In this chapter we consider generalized pattern matching, in which a set of patterns (rather than a single pattern) is given. We assume here that the pattern is a pair of sets of words (W0, W), where Wi consists of the sets Wi ⊂ Ami (i.e., all words in Wi have a fixed length mi). The set W0 is called the forbidden set. For W0 = ∅ one is interested in the number of pattern occurrences On(W), defined as the number of patterns from W occurring in a text generated by a (random) source. Another parameter of interest is the number of positions in where a pattern from W appears (clearly, several patterns may occur at the same positions but words from Wi must occur in different locations); this quantity we denote as Πn. If we define as the number of positions where a word from Wi occurs, then

Notice that at any given position of the text and for a given i only one word from Wi can occur.

For W0 ≠ ∅ one studies the number of occurrences On(W) under the condition that, that is, there is no occurrence of a pattern from W0 in the text. This could be called constrained pattern matching since one restricts the text to those strings that do not contain strings from W0. A simple version of constrained pattern matching was discussed in Chapter 3 (see also Exercises 3.3, 3.6, and 3.10).

In this chapter we first present an analysis of generalized pattern matching with W0 = ∅ and d = 1, which we call the reduced pattern set (i.e., no pattern is a substring of another pattern).