: program-data duality, infinitely large computational media, and the capability for negation.
• The findings offer practical implications for designing complex adaptive systems and AI by understanding the role of self-reference in emergent phenomena, while also contributing theoretically to computation and complexity science.

Overview of Self-referential Basis of Undecidable Dynamics

In the field of complex systems and computation, the intersections between formal logic, algorithms, and dynamical systems are pivotal for understanding fundamental issues of undecidability, universality, and self-reference. The paper "Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos" by Mikhail Prokopenko et al. explores these intersections by comparing recursive formal systems, Turing machines, and Cellular Automata (CAs). The authors elaborate on how undecidable dynamics can emerge in computational frameworks and clarify existing interconnections through the lens of diagonalization and self-reference.

Conceptual Foundations

The exploration begins by establishing the core concepts of universality and undecidability within computational systems. Turing's Halting Problem is noted as the quintessential example of undecidability in computability theory, where no algorithm can determine, for all inputs, whether a Turing machine will eventually halt. Analogously, Gödel's Incompleteness Theorems reveal that self-referential formal systems like arithmetic are inherently incomplete, as they cannot prove all truths expressible within their own language.

The paper illustrates that self-reference, a philosophical insight epitomized by The Liar Paradox, is the critical underpinning facilitating undecidability across different frameworks—formal systems, Turing machines, and CAs. The authors highlight that self-reference, often realized through diagonalization arguments, engenders contradictions capable of proving undecidability.

Cellular Automata and Undecidability

The authors draw attention to Cellular Automata, a class of discrete dynamical systems valued for their ability to simulate complex computations and physical processes. Rule 110, an Elementary Cellular Automaton, is emphasized due to its proven universality akin to a Turing machine, thereby capable of generating undecidable dynamics. The paper discusses how Cellular Automata exhibit ordered, chaotic, and complex dynamics—where universality is typically found at the "edge of chaos," a transitional phase possessing rich computational features.

Numerical Results and Claims

One focal argument realizes that the undecidable nature of CAs can be demonstrated using diagonalization methods akin to those employed in proving the Halting Problem for Turing machines. The authors constructively demonstrate elements of Gödel's proof for CAs, articulating that diagonalization can be decomposed into components mirroring the self-referential cascade found in both Gödel's and Turing's frameworks.

Significantly, the paper proposes that the capability to generate undecidable dynamics is fundamentally based on:

1. Program-data duality: The capacity of a computational system to treat code as data and vice-versa.
2. Infinitely large computational media: The necessity for unbounded resources (analogous to the infinite tape of Turing machines).
3. Negation: Implementing logical negation within the formal system to express contradictions.

Implications and Future Directions

The implications of this work extend both practically and theoretically. Practically, recognizing the self-referential basis of undecidability can refine how we design complex adaptive systems and computing architectures. Theoretically, it underscores undecidability's foundational role in emergent phenomena and problem-solving within computational systems. As such mechanisms empower novelty and innovation, they open avenues for research in computational creativity and the development of systems capable of generating new knowledge dynamically.

The future development of AI and complex systems must consider these insights, exploring how self-reference can be leveraged for designing intelligent systems that better mimic natural adaptability and complexity. Further research into the diagonalization concept across different computational frameworks could yield additional insights into the fabric of computation and its constraints.

In summary, Prokopenko et al.'s paper provides an impactful synthesis on the fundamental role of undecidability in computation, tying together classical problems with contemporary research domains. This insightful analysis offers a converging point for multidimensional research spanning computation, complexity science, and philosophical logic.