Learning Deterministic One-Counter Automata in Polynomial Time

We give an active learning algorithm for deterministic one-counter automata (DOCAs) where the learner can ask the teacher membership and minimal equivalence queries. The algorithm called OL* learns a DOCA in time polynomial in the size of the smallest DOCA, recognising the target language. All existing algorithms for learning DOCAs, even for the subclasses of deterministic real-time one-counter automata (DROCAs) and visibly one-counter automata (VOCAs), in the worst case, run in exponential time with respect to the size of the DOCA under learning. Furthermore, previous learning algorithms are ``grey-box'' algorithms relying on an additional query type - counter value query - where the teacher returns the counter value reached on reading a given word. In contrast, our algorithm is a ``black-box'' algorithm. It is known that the minimisation of VOCAs is NP-hard. However, OL* can be used for approximate minimisation of DOCAs. In this case, the output size is at most polynomial in the size of a minimal DOCA.