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

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Mathematical Foundations of Computer Science 2007 (MFCS 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4708))

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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.

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Authors and Affiliations

  1. IMADA, Syddansk Universitet, Campusvej 55, 5230 Odense M, Denmark

    Klaus Meer

  2. University of Paderborn,  

    Martin Ziegler

Authors
  1. Klaus Meer
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  2. Martin Ziegler
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Luděk Kučera Antonín Kučera

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Meer, K., Ziegler, M. (2007). Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation. In: Kučera, L., Kučera, A. (eds) Mathematical Foundations of Computer Science 2007. MFCS 2007. Lecture Notes in Computer Science, vol 4708. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74456-6_64

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  • DOI: https://doi.org/10.1007/978-3-540-74456-6_64

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74455-9

  • Online ISBN: 978-3-540-74456-6

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