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We construct examples of finitely generated decidable group presentationsthat satisfy certain combinations of solvability for the word problem,solvability for the bounded word problem, and computablity for the Dehnfunction. We prove that no finitely generated decidable presentations existsatisfying the combinations for which we do not provide examples. Thepresentations we construct are also minimal. These constructions answer an openquestion asked by R.I. Grigorchuk and S.V. Ivanov. Our approach uses machinerydeveloped by Birget, Ol'shanskii, Rips, and Sapir for constructing finite grouppresentations that simulate Turing machines. We generalize this machinery toconstruct finitely generated decidable group presentations that simulatecomputing objects similar to oracle Turing machines.

Dehn Functions, the Word Problem, and the Bounded Word Problem For Decidable Group Presentations