Learning One-variable Pattern Languages Very Eeciently on Average, in Parallel, and by Asking Queries

A pattern is a nite string of constant and variable symbols. The language generated by a pattern is the set of all strings of constant symbols which can be obtained from the pattern by substituting non-empty strings for variables. We study the learnability of one-variable pattern languages in the limit with respect to the update time needed for computing a new single hypothesis and the expected total learning time taken until convergence to a correct hypothesis. Our results are as follows. First, we design a consistent and set-driven learner that, using the concept of descriptive patterns, achieves update time O(n 2 log n), where n is the size of the input sample. The best previously known algorithm for computing descriptive one-variable patterns requires time O(n 4 log n) (cf. Angluin 2]). Second, we give a parallel version of this algorithm that requires time O(log n) and O(n 3 = log n) processors on an EREW-PRAM. Third, using a modiied version of the sequential algorithm as a subroutine, we devise a learning algorithm for one-variable patterns whose expected total learning time is O(` 2 log`) provided the sample strings are drawn from the target language according to a probability distribution with expected string length`. The probability distribution must be such that strings of equal length have equal probability, but can be arbitrary otherwise. Thus, we establish the rst algorithm for learning one-variable pattern languages having an expected total learning time that provably diiers from the update time by a constant factor only. Finally, we show how the algorithm for descriptive one-variable patterns can be used for learning one-variable patterns with a polynomial number of superset queries with respect to the one-variable patterns as query language.