Computing All ℓ-Cover Automata Fast

Given a language L and a number l, an l-cover automaton for L is a DFA M such that its language coincides with L on all words of length at most l. It is known that an equivalent minimal l-cover automaton can be constructed in time O(n log n), where n is the number of states of M. This is achieved by a clever and sophisticated variant of HOPCROFT'S algorithm, which computes the l-similarity inside the main algorithm. This contribution presents an alternative simple algorithm with running time O(n log n), in which the computation is split into three phases. First, a compact representation of the gap table is created. Second, this representation is enriched with information about the length of a shortest word leading to the states. These two steps are independent of the parameter l. Third, the l-similarity is extracted by simple comparisons against l. In particular, this approach allows the calculation of all the sizes of minimal l-cover automata (for all valid l) in the same time bound.