  The problem of k-minimisation for a DFA M is the computation of a smallest
DFA N (where the size |M| of a DFA M is the size of the domain of the
transition function) such that their recognized languages differ only on words
of length less than k. The previously best algorithm, which runs in time O(|M|
log^2 n) where n is the number of states, is extended to DFAs with partial
transition functions. Moreover, a faster O(|M| log n) algorithm for DFAs that
recognise finite languages is presented. In comparison to the previous
algorithm for total DFAs, the new algorithm is much simpler and allows the
calculation of a k-minimal DFA for each k in parallel. Secondly, it is
demonstrated that calculating the least number of introduced errors is hard:
Given a DFA M and numbers k and m, it is NP-hard to decide whether there exists
a k-minimal DFA N differing from DFA M on at most m words. A similar result
holds for hyper-minimisation of DFAs in general: Given a DFA M and numbers s
and m, it is NP-hard to decide whether there exists a DFA N with at most s
states such that DFA M and N differ on at msot m words.
