Learning Languages from Positive Data and a Finite Number of Queries

A computational model for learning languages in the limit from full positive data and a bounded number of queries to the teacher (oracle) is introduced and explored. Equivalence, superset, and subset queries are considered (for the latter one we consider also a variant when the learner tests every conjecture, but the number of negative answers is uniformly bounded). If the answer is negative, the teacher may provide a counterexample. We consider several types of counterexamples: arbitrary, least counterexamples, the ones whose size is bounded by the size of positive data seen so far, and no counterexamples. A number of hierarchies based on the number of queries (answers) and types of answers/counterexamples is established. Capabilities of learning with different types of queries are compared. In most cases, one or two queries of one type can sometimes do more than any bounded number of queries of another type. Still, surprisingly, a finite number of subset queries is sufficient to simulate the same number of equivalence queries when behaviourally correct learners do not receive counterexamples and may have unbounded number of errors in almost all conjectures.