Termination proofs of rewrite systems using the Knuth-Bendix ordering, kbo, imply multiply-recursive bounds on the lengths of derivations [HL89]. Hofbauer has also given bounds for some restrictions of kbo. The rst one is a restriction to the use of positive weights, the second is a restriction on the signature which is only allowed to contain at most unary symbols. In both cases, Hofbauer proves the derivation lengths to be bounded exponentially. We prove that one restriction correspond exactly to computations in Linspace on Turing machine and the second one to computation in Linspace or Etime on a unary alphabet. Under some further considerations, we give a caracterisation of Espace with the help of a variant of the second type of computations. 1 Introduction The theme of our investigations is that of complexity characterisations of classes of functions computed by rewrite systems. Comparisons are achieved with respect to computations with Turing machines. Actually, we c..