2006/04/07 by Martin Ziegler, Ziegler, Martin, Klaus Meer +1
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #F.1.1 #F.4.1 #F.4.2 #FOS: Computer and information sciences #Geometric and Algebraic Topology #Logic in Computer Science (cs.LO) #Symbolic Computation (cs.SC) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.cs/0604032
openalex publication_date 2006/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.