Average-case optimal one-variable pattern language learner

Abstract

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 defining 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 finitely inferred from positive data nor PAC-learned, our approach can be extended to a probabilistic finite learner that exactly infers all one-variable pattern languages from positive data with high confidence. It is a long-standing open problem whether or not pattern languages can be learned in cases that empty substitutions for the pattern variables are also allowed. 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.