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

TL;DR

This paper investigates the fundamental role of self-reference and diagonalization in generating undecidable dynamics across formal systems, algorithms, and cellular automata, highlighting their interconnected nature and underlying principles.

Contribution

It provides a detailed comparison of recursive systems, Turing machines, and cellular automata, clarifying how self-reference and diagonalization lead to undecidability in these frameworks.

Findings

Diagonalization applies to cellular automata, illustrating G"odel's proof in distributed computation.

Undecidable dynamics arise from program-data duality, infinite media access, and negation capability.

Distinction between computational universality and undecidability is clarified through adapted G"odel's proof.

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.