Alphabet (formal languages)

In formal language theory, an alphabet, sometimes called a vocabulary, is a non-empty set of indivisible symbols/glyphs, typically thought of as representing letters, characters, digits, phonemes, or even words. Alphabets in this technical sense of a set are used in a diverse range of fields including logic, mathematics, computer science, and linguistics. An alphabet may have any cardinality ("size") and depending on its purpose maybe be finite (e.g., the alphabet of letters "a" through "z"), countable (e.g., ), or even uncountable (e.g., ). Strings, also known as "words" or "sentences", over an alphabet are defined as a sequence of the symbols from the alphabet set. For example, the alphabet of lowercase letters "a" through "z" can be used to form English words like "iceberg" while the alphabet of both upper and lower case letters can also be used to form proper names like "Wikipedia". A common alphabet is {0,1}, the binary alphabet, and a "00101111" is an example of a binary string. Infinite sequence of symbols may be considered as well (see Omega language). It is often necessary for practical purposes to restrict the symbols in an alphabet so that they are unambiguous when interpreted. For instance, if the two-member alphabet is {00,0}, a string written on paper as "000" is ambiguous because it is unclear if it is a sequence of three "0" symbols, a "00" followed by a "0", or a "0" followed by a "00". If L is a formal language, i.e. a (possibly infinite) set of finite-length strings, the alphabet of L is the set of all symbols that may occur in any string in L. For example, if L is the set of all variable identifiers in the programming language C, Ls alphabet is the set { a, b, c, ..., x, y, z, A, B, C, ..., X, Y, Z, 0, 1, 2, ..., 7, 8, 9, _ }. Given an alphabet , the set of all strings of length over the alphabet is indicated by . The set of all finite strings (regardless of their length) is indicated by the Kleene star operator as , and is also called the Kleene closure of .