Summary
A finitely presented (f.p.) group has a K-decidable word problem for a complexity class K if the group operations are K-functions. Using a construction of Boone and Britton [15] we show that for two complexity classes K and K with K a f.p. group can be constructed with K but not K-decidable word problem.
As a consequence of this result it is shown that there is a f.p. K-decidable group with generalized word problem of complexity K, there is no universal K-decidable f.p. group, there is a r.p. not K-decidable simple group (K≠ℛ).
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Avenhaus, J., Madlener, K. Subrekursive Komplexität bei Gruppen. Acta Informatica 9, 87–104 (1977). https://doi.org/10.1007/BF00263767
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DOI: https://doi.org/10.1007/BF00263767