Hyper-minimisation made efficient

Abstract

We consider a problem of hyper-minimisation of an automaton [2,3]: given a DFA M we want to compute a smallest automaton N such that the language L(M) ΔL(N) is finite, where Δ denotes the symmetric difference. We improve the previously O(|∑|n2) known solution by giving an expected O(|δ|log2 n)time algorithm for this problem, where |δ| is the size of the (potentially partial) transition function. We also give a slightly slower deterministic O(|δ|log2 n)version of the algorithm. Then we introduce a similar problem of k-minimisation: for an automaton M and number k we want to find a smallest automaton N such that L(M) ΔL(N)⊆ ∑