How Redundant Is Your Universal Computation Device?

Abstract

Given a computational model ${\cal M}$, and a “reasonable” encoding function ${\cal C}: {\cal M} \to \{0,1\}^\ast$ that encodes any computation device M of ${\cal M}$ as a finite bit string, we define the description size of M (under the encoding ${\cal C}$) as the length of ${\cal C}(M)$. The description size of the entire class ${\cal M}$ (under the encoding ${\cal C}$) can then be defined as the length of the shortest bit string that encodes a universal device of ${\cal M}$. In this paper we propose the description size as a complexity measure that allows to compare different computational models. We compute upper bounds to the description size of deterministic register machines, Turing machines, spiking neural P systems and UREM P systems. By comparing these sizes, we provide a first partial answer to the following intriguing question: what is the minimal (description) size of a universal computation device?