Complexity and real computation
Complexity and real computation
Computable analysis: an introduction
Computable analysis: an introduction
The P-DNP problem for infinite Abelian groups
Journal of Complexity
Quantum automata and algebraic groups
Journal of Symbolic Computation
Uncomputability below the real halting problem
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
An explicit solution to post's problem over the reals
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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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 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 semialgebraic 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.