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

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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 a main difference to discrete groups these groups may be generated by uncountably many generators with index running over certain sets of real numbers. We study the word problem for such groups within the Blum-Shub-Smale (BSS) model of real number computation. The main result establishes the word problem to be computationally equivalent to 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.
Publisher
SPRINGER
Issue Date
2009-10
Language
English
Article Type
Article
Keywords

ABELIAN-GROUPS; NP

Citation

FOUNDATIONS OF COMPUTATIONAL MATHEMATICS, v.9, no.5, pp.599 - 609

ISSN
1615-3375
DOI
10.1007/s10208-009-9048-2
URI
http://hdl.handle.net/10203/203392
Appears in Collection
CS-Journal Papers(저널논문)
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