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

Por • 4 may, 2018 • Sección: Ciencia y tecnología

Mikhail Prokopenko, Michael Harré, Joseph Lizier, Fabio Boschetti, Pavlos Peppas, Stuart Kauffman

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 Godel’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 Godel’s proof distinguish between computational universality and undecidability, and show how the diagonalization argument exploits, on several levels, the self-referential basis of undecidability.

arXiv:1711.02456v1 [cs.LO]

Logic in Computer Science (cs.LO); Formal Languages and Automata Theory (cs.FL); Cellular Automata and Lattice Gases (nlin.CG)

Post to Twitter

Escribe un comentario