On the parameterised complexity of learning patterns

Angluin (1980) showed that there is a consistent and conservative learner for the class of non-erasing pattern languages; however, most of these learners are NP-hard. In the current work, the complexity of consistent polynomial time learners for the class of non-erasing pattern languages is revisited, with the goal to close one gap left by Angluin, namely the question on what happens if the learner is not required to output each time a consistent pattern of maximum possible length. It is shown that consistent learners are non-uniformly W[1]-hard inside the fixed-parameter hierarchy of Downey and Fellows (1999), and that there is also a W[1]-complete such learner. Only when one requires that the learner is in addition both, conservative and class-preserving, then one can show that the learning task is NP-hard for certain alphabet-sizes.