An Average-Case Optimal One-Variable Pattern Language Learner

A new algorithm for learning one-variable pattern languages from positive data is proposed and analyzed with respect to its average-case behavior. We consider the total learning time that takes into account all operations till convergence to a correct hypothesis is achieved. For almost all meaningful distributions deening how the pattern variable is replaced by a string to generate random examples of the target pattern language, it is shown that this algorithm converges within an expected constant number of rounds and a total learning time that is linear in the pattern length. Thus, our solution is average-case optimal in a strong sense. Though one-variable pattern languages can neither be nitely inferred from positive data nor PAC-learned, our approach can also be extended to a probabilistic nite learner that exactly infers all one-variable pattern languages from positive data with high conn-dence. It is a long standing open problem whether pattern languages can be learned in case that substitutions of pattern variables by the empty string can also occur. Our learning strategy can be generalized to this situation as well. Finally, we show some experimental results for the behavior of this new learning algorithm in practice.