Unitsoccurs

Unitsoccurs is a term used in the study of symbolic sequences to describe the presence of a predefined unit within a larger sequence. It is used in areas such as formal languages, text processing, bioinformatics, and pattern mining to formalize questions about whether and where a unit appears.

Formally, let A be a finite alphabet and let U be a set of units, where each unit u in U is a finite string over A. For a sequence s in A*, we say that u occurs at position i if the substring s[i .. i+|u|-1] equals u. The relation unitsoccurs(s, u, i) holds precisely when this equality is true. The predicate unitsoccurs(s, u) can denote that there exists at least one position i such that unitsoccurs(s, u, i) holds. Variants may specify the number of occurrences, the non-overlapping constraint, or a maximum window length where occurrences must be found.

Example: with alphabet {a, b}, U = {"ab", "ba"} and s = "ababca", the unit "ab" occurs at positions 1 and 3, while "ba" occurs at positions 2 and 4. Different indexing conventions may be used (1-based or 0-based).

Applications include designing and analyzing text search algorithms, implementing regular-expression-like pattern matchers, scanning DNA or protein sequences for motifs, and identifying recurring units in data mining. Related concepts are substrings, n-grams, and motifs. The term serves as a concise label for the basic predicate of recognizing unit occurrences within sequences.