Real Computational Universality: The Word Problem for a class of groups with infinite presentation

TL;DR

This paper extends the classical word problem in group theory to uncountably generated groups with real number indices, demonstrating its computational universality within the Blum-Shub-Smale model of real computation.

Contribution

It introduces a real extension of the word problem for groups with uncountable generators, establishing its completeness for the BSS model of computation.

Findings

The real word problem is semi-decidable and reducible to the Halting Problem.

It is the first example of a problem that is computationally universal in the BSS model.

Extends computational group theory into the realm of real number computation.

Abstract

The word problem for discrete groups is well-known to be undecidable by a Turing Machine; more precisely, it is reducible both to and from and thus equivalent to the discrete Halting Problem. The present work introduces and studies a real extension of the word problem for a certain class of groups which are presented as quotient groups of a free group and a normal subgroup. Most important, the free group will be generated by an uncountable set of generators with index running over certain sets of real numbers. This allows to include many mathematically important groups which are not captured in the framework of the classical word problem. Our contribution extends computational group theory from the discrete to the Blum-Shub-Smale (BSS) model of real number computation. We believe this to be an interesting step towards applying BSS theory, in addition to semi-algebraic geometry, also to further areas of mathematics. The main result establishes the word problem for such groups to be not only semi-decidable (and thus reducible FROM) but also reducible TO the Halting Problem for such machines. It thus provides the first non-trivial example of a problem COMPLETE, that is, computationally universal for this model.