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Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation

(Extended Abstract)
  • Klaus Meer
  • Martin Ziegler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4708)

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. As main difference with discrete groups, these groups may be generated by uncountably many generators with index running over certain sets of real numbers. This includes a variety of groups which are not captured by the finite 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. It provides a 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 to) but also reducible from the Halting Problem for such machines. It thus gives the first non-trivial example of a problem complete, that is, computationally universal for this model.

Keywords

Word Problem Turing Machine Free Product Real Group Combinatorial Group Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Klaus Meer
    • 1
  • Martin Ziegler
    • 2
  1. 1.IMADA, Syddansk Universitet, Campusvej 55, 5230 Odense MDenmark
  2. 2.University of Paderborn 

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