Simple DFA are Polynomially Probably Exactly Learnable from Simple Examples

Efficient learning of DFA is a challenging research problem in grammatical inference. Both exact and approximate (in the PAC sense) identifiability of DFA from examples is known to be hard. Pitt, in his seminal paper posed the following open research problem: "Are DFA PAC-identifiable if examples are drawn from the uniform distribution, or some other known simple distribution ?" (Pitt, 1989). We demonstrate that the class of simple DFA (i.e., DFA whose canonical representations have logarithmic Kolmogorov complexity) is efficiently probably exactly learnable under the Solomonoff Levin universal distribution m (wherein an instance x with Kolmogorov complexity K(x) is sampled with probability that is proportional to 2 \GammaK(x) ). The simple distribution independent learning theorem states that a concept class is learnable under the universal distribution m iff it is learnable under the entire class of simple distributions provided the examples are drawn accordin..