and Thomas Zeugmann
Source: Electronic Colloquium on Computational Complexity, Report TR98-069, December 08, 1998.
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 also 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 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.
*This work was performed while this author was visiting the Department of Informatics at Kyushu University and was supported by the Japanese Society for the Promotion of Science under Grant JSPS 29716102.
Note that we have implemented the learning algorithms presented in this paper, and you may try them out using our Average Case Optimal 1-Variable Pattern Language Learning page which is also mirrored in Japan.