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.
KeywordsWord problem for groups Computational universality Blum–Shub–Smale model Real halting problem
Mathematics Subject Classification (2000)20F10 68Q17 68Q10
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- 2.L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation (Springer, New York, 1998). Google Scholar
- 12.S. Lang, SL 2(ℝ) (Springer, New York, 1985). Google Scholar
- 16.C.F. Miller III, Decision problems for groups—survey and reflections, in Algorithms and Classification in Combinatorial Group Theory, ed. by G. Baumslag, C.F. Miller III. Math. Sci. Res. Inst. Publ., vol. 1 (Springer, New York, 1992), pp. 1–59. Google Scholar
- 18.P.S. Novikov, On the algorithmic unsolvability of the word problem in group theory. Tr. Mat. Inst. Steklov 44 (1955). Google Scholar