 One of the major results in the subject was obtained by Sapir, Birget and Rips who showed that most "reasonable" time complexity functions of Turing machines can be realized, up to natural equivalence, as Dehn functions of finitely presented groups.Let
be a finite group presentation where the "X" is a finite alphabet and where "R" ⊆ "F"("X") is a finite set of cyclically reduced words.Let "w" ∈ "F"("X") be a "relation" in "G", that is, a freely reduced word such that "w" = 1 in "G".