# Learning one-variable pattern languages very efficiently on average, in parallel, and by asking queries

## Abstract

A pattern is a string of constant and variable symbols. The language generated by a pattern π is the set of all strings of constant symbols which can be obtained from π 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 guess and the *expected total learning time* taken until convergence to a correct hypothesis. The results obtained are threefold. 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 to compute descriptive one-variable patterns requires time *O*(*n*^{4} log *n*) (cf. Angluin [1]). Second, we give a parallel version of this algorithm requiring time *O*(log *n*) and *O*(*n*^{3}/log *n*) processors on an EREW-PRAM. Third, we devise a one-variable pattern learner whose *expected total learning time* is *O*(l^{2} log l) provided the sample strings are drawn from the target language according to a probability distribution *D* with expected string length l. The distribution *D* must be such that strings of equal length have equal probability, but can be arbitrary otherwise. Thus, we establish the first one-variable pattern learner having an expected total learning time that *provably* differs from the update time by a *constant factor* only.

Finally, we apply the algorithm for finding descriptive one-variable patterns to learn one-variable patterns with a polynomial number of superset queries with respect to the one-variable patterns as *query language*.

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