On Universality in Real Computation

TL;DR

This paper explores the limits of universality in real computation models, showing that Turing universality is confined to specific degrees and introducing the concepts of universal relativity and jumps across hierarchical levels.

Contribution

It introduces the idea that Turing universality is restricted to certain degrees and proposes a new framework of universal relativity and jumps in the arithmetical and analytical hierarchies.

Findings

Turing universality is only possible at specific Turing degrees.

Universality at higher degrees is ambiguous and not well-defined.

Introduces the concepts of universal relativity and jumps across hierarchical levels.

Abstract

Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary non-computable number or non-recursive function. In this paper we show that Turing universality is only possible at every Turing degree but not over all, in that sense universality at the first level is elegantly well defined while universality at higher degrees is at least ambiguous. We propose a concept of universal relativity and universal jump between levels in the arithmetical and analytical hierarchy.