Real Computational Universality: The Word Problem for a class of groups with infinite presentation
Abstract
The word problem for discrete groups is wellknown 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 BlumShubSmale (BSS) model of real number computation. We believe this to be an interesting step towards applying BSS theory, in addition to semialgebraic geometry, also to further areas of mathematics. The main result establishes the word problem for such groups to be not only semidecidable (and thus reducible FROM) but also reducible TO the Halting Problem for such machines. It thus provides the first nontrivial example of a problem COMPLETE, that is, computationally universal for this model.
 Publication:

arXiv eprints
 Pub Date:
 April 2006
 arXiv:
 arXiv:cs/0604032
 Bibcode:
 2006cs........4032Z
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Symbolic Computation;
 F.1.1;
 F.4.1;
 F.4.2
 EPrint:
 corrected Section 4.5