On Minimising Automata with Errors

The problem of k-minimisation for a DFA M is the computation of a smallest DFAN (where the size |M | of a DFAM is the size of the domain of the transition function) such that L(M)4L(N) ⊆ Σ, which means that their recognized languages differ only on words of length less than k. The previously best algorithm, which runs in time O(|M | log n) where n is the number of states, is extended to DFAs with partial transition functions. Moreover, a faster O(|M | logn) 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 with |L(M)4L(N)| ≤ m. 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 |L(M)4L(N)| ≤ m.