The tolerant learning hierarchy for c.e. sets
Learning from positive data models a scenario in which a learner receives longer and longer initial segments of an enumeration of an unknown set and repeatedly outputs a conjecture about it. Classical identification in the limit requires the learner to eventually stabilise to a hypothesis naming the target exactly, while anomalous learning relaxes this requirement by allowing finitely many errors in the final hypothesis. We develop a natural generalisation for computably enumerable (c.e.) sets in which correctness is determined by an arbitrary equivalence relation on their indices that respects equality of the sets they enumerate. We call the resulting framework tolerant learning. Exact learning and finite-anomaly learning arise as special cases. By comparing tolerance relations according to the families they make learnable, we obtain a new degree structure, the tolerant learning hierarchy. We extend the classical methods of locking sequences and tell-tales to this setting and derive a combinatorial characterisation of those tolerance relations that are universal (that is, under which every family of nonempty computably enumerable sets is learnable). We then show that the hierarchy is a bounded lower semilattice but not an upper semilattice, is atomless and coatomless, and contains chains and antichains of size continuum. We also analyse natural benchmark relations, including c.e.-index analogues of standard equivalence relations from descriptive set theory, and establish substantial incomparability among their learning powers. Finally, we separate non-effective from computable tolerant learning by exhibiting a universal relation that is not computably universal. These results show that semantic tolerance gives rise to a genuine degree theory at the interface of algorithmic learning and the theory of equivalence relations.