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

@article{Meer2006RealCU,
  title={Real Computational Universality: The Word Problem for a Class of Groups with Infinite Presentation},
  author={Klaus Meer and Martin Ziegler},
  journal={Foundations of Computational Mathematics},
  year={2006},
  volume={9},
  pages={599-609}
}
  • K. MeerM. Ziegler
  • Published 7 April 2006
  • Mathematics
  • Foundations of Computational Mathematics
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… 
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References

SHOWING 1-10 OF 45 REFERENCES

Uncomputability Below the Real Halting Problem

A real variant of the classical word problem for groups is presented which is established reducible to and from the real Halting Problem and how additional real constants increase the power of a BSS machine is analyzed.

On a theory of computation and complexity over the real numbers: $NP$- completeness, recursive functions and universal machines

We present a model for computation over the reals or an arbitrary (ordered) ring R. In this general setting, we obtain universal machines, partial recursive functions, as well as JVP-complete

Groups, the Theory of Ends, and Context-Free Languages

Recursiveness over the Complex Numbers is Time-Bounded

The aim of this note is to show that over the complex numbers the situation regarding time bounded recursive sets turns out to be the Boolean one, i.e. without running time bounds for any machine solving them.

An explicit solution to Post's Problem over the reals

Ordered rings over which output sets are recursively enumerable sets

In a recent paper [BSS], L. Blum, M. Shub, and S. Smale developed a theory of computation over the reals and over commutative ordered rings ; in 9 of [BSS] they showed that over the reals (and over

Groups and computation

Computing composition series in primitive groups by L. Babai, E. M. Luks, and A. Seress Computing blocks of imprimitivity for small-base groups in nearly linear time by R. Beals Fast Fourier

Groups with Context‐Free Co‐Word Problem

It is proved that languages with certain purely arithmetical properties cannot be context‐free by proving that the only examples amongst polycyclic groups or the Baumslag–Solitar groups are virtually abelian.

The P-DNP problem for infinite Abelian groups

Abstract We consider a uniform model of computation for groups. This is a generalization of the Blum–Shub–Smale model over the additive group of real numbers. We show that the inequalities P ≠ DNP

Subgroups of finitely presented groups

  • G. Higman
  • Mathematics
    Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1961
The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It