Finding a Good Collection of Patterns Covering a Set of Sequences iiContents

The papers in the series are intended for internal use and are distributed by the author. Copies may be ordered from the library of Department of Computer Science. Abstract. We consider a problem of learning of unions of pattern languages from positive examples. We consider three diierent classes of patterns-regular patterns, substring patterns and the so called PROSITE patterns. By regular patterns we understand patterns where each variable symbol can appear only once. By substring patterns we understand a subclass of regular patterns of the type xxy, where x and y are variables and is a string of constant symbols. The PROSITE patterns is a class of patterns used for classiication of bio-sequences in PROSITE database. We present an algorithm which, given a set of sequences, nds a `good' collection of patterns`covering' this set. The notion of a `good covering' is deened as the most probable collection of patterns likely to produce the examples in some simple and natural probabilistic model. We show that this criterion is equivalent to the so called Minimum Description Length (MDL) principle. We present a polynomial-time algorithm for approximating the optimal cover within a logarithmic factor and prove its performance guarantees. In the case of substring patterns the running time of the algorithm is almost linear.