Alphabet (computer science)

In formal language theory, a string is defined as a finite sequence of members of an underlying base set; this set is called the alphabet of a string or collection of strings. The members of the set are called symbols, and are typically thought of as representing letters, characters, or digits. For example, a common alphabet is {0,1}, the binary alphabet, and a binary string is a string drawn from the alphabet {0,1}. An infinite sequence of letters may be constructed from elements of an alphabet as well.

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, L’s 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 ${\displaystyle \Sigma }$, the set of all strings over the alphabet ${\displaystyle \Sigma }$ of length ${\displaystyle n}$ is indicated by ${\displaystyle \Sigma ^{n}}$. The set ${\textstyle \bigcup _{i\in \mathbb {N} }\Sigma ^{i}}$ of all finite strings (regardless of their length) is indicated by the Kleene star operator as ${\displaystyle \Sigma ^{*}}$, and is also called the Kleene closure of ${\displaystyle \Sigma }$. The notation ${\displaystyle \Sigma ^{\omega }}$ indicates the set of all infinite sequences over the alphabet ${\displaystyle \Sigma }$, and ${\displaystyle \Sigma ^{\infty }}$ indicates the set ${\displaystyle \Sigma ^{\ast }\cup \Sigma ^{\omega }}$ of all finite or infinite sequences.

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