Simple DFA are Polynomially Probably Exactly Learnable from Simple Examples

E cient learning of DFA is a challenging research problem in grammatical inference Both exact and approximate in the PAC sense identi ability of DFA from examples is known to be hard Pitt in his semi nal paper posed the following open research problem Are DFA PAC identi able if ex amples are drawn from the uniform distribu tion or some other known simple distribu tion Pitt We demonstrate that the class of simple DFA i e DFA whose canon ical representations have logarithmic Kol mogorov complexity is e ciently probably exactly learnable under the Solomono Levin universal distributionm wherein an instance x with Kolmogorov complexity K x is sam pled with probability that is proportional to K x The simple distribution independent learning theorem states that a concept class is learnable under the universal distribution m i it is learnable under the entire class of simple distributions provided the examples are drawn according to the universal distribu tion Li Vit anyi The class of simple distributions includes all computable distribu tions Thus it follows that the class of simple DFA is learnable under a su ciently general class of distributions