Inferring regular languages and ω-languages

In 1987 Angluin proposed an algorithm, termed L^⁎ for inferring an unknown regular language using membership and equivalence queries. This algorithm has found many applications, amongst which in the area of system design and verification. These applications challenge the state-of-the art solutions in various directions, in particular, scaling or working with more succinct representations, and dealing with ω-languages, the main model for reasoning about reactive systems. Both extensions confront a similar difficulty. Inference algorithms typically rely on the correspondence between the automata states and the right congruence, henceforth, the residuality property. DFAs enjoy the residuality property (as stated by the Myhill–Nerode Theorem) but more succinct representations such as non-deterministic and alternating finite automata (NFAs and AFAs) in general do not. The situation in the ω-languages realm is even worse, since none of the traditional automata that can express all regular ω-languages enjoys the residuality property. This paper surveys residual models for regular languages and ω-languages and the learning algorithms that can infer these models.